A new solution procedure for a nonlinear infinite beam equation of motion

NASA Astrophysics Data System (ADS)

Jang, T. S.

2016-10-01

Our goal of this paper is of a purely theoretical question, however which would be fundamental in computational partial differential equations: Can a linear solution-structure for the equation of motion for an infinite nonlinear beam be directly manipulated for constructing its nonlinear solution? Here, the equation of motion is modeled as mathematically a fourth-order nonlinear partial differential equation. To answer the question, a pseudo-parameter is firstly introduced to modify the equation of motion. And then, an integral formalism for the modified equation is found here, being taken as a linear solution-structure. It enables us to formulate a nonlinear integral equation of second kind, equivalent to the original equation of motion. The fixed point approach, applied to the integral equation, results in proposing a new iterative solution procedure for constructing the nonlinear solution of the original beam equation of motion, which consists luckily of just the simple regular numerical integration for its iterative process; i.e., it appears to be fairly simple as well as straightforward to apply. A mathematical analysis is carried out on both natures of convergence and uniqueness of the iterative procedure by proving a contractive character of a nonlinear operator. It follows conclusively,therefore, that it would be one of the useful nonlinear strategies for integrating the equation of motion for a nonlinear infinite beam, whereby the preceding question may be answered. In addition, it may be worth noticing that the pseudo-parameter introduced here has double roles; firstly, it connects the original beam equation of motion with the integral equation, second, it is related with the convergence of the iterative method proposed here.

Using EIGER for Antenna Design and Analysis

NASA Technical Reports Server (NTRS)

Champagne, Nathan J.; Khayat, Michael; Kennedy, Timothy F.; Fink, Patrick W.

2007-01-01

EIGER (Electromagnetic Interactions GenERalized) is a frequency-domain electromagnetics software package that is built upon a flexible framework, designed using object-oriented techniques. The analysis methods used include moment method solutions of integral equations, finite element solutions of partial differential equations, and combinations thereof. The framework design permits new analysis techniques (boundary conditions, Green#s functions, etc.) to be added to the software suite with a sensible effort. The code has been designed to execute (in serial or parallel) on a wide variety of platforms from Intel-based PCs and Unix-based workstations. Recently, new potential integration scheme s that avoid singularity extraction techniques have been added for integral equation analysis. These new integration schemes are required for facilitating the use of higher-order elements and basis functions. Higher-order elements are better able to model geometrical curvature using fewer elements than when using linear elements. Higher-order basis functions are beneficial for simulating structures with rapidly varying fields or currents. Results presented here will demonstrate curren t and future capabilities of EIGER with respect to analysis of installed antenna system performance in support of NASA#s mission of exploration. Examples include antenna coupling within an enclosed environment and antenna analysis on electrically large manned space vehicles.

First integrals and parametric solutions of third-order ODEs admitting {\\mathfrak{sl}(2, {R})}

NASA Astrophysics Data System (ADS)

Ruiz, A.; Muriel, C.

2017-05-01

A complete set of first integrals for any third-order ordinary differential equation admitting a Lie symmetry algebra isomorphic to sl(2, {R}) is explicitly computed. These first integrals are derived from two linearly independent solutions of a linear second-order ODE, without additional integration. The general solution in parametric form can be obtained by using the computed first integrals. The study includes a parallel analysis of the four inequivalent realizations of sl(2, {R}) , and it is applied to several particular examples. These include the generalized Chazy equation, as well as an example of an equation which admits the most complicated of the four inequivalent realizations.

Dispersive optical solitons and modulation instability analysis of Schrödinger-Hirota equation with spatio-temporal dispersion and Kerr law nonlinearity

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi; Baleanu, Dumitru

2018-01-01

This paper obtains the dark, bright, dark-bright or combined optical and singular solitons to the perturbed nonlinear Schrödinger-Hirota equation (SHE) with spatio-temporal dispersion (STD) and Kerr law nonlinearity in optical fibers. The integration algorithm is the Sine-Gordon equation method (SGEM). Furthermore, the modulation instability analysis (MI) of the equation is studied based on the standard linear-stability analysis and the MI gain spectrum is got.

Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

NASA Astrophysics Data System (ADS)

Tisdell, C. C.

2017-08-01

Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using 'exact methods', illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

Lie symmetry analysis, conservation laws, solitary and periodic waves for a coupled Burger equation

NASA Astrophysics Data System (ADS)

Xu, Mei-Juan; Tian, Shou-Fu; Tu, Jian-Min; Zhang, Tian-Tian

2017-01-01

Under investigation in this paper is a generalized (2 + 1)-dimensional coupled Burger equation with variable coefficients, which describes lots of nonlinear physical phenomena in geophysical fluid dynamics, condense matter physics and lattice dynamics. By employing the Lie group method, the symmetry reductions and exact explicit solutions are obtained, respectively. Based on a direct method, the conservations laws of the equation are also derived. Furthermore, by virtue of the Painlevé analysis, we successfully obtain the integrable condition on the variable coefficients, which plays an important role in further studying the integrability of the equation. Finally, its auto-Bäcklund transformation as well as some new analytic solutions including solitary and periodic waves are also presented via algebraic and differential manipulation.

Optical solitons and modulation instability analysis of an integrable model of (2+1)-Dimensional Heisenberg ferromagnetic spin chain equation

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi; Baleanu, Dumitru

2017-12-01

This paper addresses the nonlinear Schrödinger type equation (NLSE) in (2+1)-dimensions which describes the nonlinear spin dynamics of Heisenberg ferromagnetic spin chains (HFSC) with anisotropic and bilinear interactions in the semiclassical limit. Two integration schemes are employed to study the equation. These are the complex envelope function ansatz and the generalized tanh methods. Dark, dark-bright or combined optical and singular soliton solutions of the equation are derived. Furthermore, the modulational instability (MI) is studied based on the standard linear-stability analysis and the MI gain is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

A constitutive material model for nonlinear finite element structural analysis using an iterative matrix approach

NASA Technical Reports Server (NTRS)

Koenig, Herbert A.; Chan, Kwai S.; Cassenti, Brice N.; Weber, Richard

1988-01-01

A unified numerical method for the integration of stiff time dependent constitutive equations is presented. The solution process is directly applied to a constitutive model proposed by Bodner. The theory confronts time dependent inelastic behavior coupled with both isotropic hardening and directional hardening behaviors. Predicted stress-strain responses from this model are compared to experimental data from cyclic tests on uniaxial specimens. An algorithm is developed for the efficient integration of the Bodner flow equation. A comparison is made with the Euler integration method. An analysis of computational time is presented for the three algorithms.

PREFACE: Symmetries and Integrability of Difference Equations

NASA Astrophysics Data System (ADS)

Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane

2007-10-01

The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations (DE), like differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, and quantum field theory. It is thus crucial to develop tools to study and solve DEs. While the theory of symmetry and integrability for differential equations is now largely well-established, this is not yet the case for discrete equations. Although over recent years there has been significant progress in the development of a complete analytic theory of difference equations, further tools are still needed to fully understand, for instance, the symmetries, asymptotics and the singularity structure of difference equations. The series of SIDE meetings on Symmetries and Integrability of Difference Equations started in 1994. Its goal is to provide a platform for an international and interdisciplinary communication for researchers working in areas associated with integrable discrete systems, such as classical and quantum physics, computer science and numerical analysis, mathematical biology and economics, discrete geometry and combinatorics, theory of special functions, etc. The previous SIDE meetings took place in Estérel near Montréal, Canada (1994), at the University of Kent in Canterbury, UK (1996), in Sabaudia near Rome, Italy (1998), at the University of Tokyo, Japan (2000), in Giens, France (2002), and in Helsinki, Finland (2004). The SIDE VII meeting was held at the University of Melbourne from 10-14 July 2006. The scientific committee consisted of Nalini Joshi (The University of Sydney), Frank W Nijhoff (University of Leeds), Reinout Quispel (La Trobe University) and Colin Rogers (University of New South Wales). The local organization was in the hands of John A G Roberts and Wolfgang K Schief. Proceedings of all the previous SIDE meetings have been published; the 1994 and 1988 meetings (edited respectively by D Levi, L Vinet and P Winternitz, and by D Levi and O Ragnisco) as volumes of the CRM Proceedings and Lecture Notes (AMS Publications), the 1996 meeting (edited by P Clarkson and F W Nijhoff) as Volume 255 in the LMS Lecture Note Series. Starting from the 1996 meeting the formula of publication has been changed to include rather selected refereed contributions submitted in response to a call for papers issued after the meetings and not restricted to their participants. Thus publications reflecting the scope of the 1996 meeting (edited by J Hietarinta, F W Nijhoff and J Satsuma) appeared in Journal of Physics A: Mathematical and General 34 48 (special issue), and of the 1998 and 2000 meetings (edited respectively by F W Nijhoff, Yu B Suris and C-M Viallet, and by J F van Diejen and R Halburd) in Journal of Nonlinear Mathematical Physics 10 (Suppl. 2) and 12 (Suppl. 2). The aim of this special issue is to benefit from the occasion offered by the SIDE VII meeting, producing an issue containing papers which represent the state-of-the-art knowledge for studying integrability and symmetry properties of difference equations. This special issue features high quality research papers and invited reviews which deal with themes that were covered by the SIDE VII conference. These are in alphabetical order: Algebraic-geometric approaches to integrability. The first section contains a paper by T Hamamoto and K Kajiwara on hypergeometric solutions to the q-Painlevé equation of type A4(1). Discrete geometry. In this category there are three papers. J Cielinski offers a geometric definition and a spectral approach on pseudospherical surfaces on time scales, while A Doliwa considers generalized isothermic lattices. The paper by U Pinkall, B Springborn and S Weiss mann is concerned with a new doubly discrete analogue of smoke ring flow and the real time simulation of fluid flow. Integrable systems in statistical physics. Under this heading there is a paper by R J Baxter on corner transfer matrices in statistical mechanics, and a paper by S Boukraa, S Hassani, J-M Maillard, B M McCoy, J-A Weil and N Zenine where the authors consider Fuchs-Painlevé elliptic representation of the Painlevé VI equation. KP lattices and differential-difference hierarchies. In this section we have seven articles. C R Gilson, J J C Nimmo and Y Ohta consider quasideterminant solutions of a non-Abelian Hirota-Miwa equation, while B Grammaticos, A Ramani, V Papageorgiou, J Satsuma and R Willox discuss the construction of lump-like solutions of the Hirota-Miwa equation. J Hietarinta and C Viallet analyze the factorization process for lattice maps searching for integrable cases, the paper by X-B Hu and G-F Yu is concerned with integrable discretizations of the (2+1)-dimensional sinh-Gordon equation, and K Kajiwara, M Mazzocco and Y Ohta consider the Hankel determinant formula of the tau-functions of the Toda equation. Finally, V G Papageorgiou and A G Tongas study Yang-Baxter maps and multi-field integrable lattice equations, and H-Y Wang, X-B Hu and H-W Tam consider the two-dimensional Leznov lattice equation with self-consistent sources. Quantum integrable systems. This category contains a paper on q-extended eigenvectors of the integral and finite Fourier transforms by N M Atakishiyev, J P Rueda and K B Wolf, and an article by S M Sergeev on quantization of three-wave equations. Random matrix theory. This section contains a paper by A V Kitaev on the boundary conditions for scaled random matrix ensembles in the bulk of the spectrum. Symmetries and conservation laws. In this section we have five articles. H Gegen, X-B Hu, D Levi and S Tsujimoto consider a difference-analogue of Davey-Stewartson system giving its discrete Gram-type determinant solution and Lax pair. The paper by D Levi, M Petrera, and C Scimiterna is about the lattice Schwarzian KDV equation and its symmetries, while O G Rasin and P E Hydon study the conservation laws for integrable difference equations. S Saito and N Saitoh discuss recurrence equations associated with invariant varieties of periodic points, and P H van der Kamp presents closed-form expressions for integrals of MKDV and sine-Gordon maps. Ultra-discrete systems. This final category contains an article by C Ormerod on connection matrices for ultradiscrete linear problems. We would like to express our sincerest thanks to all contributors, and to everyone involved in compiling this special issue.

Analysis of Three-Dimensional, Nonlinear Development of Wave-Like Structure in a Compressible Round Jet

NASA Technical Reports Server (NTRS)

Dahl, Milo D.; Mankbadi, Reda R.

2002-01-01

An analysis of the nonlinear development of the large-scale structures or instability waves in compressible round jets was conducted using the integral energy method. The equations of motion were decomposed into two sets of equations; one set governing the mean flow motion and the other set governing the large-scale structure motion. The equations in each set were then combined to derive kinetic energy equations that were integrated in the radial direction across the jet after the boundary-layer approximations were applied. Following the application of further assumptions regarding the radial shape of the mean flow and the large structures, equations were derived that govern the nonlinear, streamwise development of the large structures. Using numerically generated mean flows, calculations show the energy exchanges and the effects of the initial amplitude on the coherent structure development in the jet.

Numerical solution of boundary-integral equations for molecular electrostatics.

PubMed

Bardhan, Jaydeep P

2009-03-07

Numerous molecular processes, such as ion permeation through channel proteins, are governed by relatively small changes in energetics. As a result, theoretical investigations of these processes require accurate numerical methods. In the present paper, we evaluate the accuracy of two approaches to simulating boundary-integral equations for continuum models of the electrostatics of solvation. The analysis emphasizes boundary-element method simulations of the integral-equation formulation known as the apparent-surface-charge (ASC) method or polarizable-continuum model (PCM). In many numerical implementations of the ASC/PCM model, one forces the integral equation to be satisfied exactly at a set of discrete points on the boundary. We demonstrate in this paper that this approach to discretization, known as point collocation, is significantly less accurate than an alternative approach known as qualocation. Furthermore, the qualocation method offers this improvement in accuracy without increasing simulation time. Numerical examples demonstrate that electrostatic part of the solvation free energy, when calculated using the collocation and qualocation methods, can differ significantly; for a polypeptide, the answers can differ by as much as 10 kcal/mol (approximately 4% of the total electrostatic contribution to solvation). The applicability of the qualocation discretization to other integral-equation formulations is also discussed, and two equivalences between integral-equation methods are derived.

On the complete and partial integrability of non-Hamiltonian systems

NASA Astrophysics Data System (ADS)

Bountis, T. C.; Ramani, A.; Grammaticos, B.; Dorizzi, B.

1984-11-01

The methods of singularity analysis are applied to several third order non-Hamiltonian systems of physical significance including the Lotka-Volterra equations, the three-wave interaction and the Rikitake dynamo model. Complete integrability is defined and new completely integrable systems are discovered by means of the Painlevé property. In all these cases we obtain integrals, which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation (ODE) solved by Painlevé transcendents. Relaxing the Painlevé property we find many partially integrable cases whose movable singularities are poles at leading order, with In( t- t0) terms entering at higher orders. In an Nth order, generalized Rössler model a precise relation is established between the partial fulfillment of the Painlevé conditions and the existence of N - 2 integrals of the motion.

The integrated Michaelis-Menten rate equation: déjà vu or vu jàdé?

PubMed

Goličnik, Marko

2013-08-01

A recent article of Johnson and Goody (Biochemistry, 2011;50:8264-8269) described the almost-100-years-old paper of Michaelis and Menten. Johnson and Goody translated this classic article and presented the historical perspective to one of incipient enzyme-reaction data analysis, including a pioneering global fit of the integrated rate equation in its implicit form to the experimental time-course data. They reanalyzed these data, although only numerical techniques were used to solve the model equations. However, there is also the still little known algebraic rate-integration equation in a closed form that enables direct fitting of the data. Therefore, in this commentary, I briefly present the integral solution of the Michaelis-Menten rate equation, which has been largely overlooked for three decades. This solution is expressed in terms of the Lambert W function, and I demonstrate here its use for global nonlinear regression curve fitting, as carried out with the original time-course dataset of Michaelis and Menten.

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

PubMed Central

Müller, Eike H.; Scheichl, Rob; Shardlow, Tony

2015-01-01

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy. PMID:27547075

Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation.

PubMed

Müller, Eike H; Scheichl, Rob; Shardlow, Tony

2015-04-08

This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis as an alternative to strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.

Mathematics for Physics

NASA Astrophysics Data System (ADS)

Stone, Michael; Goldbart, Paul

2009-07-01

Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

Integrability of the coupled cubic-quintic complex Ginzburg-Landau equations and multiple-soliton solutions via mathematical methods

NASA Astrophysics Data System (ADS)

Selima, Ehab S.; Seadawy, Aly R.; Yao, Xiaohua; Essa, F. A.

2018-02-01

This paper is devoted to study the (1+1)-dimensional coupled cubic-quintic complex Ginzburg-Landau equations (cc-qcGLEs) with complex coefficients. This equation can be used to describe the nonlinear evolution of slowly varying envelopes of periodic spatial-temporal patterns in a convective binary fluid. Dispersion relation and properties of cc-qcGLEs are constructed. Painlevé analysis is used to check the integrability of cc-qcGLEs and to establish the Bäcklund transformation form. New traveling wave solutions and a general form of multiple-soliton solutions of cc-qcGLEs are obtained via the Bäcklund transformation and simplest equation method with Bernoulli, Riccati and Burgers’ equations as simplest equations.

A boundary integral approach to the scattering of nonplanar acoustic waves by rigid bodies

NASA Technical Reports Server (NTRS)

Gallman, Judith M.; Myers, M. K.; Farassat, F.

1990-01-01

The acoustic scattering of an incident wave by a rigid body can be described by a singular Fredholm integral equation of the second kind. This equation is derived by solving the wave equation using generalized function theory, Green's function for the wave equation in unbounded space, and the acoustic boundary condition for a perfectly rigid body. This paper will discuss the derivation of the wave equation, its reformulation as a boundary integral equation, and the solution of the integral equation by the Galerkin method. The accuracy of the Galerkin method can be assessed by applying the technique outlined in the paper to reproduce the known pressure fields that are due to various point sources. From the analysis of these simpler cases, the accuracy of the Galerkin solution can be inferred for the scattered pressure field caused by the incidence of a dipole field on a rigid sphere. The solution by the Galerkin technique can then be applied to such problems as a dipole model of a propeller whose pressure field is incident on a rigid cylinder. This is the groundwork for modeling the scattering of rotating blade noise by airplane fuselages.

A New Formulation of Time Domain Boundary Integral Equation for Acoustic Wave Scattering in the Presence of a Uniform Mean Flow

NASA Technical Reports Server (NTRS)

Hu, Fang; Pizzo, Michelle E.; Nark, Douglas M.

2017-01-01

It has been well-known that under the assumption of a constant uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation, in both the time domain and the frequency domain. Compared with solving partial differential equations, numerical methods based on the boundary integral equation have the advantage of a reduced spatial dimension and, hence, requiring only a surface mesh. However, the constant uniform mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the uniform mean flow. In this paper, we argue that the proper boundary condition for the acoustic wave should not have its normal velocity be zero everywhere on the solid surfaces, as has been applied in the literature. A careful study of the acoustic energy conservation equation is presented that shows such a boundary condition in fact leads to erroneous source or sink points on solid surfaces not aligned with the mean flow. A new solid wall boundary condition is proposed that conserves the acoustic energy and a new time domain boundary integral equation is derived. In addition to conserving the acoustic energy, another significant advantage of the new equation is that it is considerably simpler than previous formulations. In particular, tangential derivatives of the solution on the solid surfaces are no longer needed in the new formulation, which greatly simplifies numerical implementation. Furthermore, stabilization of the new integral equation by Burton-Miller type reformulation is presented. The stability of the new formulation is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the new formulation.

Technological Literacy Learning with Cumulative and Stepwise Integration of Equations into Electrical Circuit Diagrams

ERIC Educational Resources Information Center

Ozogul, G.; Johnson, A. M.; Moreno, R.; Reisslein, M.

2012-01-01

Technological literacy education involves the teaching of basic engineering principles and problem solving, including elementary electrical circuit analysis, to non-engineering students. Learning materials on circuit analysis typically rely on equations and schematic diagrams, which are often unfamiliar to non-engineering students. The goal of…

Application of boundary integral equations to elastoplastic problems

NASA Technical Reports Server (NTRS)

Mendelson, A.; Albers, L. U.

1975-01-01

The application of boundary integral equations to elastoplastic problems is reviewed. Details of the analysis as applied to torsion problems and to plane problems is discussed. Results are presented for the elastoplastic torsion of a square cross section bar and for the plane problem of notched beams. A comparison of different formulations as well as comparisons with experimental results are presented.

Opening of an interface flaw in a layered elastic half-plane under compressive loading

NASA Technical Reports Server (NTRS)

Kennedy, J. M.; Fichter, W. B.; Goree, J. G.

1984-01-01

A static analysis is given of the problem of an elastic layer perfectly bonded, except for a frictionless interface crack, to a dissimilar elastic half-plane. The free surface of the layer is loaded by a finite pressure distribution directly over the crack. The problem is formulated using the two dimensional linear elasticity equations. Using Fourier transforms, the governing equations are converted to a pair of coupled singular integral equations. The integral equations are reduced to a set of simultaneous algebraic equations by expanding the unknown functions in a series of Jacobi polynomials and then evaluating the singular Cauchy-type integrals. The resulting equations are found to be ill-conditioned and, consequently, are solved in the least-squares sense. Results from the analysis show that, under a normal pressure distribution on the free surface of the layer and depending on the combination of geometric and material parameters, the ends of the crack can open. The resulting stresses at the crack-tips are singular, implying that crack growth is possible. The extent of the opening and the crack-top stress intensity factors depend on the width of the pressure distribution zone, the layer thickness, and the relative material properties of the layer and half-plane.

Computerized power supply analysis: State equation generation and terminal models

NASA Technical Reports Server (NTRS)

Garrett, S. J.

1978-01-01

To aid engineers that design power supply systems two analysis tools that can be used with the state equation analysis package were developed. These tools include integration routines that start with the description of a power supply in state equation form and yield analytical results. The first tool uses a computer program that works with the SUPER SCEPTRE circuit analysis program and prints the state equation for an electrical network. The state equations developed automatically by the computer program are used to develop an algorithm for reducing the number of state variables required to describe an electrical network. In this way a second tool is obtained in which the order of the network is reduced and a simpler terminal model is obtained.

Efficient sensitivity analysis method for chaotic dynamical systems

NASA Astrophysics Data System (ADS)

Liao, Haitao

2016-05-01

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.

Numerical solution of linear and nonlinear Fredholm integral equations by using weighted mean-value theorem.

PubMed

Altürk, Ahmet

2016-01-01

Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.

Variational objective analysis for cyclone studies

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.

1989-01-01

Significant accomplishments during 1987 to 1988 are summarized with regard to each of the major project components. Model 1 requires satisfaction of two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. Model 2 requires satisfaction of model 1 plus the thermodynamic equation for a dry atmosphere. Model 3 requires satisfaction of model 2 plus the radiative transfer equation. Model 4 requires satisfaction of model 3 plus a moisture conservation equation and a parameterization for moist processes.

Master equations and the theory of stochastic path integrals

NASA Astrophysics Data System (ADS)

Weber, Markus F.; Frey, Erwin

2017-04-01

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Master equations and the theory of stochastic path integrals.

PubMed

Weber, Markus F; Frey, Erwin

2017-04-01

This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon expanding the forward and the backward path integrals around stationary paths, we then discuss and extend a recent method for the computation of rare event probabilities. Besides, we also derive path integral representations for processes with continuous state spaces whose forward and backward master equations admit Kramers-Moyal expansions. A truncation of the backward expansion at the level of a diffusion approximation recovers a classic path integral representation of the (backward) Fokker-Planck equation. One can rewrite this path integral in terms of an Onsager-Machlup function and, for purely diffusive Brownian motion, it simplifies to the path integral of Wiener. To make this review accessible to a broad community, we have used the language of probability theory rather than quantum (field) theory and do not assume any knowledge of the latter. The probabilistic structures underpinning various technical concepts, such as coherent states, the Doi-shift, and normal-ordered observables, are thereby made explicit.

Soliton structure versus singularity analysis: Third-order completely intergrable nonlinear differential equations in 1 + 1-dimensions

NASA Astrophysics Data System (ADS)

Fuchssteiner, Benno; Carillo, Sandra

1989-01-01

Bäcklund transformations between all known completely integrable third-order differential equations in (1 + 1)-dimensions are established and the corresponding transformations formulas for their hereditary operators and Hamiltonian formulations are exhibited. Some of these Bäcklund transformations are not injective; therefore additional non-commutative symmetry groups are found for some equations. These non-commutative symmetry groups are classified as having a semisimple part isomorphic to the affine algebra A(1)1. New completely integrable third-order integro-differential equations, some depending explicitly on x, are given. These new equations give rise to nonin equation. Connections between the singularity equations (from the Painlevé analysis) and the nonlinear equations for interacting solitons are established. A common approach to singularity analysis and soliton structure is introduced. The Painlevé analysis is modified in such a sense that it carries over directly and without difficulty to the time evolution of singularity manifolds of equations like the sine-Gordon and nonlinear Schrödinger equation. A method to recover the Painlevé series from its constant level term is exhibit. The soliton-singularity transform is recognized to be connected to the Möbius group. This gives rise to a Darboux-like result for the spectral properties of the recursion operator. These connections are used in order to explain why poles of soliton equations move like trajectories of interacting solitons. Furthermore it is explicitly computed how solitons of singularity equations behave under the effect of this soliton-singularity transform. This then leads to the result that only for scaling degrees α = -1 and α = -2 the usual Painlevé analysis can be carried out. A new invariance principle, connected to kernels of differential operators is discovered. This new invariance, for example, connects the explicit solutions of the Liouville equation with the Miura transform. Simple methods are exhibited which allow to compute out of N-soliton solutions of the KdV (Bargman potentials) explicit solutions of equations like the Harry Dym equation. Certain solutions are plotted.

Integrals and integral equations in linearized wing theory

NASA Technical Reports Server (NTRS)

Lomax, Harvard; Heaslet, Max A; Fuller, Franklyn B

1951-01-01

The formulas of subsonic and supersonic wing theory for source, doublet, and vortex distributions are reviewed and a systematic presentation is provided which relates these distributions to the pressure and to the vertical induced velocity in the plane of the wing. It is shown that care must be used in treating the singularities involved in the analysis and that the order of integration is not always reversible. Concepts suggested by the irreversibility of order of integration are shown to be useful in the inversion of singular integral equations when operational techniques are used. A number of examples are given to illustrate the methods presented, attention being directed to supersonic flight speed.

Theoretical analysis of linearized acoustics and aerodynamics of advanced supersonic propellers

NASA Technical Reports Server (NTRS)

Farassat, F.

1985-01-01

The derivation of a formula for prediction of the noise of supersonic propellers using time domain analysis is presented. This formula is a solution of the Ffowcs Williams-Hawkings equation and does not have the Doppler singularity of some other formulations. The result presented involves some surface integrals over the blade and line integrals over the leading and trailing edges. The blade geometry, motion and surface pressure are needed for noise calculation. To obtain the blade surface pressure, the observer is moved onto the blade surface and a linear singular integral equation is derived which can be solved numerically. Two examples of acoustic calculations using a computer program are currently under development.

Separation of variables in the special diagonal Hamilton-Jacobi equation: Application to the dynamical problem of a particle constrained on a moving surface

NASA Technical Reports Server (NTRS)

Blanchard, D. L.; Chan, F. K.

1973-01-01

For a time-dependent, n-dimensional, special diagonal Hamilton-Jacobi equation a necessary and sufficient condition for the separation of variables to yield a complete integral of the form was established by specifying the admissible forms in terms of arbitrary functions. A complete integral was then expressed in terms of these arbitrary functions and also the n irreducible constants. As an application of the results obtained for the two-dimensional Hamilton-Jacobi equation, analysis was made for a comparatively wide class of dynamical problems involving a particle moving in Euclidean three-dimensional space under the action of external forces but constrained on a moving surface. All the possible cases in which this equation had a complete integral of the form were obtained and these are tubulated for reference.

Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture

NASA Astrophysics Data System (ADS)

Davey, K.; Darvizeh, R.

2016-09-01

Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine-Hugoniot conditions for fracture are established along with extended forms of J -integrals.

Nonlinear truncation error analysis of finite difference schemes for the Euler equations

NASA Technical Reports Server (NTRS)

Klopfer, G. H.; Mcrae, D. S.

1983-01-01

It is pointed out that, in general, dissipative finite difference integration schemes have been found to be quite robust when applied to the Euler equations of gas dynamics. The present investigation considers a modified equation analysis of both implicit and explicit finite difference techniques as applied to the Euler equations. The analysis is used to identify those error terms which contribute most to the observed solution errors. A technique for analytically removing the dominant error terms is demonstrated, resulting in a greatly improved solution for the explicit Lax-Wendroff schemes. It is shown that the nonlinear truncation errors are quite large and distributed quite differently for each of the three conservation equations as applied to a one-dimensional shock tube problem.

Ignition of a combustible half space

NASA Technical Reports Server (NTRS)

Olmstead, W. E.

1983-01-01

A half space of combustible material is subjected to an arbitrary energy flux at the boundary where convection heat loss is also allowed. An asymptotic analysis of the temperature growth reveals two conditions necessary for ignition to occur. Cases of both large and order unity Lewis number are shown to lead to a nonlinear integral equation governing the thermal runaway. Some global and asymptotic properties of the integral equation are obtained.

Generalized recursive solutions to Ornstein-Zernike integral equations

NASA Astrophysics Data System (ADS)

Rossky, Peter J.; Dale, William D. T.

1980-09-01

Recursive procedures for the solution of a class of integral equations based on the Ornstein-Zernike equation are developed; the hypernetted chain and Percus-Yevick equations are two special cases of the class considered. It is shown that certain variants of the new procedures developed here are formally equivalent to those recently developed by Dale and Friedman, if the new recursive expressions are initialized in the same way as theirs. However, the computational solution of the new equations is significantly more efficient. Further, the present analysis leads to the identification of various graphical quantities arising in the earlier study with more familiar quantities related to pair correlation functions. The analysis is greatly facilitated by the use of several identities relating simple chain sums whose graphical elements can be written as a sum of two or more parts. In particular, the use of these identities permits renormalization of the equivalent series solution to the integral equation to be directly incorporated into the recursive solution in a straightforward manner. Formulas appropriate to renormalization with respect to long and short range parts of the pair potential, as well as more general components of the direct correlation function, are obtained. To further illustrate the utility of this approach, we show that a simple generalization of the hypernetted chain closure relation for the direct correlation function leads directly to the reference hypernetted chain (RHNC) equation due to Lado. The form of the correlation function used in the exponential approximation of Andersen and Chandler is then seen to be equivalent to the first estimate obtained from a renormalized RHNC equation.

Fourier Spectroscopy: A Simple Analysis Technique

ERIC Educational Resources Information Center

Oelfke, William C.

1975-01-01

Presents a simple method of analysis in which the student can integrate, point by point, any interferogram to obtain its Fourier transform. The manual technique requires no special equipment and is based on relationships that most undergraduate physics students can derive from the Fourier integral equations. (Author/MLH)

Implementation of Laminate Theory Into Strain Rate Dependent Micromechanics Analysis of Polymer Matrix Composites

NASA Technical Reports Server (NTRS)

Goldberg, Robert K.

2000-01-01

A research program is in progress to develop strain rate dependent deformation and failure models for the analysis of polymer matrix composites subject to impact loads. Previously, strain rate dependent inelastic constitutive equations developed to model the polymer matrix were implemented into a mechanics of materials based micromechanics method. In the current work, the computation of the effective inelastic strain in the micromechanics model was modified to fully incorporate the Poisson effect. The micromechanics equations were also combined with classical laminate theory to enable the analysis of symmetric multilayered laminates subject to in-plane loading. A quasi-incremental trapezoidal integration method was implemented to integrate the constitutive equations within the laminate theory. Verification studies were conducted using an AS4/PEEK composite using a variety of laminate configurations and strain rates. The predicted results compared well with experimentally obtained values.

Concepts for radically increasing the numerical convergence rate of the Euler equations

NASA Technical Reports Server (NTRS)

Nixon, David; Tzuoo, Keh-Lih; Caruso, Steven C.; Farshchi, Mohammad; Klopfer, Goetz H.; Ayoub, Alfred

1987-01-01

Integral equation and finite difference methods have been developed for solving transonic flow problems using linearized forms of the transonic small disturbance and Euler equations. A key element is the use of a strained coordinate system in which the shock remains fixed. Additional criteria are developed to determine the free parameters in the coordinate straining; these free parameters are functions of the shock location. An integral equation analysis showed that the shock is located by ensuring that no expansion shocks exist in the solution. The expansion shock appears as oscillations in the solution near the sonic line, and the correct shock location is determined by removing these oscillations. A second objective was to study the ability of the Euler equation to model separated flow.

Method of mechanical quadratures for solving singular integral equations of various types

NASA Astrophysics Data System (ADS)

Sahakyan, A. V.; Amirjanyan, H. A.

2018-04-01

The method of mechanical quadratures is proposed as a common approach intended for solving the integral equations defined on finite intervals and containing Cauchy-type singular integrals. This method can be used to solve singular integral equations of the first and second kind, equations with generalized kernel, weakly singular equations, and integro-differential equations. The quadrature rules for several different integrals represented through the same coefficients are presented. This allows one to reduce the integral equations containing integrals of different types to a system of linear algebraic equations.

Solving Modal Equations of Motion with Initial Conditions Using MSC/NASTRAN DMAP. Part 2; Coupled Versus Uncoupled Integration

NASA Technical Reports Server (NTRS)

Barnett, Alan R.; Ibrahim, Omar M.; Abdallah, Ayman A.; Sullivan, Timothy L.

1993-01-01

By utilizing MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) in an existing NASA Lewis Research Center coupled loads methodology, solving modal equations of motion with initial conditions is possible using either coupled (Newmark-Beta) or uncoupled (exact mode superposition) integration available within module TRD1. Both the coupled and newly developed exact mode superposition methods have been used to perform transient analyses of various space systems. However, experience has shown that in most cases, significant time savings are realized when the equations of motion are integrated using the uncoupled solver instead of the coupled solver. Through the results of a real-world engineering analysis, advantages of using the exact mode superposition methodology are illustrated.

A study of the use of abstract types for the representation of engineering units in integration and test applications

NASA Technical Reports Server (NTRS)

Johnson, Charles S.

1986-01-01

Physical quantities using various units of measurement can be well represented in Ada by the use of abstract types. Computation involving these quantities (electric potential, mass, volume) can also automatically invoke the computation and checking of some of the implicitly associable attributes of measurements. Quantities can be held internally in SI units, transparently to the user, with automatic conversion. Through dimensional analysis, the type of the derived quantity resulting from a computation is known, thereby allowing dynamic checks of the equations used. The impact of the possible implementation of these techniques in integration and test applications is discussed. The overhead of computing and transporting measurement attributes is weighed against the advantages gained by their use. The construction of a run time interpreter using physical quantities in equations can be aided by the dynamic equation checks provided by dimensional analysis. The effects of high levels of abstraction on the generation and maintenance of software used in integration and test applications are also discussed.

Stability of the iterative solutions of integral equations as one phase freezing criterion.

PubMed

Fantoni, R; Pastore, G

2003-10-01

A recently proposed connection between the threshold for the stability of the iterative solution of integral equations for the pair correlation functions of a classical fluid and the structural instability of the corresponding real fluid is carefully analyzed. Direct calculation of the Lyapunov exponent of the standard iterative solution of hypernetted chain and Percus-Yevick integral equations for the one-dimensional (1D) hard rods fluid shows the same behavior observed in 3D systems. Since no phase transition is allowed in such 1D system, our analysis shows that the proposed one phase criterion, at least in this case, fails. We argue that the observed proximity between the numerical and the structural instability in 3D originates from the enhanced structure present in the fluid but, in view of the arbitrary dependence on the iteration scheme, it seems uneasy to relate the numerical stability analysis to a robust one-phase criterion for predicting a thermodynamic phase transition.

Error and Complexity Analysis for a Collocation-Grid-Projection Plus Precorrected-FFT Algorithm for Solving Potential Integral Equations with LaPlace or Helmholtz Kernels

NASA Technical Reports Server (NTRS)

Phillips, J. R.

1996-01-01

In this paper we derive error bounds for a collocation-grid-projection scheme tuned for use in multilevel methods for solving boundary-element discretizations of potential integral equations. The grid-projection scheme is then combined with a precorrected FFT style multilevel method for solving potential integral equations with 1/r and e(sup ikr)/r kernels. A complexity analysis of this combined method is given to show that for homogeneous problems, the method is order n natural log n nearly independent of the kernel. In addition, it is shown analytically and experimentally that for an inhomogeneity generated by a very finely discretized surface, the combined method slows to order n(sup 4/3). Finally, examples are given to show that the collocation-based grid-projection plus precorrected-FFT scheme is competitive with fast-multipole algorithms when considering realistic problems and 1/r kernels, but can be used over a range of spatial frequencies with only a small performance penalty.

A wideband FMBEM for 2D acoustic design sensitivity analysis based on direct differentiation method

NASA Astrophysics Data System (ADS)

Chen, Leilei; Zheng, Changjun; Chen, Haibo

2013-09-01

This paper presents a wideband fast multipole boundary element method (FMBEM) for two dimensional acoustic design sensitivity analysis based on the direct differentiation method. The wideband fast multipole method (FMM) formed by combining the original FMM and the diagonal form FMM is used to accelerate the matrix-vector products in the boundary element analysis. The Burton-Miller formulation is used to overcome the fictitious frequency problem when using a single Helmholtz boundary integral equation for exterior boundary-value problems. The strongly singular and hypersingular integrals in the sensitivity equations can be evaluated explicitly and directly by using the piecewise constant discretization. The iterative solver GMRES is applied to accelerate the solution of the linear system of equations. A set of optimal parameters for the wideband FMBEM design sensitivity analysis are obtained by observing the performances of the wideband FMM algorithm in terms of computing time and memory usage. Numerical examples are presented to demonstrate the efficiency and validity of the proposed algorithm.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liao, Haitao, E-mail: liaoht@cae.ac.cn

The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results inmore » an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.« less

Stochastic theory of polarized light in nonlinear birefringent media: An application to optical rotation

NASA Astrophysics Data System (ADS)

Tsuchida, Satoshi; Kuratsuji, Hiroshi

2018-05-01

A stochastic theory is developed for the light transmitting the optical media exhibiting linear and nonlinear birefringence. The starting point is the two-component nonlinear Schrödinger equation (NLSE). On the basis of the ansatz of “soliton” solution for the NLSE, the evolution equation for the Stokes parameters is derived, which turns out to be the Langevin equation by taking account of randomness and dissipation inherent in the birefringent media. The Langevin equation is converted to the Fokker-Planck (FP) equation for the probability distribution by employing the technique of functional integral on the assumption of the Gaussian white noise for the random fluctuation. The specific application is considered for the optical rotation, which is described by the ellipticity (third component of the Stokes parameters) alone: (i) The asymptotic analysis is given for the functional integral, which leads to the transition rate on the Poincaré sphere. (ii) The FP equation is analyzed in the strong coupling approximation, by which the diffusive behavior is obtained for the linear and nonlinear birefringence. These would provide with a basis of statistical analysis for the polarization phenomena in nonlinear birefringent media.

Thermoeconomic analysis of an integrated multi-effect desalination thermal vapor compression (MED-TVC) system with a trigeneration system using triple-pressure HRSG

NASA Astrophysics Data System (ADS)

Ghaebi, Hadi; Abbaspour, Ghader

2018-05-01

In this research, thermoeconomic analysis of a multi-effect desalination thermal vapor compression (MED-TVC) system integrated with a trigeneration system with a gas turbine prime mover is carried out. The integrated system comprises of a compressor, a combustion chamber, a gas turbine, a triple-pressure (low, medium and high pressures) heat recovery steam generator (HRSG) system, an absorption chiller cycle (ACC), and a multi-effect desalination (MED) system. Low pressure steam produced in the HRSG is used to drive absorption chiller cycle, medium pressure is used in desalination system and high pressure superheated steam is used for heating purposes. For thermodynamic and thermoeconomic analysis of the proposed integrated system, Engineering Equation Solver (EES) is used by employing mass, energy, exergy, and cost balance equations for each component of system. The results of the modeling showed that with the new design, the exergy efficiency in the base design will increase to 57.5%. In addition, thermoeconomic analysis revealed that the net power, heating, fresh water and cooling have the highest production cost, respectively.

Theoretical Prediction of Pressure Distributions on Nonlifting Airfoils at High Subsonic Speeds

NASA Technical Reports Server (NTRS)

Spreiter, John R; Alksne, Alberta

1955-01-01

Theoretical pressure distributions on nonlifting circular-arc airfoils in two-dimensional flows with high subsonic free-stream velocity are found by determining approximate solutions, through an iteration process, of an integral equation for transonic flow proposed by Oswatitsch. The integral equation stems directly from the small-disturbance theory for transonic flow. This method of analysis possesses the advantage of remaining in the physical, rather than the hodograph, variable and can be applied in airfoils having curved surfaces. After discussion of the derivation of the integral equation and qualitative aspects of the solution, results of calculations carried out for circular-arc airfoils in flows with free-stream Mach numbers up to unity are described. These results indicate most of the principal phenomena observed in experimental studies.

An integral turbulent kinetic energy analysis of free shear flows

NASA Technical Reports Server (NTRS)

Peters, C. E.; Phares, W. J.

1973-01-01

Mixing of coaxial streams is analyzed by application of integral techniques. An integrated turbulent kinetic energy (TKE) equation is solved simultaneously with the integral equations for the mean flow. Normalized TKE profile shapes are obtained from incompressible jet and shear layer experiments and are assumed to be applicable to all free turbulent flows. The shear stress at the midpoint of the mixing zone is assumed to be directly proportional to the local TKE, and dissipation is treated with a generalization of the model developed for isotropic turbulence. Although the analysis was developed for ducted flows, constant-pressure flows were approximated with the duct much larger than the jet. The axisymmetric flows under consideration were predicted with reasonable accuracy. Fairly good results were also obtained for the fully developed two-dimensional shear layers, which were computed as thin layers at the boundary of a large circular jet.

Applications of Ergodic Theory to Coverage Analysis

NASA Technical Reports Server (NTRS)

Lo, Martin W.

2003-01-01

The study of differential equations, or dynamical systems in general, has two fundamentally different approaches. We are most familiar with the construction of solutions to differential equations. Another approach is to study the statistical behavior of the solutions. Ergodic Theory is one of the most developed methods to study the statistical behavior of the solutions of differential equations. In the theory of satellite orbits, the statistical behavior of the orbits is used to produce 'Coverage Analysis' or how often a spacecraft is in view of a site on the ground. In this paper, we consider the use of Ergodic Theory for Coverage Analysis. This allows us to greatly simplify the computation of quantities such as the total time for which a ground station can see a satellite without ever integrating the trajectory, see Lo 1,2. More over, for any quantity which is an integrable function of the ground track, its average may be computed similarly without the integration of the trajectory. For example, the data rate for a simple telecom system is a function of the distance between the satellite and the ground station. We show that such a function may be averaged using the Ergodic Theorem.

Exact and Approximate Solutions for the Decades-Old Michaelis-Menten Equation: Progress-Curve Analysis through Integrated Rate Equations

ERIC Educational Resources Information Center

Golicnik, Marko

2011-01-01

The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate "V", and the Michaelis constant "K"[subscript M]) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to…

General pulsed-field gradient signal attenuation expression based on a fractional integral modified-Bloch equation

NASA Astrophysics Data System (ADS)

Lin, Guoxing

2018-10-01

Anomalous diffusion has been investigated in many polymer and biological systems. The analysis of PFG anomalous diffusion relies on the ability to obtain the signal attenuation expression. However, the general analytical PFG signal attenuation expression based on the fractional derivative has not been previously reported. Additionally, the reported modified-Bloch equations for PFG anomalous diffusion in the literature yielded different results due to their different forms. Here, a new integral type modified-Bloch equation based on the fractional derivative for PFG anomalous diffusion is proposed, which is significantly different from the conventional differential type modified-Bloch equation. The merit of the integral type modified-Bloch equation is that the original properties of the contributions from linear or nonlinear processes remain unchanged at the instant of the combination. From the modified-Bloch equation, the general solutions are derived, which includes the finite gradient pulse width (FGPW) effect. The numerical evaluation of these PFG signal attenuation expressions can be obtained either by the Adomian decomposition, or a direct integration method that is fast and practicable. The theoretical results agree with the continuous-time random walk (CTRW) simulations performed in this paper. Additionally, the relaxation effect in PFG anomalous diffusion is found to be different from that in PFG normal diffusion. The new modified-Bloch equations and their solutions provide a fundamental tool to analyze PFG anomalous diffusion in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI).

Dynamical property analysis of fractionally damped van der pol oscillator and its application

NASA Astrophysics Data System (ADS)

Zhong, Qiuhui; Zhang, Chunrui

2012-01-01

In this paper, the fractionally damped van der pol equation was studied. Firstly, the fractionally damped van der pol equation was transformed into a set of integer order equations. Then the Lyapunov exponents diagram was given. Secondly, it was transformed into a set of fractional integral equations and solved by a predictor-corrector method. The time domain diagrams and phase trajectory were used to describe the dynamic behavior. Finally, the fractionally damped van der pol equation was used to detect a weak signal.

Transport equations for partially ionized reactive plasma in magnetic field

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhdanov, V. M.; Stepanenko, A. A.

2016-06-08

Transport equations for partially ionized reactive plasma in magnetic field taking into account the internal degrees of freedom and electronic excitation of plasma particles are derived. As a starting point of analysis the kinetic equation with a binary collision operator written in the Wang-Chang and Uhlenbeck form and with a reactive collision integral allowing for arbitrary chemical reactions is used. The linearized variant of Grad’s moment method is applied to deduce the systems of moment equations for plasma and also full and reduced transport equations for plasma species nonequilibrium parameters.

Integrable multi-component generalization of a modified short pulse equation

NASA Astrophysics Data System (ADS)

Matsuno, Yoshimasa

2016-11-01

We propose a multi-component generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng's two-component SP equation. Above all, we address the two-component system in depth. We obtain the Lax pair, an infinite number of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the two-component system exhibits cusp solitons and breathers for which the detailed analysis is performed. Specifically, we explore the interaction process of two cusp solitons and derive the formula for the phase shift. While cusp solitons are singular solutions, smooth breather solutions are shown to exist, provided that the parameters characterizing the solutions satisfy certain conditions. Last, we discuss the relation between the proposed system and existing two-component SP equations.

The evolution of methods for noise prediction of high speed rotors and propellers in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1986-01-01

Linear wave equation models which have been used over the years at NASA Langley for describing noise emissions from high speed rotating blades are summarized. The noise sources are assumed to lie on a moving surface, and analysis of the situation has been based on the Ffowcs Williams-Hawkings (FW-H) equation. Although the equation accounts for two surface and one volume source, the NASA analyses have considered only the surface terms. Several variations on the FW-H model are delineated for various types of applications, noting the computational benefits of removing the frequency dependence of the calculations. Formulations are also provided for compact and noncompact sources, and features of Long's subsonic integral equation and Farassat's high speed integral equation are discussed. The selection of subsonic or high speed models is dependent on the Mach number of the blade surface where the source is located.

Optical solitons, explicit solutions and modulation instability analysis with second-order spatio-temporal dispersion

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Isa Aliyu, Aliyu; Yusuf, Abdullahi; Baleanu, Dumitru

2017-12-01

This paper obtains the dark, bright, dark-bright or combined optical and singular solitons to the nonlinear Schrödinger equation (NLSE) with group velocity dispersion coefficient and second-order spatio-temporal dispersion coefficient, which arises in photonics and waveguide optics and in optical fibers. The integration algorithm is the sine-Gordon equation method (SGEM). Furthermore, the explicit solutions of the equation are derived by considering the power series solutions (PSS) theory and the convergence of the solutions is guaranteed. Lastly, the modulation instability analysis (MI) is studied based on the standard linear-stability analysis and the MI gain spectrum is obtained.

Integrated force method versus displacement method for finite element analysis

NASA Technical Reports Server (NTRS)

Patnaik, S. N.; Berke, L.; Gallagher, R. H.

1991-01-01

A novel formulation termed the integrated force method (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EEs) are integrated with the global compatibility conditions (CCs) to form the governing set of equations. In IFM the CCs are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in constrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this report IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost.

An efficient computational method for solving nonlinear stochastic Itô integral equations: Application for stochastic problems in physics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Heydari, M.H., E-mail: heydari@stu.yazd.ac.ir; The Laboratory of Quantum Information Processing, Yazd University, Yazd; Hooshmandasl, M.R., E-mail: hooshmandasl@yazd.ac.ir

Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Errormore » analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.« less

Integrated force method versus displacement method for finite element analysis

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Berke, Laszlo; Gallagher, Richard H.

1990-01-01

A novel formulation termed the integrated force method (IFM) has been developed in recent years for analyzing structures. In this method all the internal forces are taken as independent variables, and the system equilibrium equations (EE's) are integrated with the global compatibility conditions (CC's) to form the governing set of equations. In IFM the CC's are obtained from the strain formulation of St. Venant, and no choices of redundant load systems have to be made, in constrast to the standard force method (SFM). This property of IFM allows the generation of the governing equation to be automated straightforwardly, as it is in the popular stiffness method (SM). In this report IFM and SM are compared relative to the structure of their respective equations, their conditioning, required solution methods, overall computational requirements, and convergence properties as these factors influence the accuracy of the results. Overall, this new version of the force method produces more accurate results than the stiffness method for comparable computational cost.

Integrability in AdS/CFT correspondence: quasi-classical analysis

NASA Astrophysics Data System (ADS)

Gromov, Nikolay

2009-06-01

In this review, we consider a quasi-classical method applicable to integrable field theories which is based on a classical integrable structure—the algebraic curve. We apply it to the Green-Schwarz superstring on the AdS5 × S5 space. We show that the proposed method reproduces perfectly the earlier results obtained by expanding the string action for some simple classical solutions. The construction is explicitly covariant and is not based on a particular parameterization of the fields and as a result is free from ambiguities. On the other hand, the finite size corrections in some particularly important scaling limit are studied in this paper for a system of Bethe equations. For the general superalgebra \\su(N|K) , the result for the 1/L corrections is obtained. We find an integral equation which describes these corrections in a closed form. As an application, we consider the conjectured Beisert-Staudacher (BS) equations with the Hernandez-Lopez dressing factor where the finite size corrections should reproduce quasi-classical results around a general classical solution. Indeed, we show that our integral equation can be interpreted as a sum of all physical fluctuations and thus prove the complete one-loop consistency of the BS equations. We demonstrate that any local conserved charge (including the AdS energy) computed from the BS equations is indeed given at one loop by the sum of the charges of fluctuations with an exponential precision for large S5 angular momentum of the string. As an independent result, the BS equations in an \\su(2) sub-sector were derived from Zamolodchikovs's S-matrix. The paper is based on the author's PhD thesis.

Acidity in DMSO from the embedded cluster integral equation quantum solvation model.

PubMed

Heil, Jochen; Tomazic, Daniel; Egbers, Simon; Kast, Stefan M

2014-04-01

The embedded cluster reference interaction site model (EC-RISM) is applied to the prediction of acidity constants of organic molecules in dimethyl sulfoxide (DMSO) solution. EC-RISM is based on a self-consistent treatment of the solute's electronic structure and the solvent's structure by coupling quantum-chemical calculations with three-dimensional (3D) RISM integral equation theory. We compare available DMSO force fields with reference calculations obtained using the polarizable continuum model (PCM). The results are evaluated statistically using two different approaches to eliminating the proton contribution: a linear regression model and an analysis of pK(a) shifts for compound pairs. Suitable levels of theory for the integral equation methodology are benchmarked. The results are further analyzed and illustrated by visualizing solvent site distribution functions and comparing them with an aqueous environment.

Integral approximations to classical diffusion and smoothed particle hydrodynamics

DOE PAGES

Du, Qiang; Lehoucq, R. B.; Tartakovsky, A. M.

2014-12-31

The contribution of the paper is the approximation of a classical diffusion operator by an integral equation with a volume constraint. A particular focus is on classical diffusion problems associated with Neumann boundary conditions. By exploiting this approximation, we can also approximate other quantities such as the flux out of a domain. Our analysis of the model equation on the continuum level is closely related to the recent work on nonlocal diffusion and peridynamic mechanics. In particular, we elucidate the role of a volumetric constraint as an approximation to a classical Neumann boundary condition in the presence of physical boundary.more » The volume-constrained integral equation then provides the basis for accurate and robust discretization methods. As a result, an immediate application is to the understanding and improvement of the Smoothed Particle Hydrodynamics (SPH) method.« less

Modification of a variational objective analysis model for new equations for pressure gradient and vertical velocity in the lower troposphere and for spatial resolution and accuracy of satellite data

NASA Technical Reports Server (NTRS)

Achtemeier, G. L.

1986-01-01

Since late 1982 NASA has supported research to develop a numerical variational model for the diagnostic assimilation of conventional and space-based meteorological data. In order to analyze the model components, four variational models are defined dividing the problem naturally according to increasing complexity. The first of these variational models (MODEL I), the subject of this report, contains the two nonlinear horizontal momentum equations, the integrated continuity equation, and the hydrostatic equation. This report summarizes the results of research (1) to improve the way the large nonmeteorological parts of the pressure gradient force are partitioned between the two terms of the pressure gradient force terms of the horizontal momentum equations, (2) to generalize the integrated continuity equation to account for variable pressure thickness over elevated terrain, and (3) to introduce horizontal variation in the precision modulus weights for the observations.

Dissolution process analysis using model-free Noyes-Whitney integral equation.

PubMed

Hattori, Yusuke; Haruna, Yoshimasa; Otsuka, Makoto

2013-02-01

Drug dissolution process of solid dosages is theoretically described by Noyes-Whitney-Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes-Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased. Copyright © 2012 Elsevier B.V. All rights reserved.

A critical analysis of the accuracy of several numerical techniques for combustion kinetic rate equations

NASA Technical Reports Server (NTRS)

Radhadrishnan, Krishnan

1993-01-01

A detailed analysis of the accuracy of several techniques recently developed for integrating stiff ordinary differential equations is presented. The techniques include two general-purpose codes EPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREK1D, and GCKP4 developed specifically to solve chemical kinetic rate equations. The accuracy study is made by application of these codes to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas-phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. To illustrate the error variation in the different combustion regimes the species are divided into three types (reactants, intermediates, and products), and error versus time plots are presented for each species type and the temperature. These plots show that CHEMEQ is the most accurate code during induction and early heat release. During late heat release and equilibration, however, the other codes are more accurate. A single global quantity, a mean integrated root-mean-square error, that measures the average error incurred in solving the complete problem is used to compare the accuracy of the codes. Among the codes examined, LSODE is the most accurate for solving chemical kinetics problems. It is also the most efficient code, in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that use of the algebraic enthalpy conservation equation to compute the temperature can be more accurate and efficient than integrating the temperature differential equation.

Coupling of Coastal Wave Transformation and Computational Fluid Dynamics Models for Seakeeping Analysis

DTIC Science & Technology

2017-04-03

setup in terms of temporal and spatial discretization . The second component was an extension of existing depth-integrated wave models to describe...equations (Abbott, 1976). Discretization schemes involve numerical dispersion and dissipation that distort the true character of the governing equations...represent a leading-order approximation of the Boussinesq-type equations. Tam and Webb (1993) proposed a wavenumber-based discretization scheme to preserve

Multiple and exact soliton solutions of the perturbed Korteweg-de Vries equation of long surface waves in a convective fluid via Painlevé analysis, factorization, and simplest equation methods.

PubMed

Selima, Ehab S; Yao, Xiaohua; Wazwaz, Abdul-Majid

2017-06-01

In this research, the surface waves of a horizontal fluid layer open to air under gravity field and vertical temperature gradient effects are studied. The governing equations of this model are reformulated and converted to a nonlinear evolution equation, the perturbed Korteweg-de Vries (pKdV) equation. We investigate the latter equation, which includes dispersion, diffusion, and instability effects, in order to examine the evolution of long surface waves in a convective fluid. Dispersion relation of the pKdV equation and its properties are discussed. The Painlevé analysis is applied not only to check the integrability of the pKdV equation but also to establish the Bäcklund transformation form. In addition, traveling wave solutions and a general form of the multiple-soliton solutions of the pKdV equation are obtained via Bäcklund transformation, the simplest equation method using Bernoulli, Riccati, and Burgers' equations as simplest equations, and the factorization method.

A unique set of micromechanics equations for high temperature metal matrix composites

NASA Technical Reports Server (NTRS)

Hopkins, D. A.; Chamis, C. C.

1985-01-01

A unique set of micromechanic equations is presented for high temperature metal matrix composites. The set includes expressions to predict mechanical properties, thermal properties and constituent microstresses for the unidirectional fiber reinforced ply. The equations are derived based on a mechanics of materials formulation assuming a square array unit cell model of a single fiber, surrounding matrix and an interphase to account for the chemical reaction which commonly occurs between fiber and matrix. A three-dimensional finite element analysis was used to perform a preliminary validation of the equations. Excellent agreement between properties predicted using the micromechanics equations and properties simulated by the finite element analyses are demonstrated. Implementation of the micromechanics equations as part of an integrated computational capability for nonlinear structural analysis of high temperature multilayered fiber composites is illustrated.

Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma

NASA Astrophysics Data System (ADS)

EL-Kalaawy, O. H.; Moawad, S. M.; Wael, Shrouk

The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions.

Variational Methods in Design Optimization and Sensitivity Analysis for Two-Dimensional Euler Equations

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Tiwari, S. N.; Smith, R. E.

1997-01-01

Variational methods (VM) sensitivity analysis employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Variational Methods in Sensitivity Analysis and Optimization for Aerodynamic Applications

NASA Technical Reports Server (NTRS)

Ibrahim, A. H.; Hou, G. J.-W.; Tiwari, S. N. (Principal Investigator)

1996-01-01

Variational methods (VM) sensitivity analysis, which is the continuous alternative to the discrete sensitivity analysis, is employed to derive the costate (adjoint) equations, the transversality conditions, and the functional sensitivity derivatives. In the derivation of the sensitivity equations, the variational methods use the generalized calculus of variations, in which the variable boundary is considered as the design function. The converged solution of the state equations together with the converged solution of the costate equations are integrated along the domain boundary to uniquely determine the functional sensitivity derivatives with respect to the design function. The determination of the sensitivity derivatives of the performance index or functional entails the coupled solutions of the state and costate equations. As the stable and converged numerical solution of the costate equations with their boundary conditions are a priori unknown, numerical stability analysis is performed on both the state and costate equations. Thereafter, based on the amplification factors obtained by solving the generalized eigenvalue equations, the stability behavior of the costate equations is discussed and compared with the state (Euler) equations. The stability analysis of the costate equations suggests that the converged and stable solution of the costate equation is possible only if the computational domain of the costate equations is transformed to take into account the reverse flow nature of the costate equations. The application of the variational methods to aerodynamic shape optimization problems is demonstrated for internal flow problems at supersonic Mach number range. The study shows, that while maintaining the accuracy of the functional sensitivity derivatives within the reasonable range for engineering prediction purposes, the variational methods show a substantial gain in computational efficiency, i.e., computer time and memory, when compared with the finite difference sensitivity analysis.

Numerical analysis of MHD Carreau fluid flow over a stretching cylinder with homogenous-heterogeneous reactions

NASA Astrophysics Data System (ADS)

Khan, Imad; Ullah, Shafquat; Malik, M. Y.; Hussain, Arif

2018-06-01

The current analysis concentrates on the numerical solution of MHD Carreau fluid flow over a stretching cylinder under the influences of homogeneous-heterogeneous reactions. Modelled non-linear partial differential equations are converted into ordinary differential equations by using suitable transformations. The resulting system of equations is solved with the aid of shooting algorithm supported by fifth order Runge-Kutta integration scheme. The impact of non-dimensional governing parameters on the velocity, temperature, skin friction coefficient and local Nusselt number are comprehensively delineated with the help of graphs and tables.

Discretization analysis of bifurcation based nonlinear amplifiers

NASA Astrophysics Data System (ADS)

Feldkord, Sven; Reit, Marco; Mathis, Wolfgang

2017-09-01

Recently, for modeling biological amplification processes, nonlinear amplifiers based on the supercritical Andronov-Hopf bifurcation have been widely analyzed analytically. For technical realizations, digital systems have become the most relevant systems in signal processing applications. The underlying continuous-time systems are transferred to the discrete-time domain using numerical integration methods. Within this contribution, effects on the qualitative behavior of the Andronov-Hopf bifurcation based systems concerning numerical integration methods are analyzed. It is shown exemplarily that explicit Runge-Kutta methods transform the truncated normalform equation of the Andronov-Hopf bifurcation into the normalform equation of the Neimark-Sacker bifurcation. Dependent on the order of the integration method, higher order terms are added during this transformation.A rescaled normalform equation of the Neimark-Sacker bifurcation is introduced that allows a parametric design of a discrete-time system which corresponds to the rescaled Andronov-Hopf system. This system approximates the characteristics of the rescaled Hopf-type amplifier for a large range of parameters. The natural frequency and the peak amplitude are preserved for every set of parameters. The Neimark-Sacker bifurcation based systems avoid large computational effort that would be caused by applying higher order integration methods to the continuous-time normalform equations.

Integral-equation based methods for parameter estimation in output pulses of radiation detectors: Application in nuclear medicine and spectroscopy

NASA Astrophysics Data System (ADS)

Mohammadian-Behbahani, Mohammad-Reza; Saramad, Shahyar

2018-04-01

Model based analysis methods are relatively new approaches for processing the output data of radiation detectors in nuclear medicine imaging and spectroscopy. A class of such methods requires fast algorithms for fitting pulse models to experimental data. In order to apply integral-equation based methods for processing the preamplifier output pulses, this article proposes a fast and simple method for estimating the parameters of the well-known bi-exponential pulse model by solving an integral equation. The proposed method needs samples from only three points of the recorded pulse as well as its first and second order integrals. After optimizing the sampling points, the estimation results were calculated and compared with two traditional integration-based methods. Different noise levels (signal-to-noise ratios from 10 to 3000) were simulated for testing the functionality of the proposed method, then it was applied to a set of experimental pulses. Finally, the effect of quantization noise was assessed by studying different sampling rates. Promising results by the proposed method endorse it for future real-time applications.

Time-symmetric integration in astrophysics

NASA Astrophysics Data System (ADS)

Hernandez, David M.; Bertschinger, Edmund

2018-04-01

Calculating the long-term solution of ordinary differential equations, such as those of the N-body problem, is central to understanding a wide range of dynamics in astrophysics, from galaxy formation to planetary chaos. Because generally no analytic solution exists to these equations, researchers rely on numerical methods that are prone to various errors. In an effort to mitigate these errors, powerful symplectic integrators have been employed. But symplectic integrators can be severely limited because they are not compatible with adaptive stepping and thus they have difficulty in accommodating changing time and length scales. A promising alternative is time-reversible integration, which can handle adaptive time-stepping, but the errors due to time-reversible integration in astrophysics are less understood. The goal of this work is to study analytically and numerically the errors caused by time-reversible integration, with and without adaptive stepping. We derive the modified differential equations of these integrators to perform the error analysis. As an example, we consider the trapezoidal rule, a reversible non-symplectic integrator, and show that it gives secular energy error increase for a pendulum problem and for a Hénon-Heiles orbit. We conclude that using reversible integration does not guarantee good energy conservation and that, when possible, use of symplectic integrators is favoured. We also show that time-symmetry and time-reversibility are properties that are distinct for an integrator.

Analysis of Some Properties of the Nonlinear Schrödinger Equation Used for Filamentation Modeling

NASA Astrophysics Data System (ADS)

Zemlyanov, A. A.; Bulygin, A. D.

2018-06-01

Properties of the integral of motion and evolution of the effective light beam radius are analyzed for the stationary model of the nonlinear Schrödinger equation describing the filamentation. It is demonstrated that within the limits of such model, filamentation is limited only by the dissipation mechanisms.

Steady and unsteady three-dimensional transonic flow computations by integral equation method

NASA Technical Reports Server (NTRS)

Hu, Hong

1994-01-01

This is the final technical report of the research performed under the grant: NAG1-1170, from the National Aeronautics and Space Administration. The report consists of three parts. The first part presents the work on unsteady flows around a zero-thickness wing. The second part presents the work on steady flows around non-zero thickness wings. The third part presents the massively parallel processing implementation and performance analysis of integral equation computations. At the end of the report, publications resulting from this grant are listed and attached.

A new integrable equation combining the modified KdV equation with the negative-order modified KdV equation: multiple soliton solutions and a variety of solitonic solutions

NASA Astrophysics Data System (ADS)

Wazwaz, Abdul-Majid

2018-07-01

A new third-order integrable equation is constructed via combining the recursion operator of the modified KdV equation (MKdV) and its inverse recursion operator. The developed equation will be termed the modified KdV-negative order modified KdV equation (MKdV-nMKdV). The complete integrability of this equation is confirmed by showing that it nicely possesses the Painlevé property. We obtain multiple soliton solutions for the newly developed integrable equation. Moreover, this equation enjoys a variety of solutions which include solitons, peakons, cuspons, negaton, positon, complexiton and other solutions.

Parabola solitons for the nonautonomous KP equation in fluids and plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yu, Xin, E-mail: yuxin@buaa.edu.cn; Sun, Zhi-Yuan

Under investigation in this paper is a nonautonomous Kadomtsev–Petviashvili (KP) equation in fluids and plasmas. The integrability of this equation is examined via the Painlevé analysis and its multi-soliton solutions are constructed. A constraint is proposed to ensure the existence of parabola solitons for such KP equation. Based on the constructed solutions, the solitonic propagation and interaction, including the elastic interaction, inelastic interaction and soliton resonance for parabola solitons, are discussed. The results might be useful for shallow water wave and rogue wave.

Parabola solitons for the nonautonomous KP equation in fluids and plasmas

NASA Astrophysics Data System (ADS)

Yu, Xin; Sun, Zhi-Yuan

2016-04-01

Under investigation in this paper is a nonautonomous Kadomtsev-Petviashvili (KP) equation in fluids and plasmas. The integrability of this equation is examined via the Painlevé analysis and its multi-soliton solutions are constructed. A constraint is proposed to ensure the existence of parabola solitons for such KP equation. Based on the constructed solutions, the solitonic propagation and interaction, including the elastic interaction, inelastic interaction and soliton resonance for parabola solitons, are discussed. The results might be useful for shallow water wave and rogue wave.

On integrability of the Killing equation

NASA Astrophysics Data System (ADS)

Houri, Tsuyoshi; Tomoda, Kentaro; Yasui, Yukinori

2018-04-01

Killing tensor fields have been thought of as describing the hidden symmetry of space(-time) since they are in one-to-one correspondence with polynomial first integrals of geodesic equations. Since many problems in classical mechanics can be formulated as geodesic problems in curved space and spacetime, solving the defining equation for Killing tensor fields (the Killing equation) is a powerful way to integrate equations of motion. Thus it has been desirable to formulate the integrability conditions of the Killing equation, which serve to determine the number of linearly independent solutions and also to restrict the possible forms of solutions tightly. In this paper, we show the prolongation for the Killing equation in a manner that uses Young symmetrizers. Using the prolonged equations, we provide the integrability conditions explicitly.

A computationally efficient modelling of laminar separation bubbles

NASA Technical Reports Server (NTRS)

Maughmer, Mark D.

1988-01-01

The goal of this research is to accurately predict the characteristics of the laminar separation bubble and its effects on airfoil performance. To this end, a model of the bubble is under development and will be incorporated in the analysis section of the Eppler and Somers program. As a first step in this direction, an existing bubble model was inserted into the program. It was decided to address the problem of the short bubble before attempting the prediction of the long bubble. In the second place, an integral boundary-layer method is believed more desirable than a finite difference approach. While these two methods achieve similar prediction accuracy, finite-difference methods tend to involve significantly longer computer run times than the integral methods. Finally, as the boundary-layer analysis in the Eppler and Somers program employs the momentum and kinetic energy integral equations, a short-bubble model compatible with these equations is most preferable.

Zdeněk Kopal: Numerical Analyst

NASA Astrophysics Data System (ADS)

Křížek, M.

2015-07-01

We give a brief overview of Zdeněk Kopal's life, his activities in the Czech Astronomical Society, his collaboration with Vladimír Vand, and his studies at Charles University, Cambridge, Harvard, and MIT. Then we survey Kopal's professional life. He published 26 monographs and 20 conference proceedings. We will concentrate on Kopal's extensive monograph Numerical Analysis (1955, 1961) that is widely accepted to be the first comprehensive textbook on numerical methods. It describes, for instance, methods for polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations with initial or boundary conditions, and numerical solution of integral and integro-differential equations. Special emphasis will be laid on error analysis. Kopal himself applied numerical methods to celestial mechanics, in particular to the N-body problem. He also used Fourier analysis to investigate light curves of close binaries to discover their properties. This is, in fact, a problem from mathematical analysis.

Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions.

PubMed

Baczewski, Andrew D; Miller, Nicholas C; Shanker, Balasubramaniam

2012-04-01

The analysis of fields in periodic dielectric structures arise in numerous applications of recent interest, ranging from photonic bandgap structures and plasmonically active nanostructures to metamaterials. To achieve an accurate representation of the fields in these structures using numerical methods, dense spatial discretization is required. This, in turn, affects the cost of analysis, particularly for integral-equation-based methods, for which traditional iterative methods require O(N2) operations, N being the number of spatial degrees of freedom. In this paper, we introduce a method for the rapid solution of volumetric electric field integral equations used in the analysis of doubly periodic dielectric structures. The crux of our method is the accelerated Cartesian expansion algorithm, which is used to evaluate the requisite potentials in O(N) cost. Results are provided that corroborate our claims of acceleration without compromising accuracy, as well as the application of our method to a number of compelling photonics applications.

The influence of wind-tunnel walls on discrete frequency noise

NASA Technical Reports Server (NTRS)

Mosher, M.

1984-01-01

This paper describes an analytical model that can be used to examine the effects of wind-tunnel walls on discrete frequency noise. First, a complete physical model of an acoustic source in a wind tunnel is described, and a simplified version is then developed. This simplified model retains the important physical processes involved, yet it is more amenable to analysis. Second, the simplified physical model is formulated as a mathematical problem. An inhomogeneous partial differential equation with mixed boundary conditions is set up and then transformed into an integral equation. The integral equation has been solved with a panel program on a computer. Preliminary results from a simple model problem will be shown and compared with the approximate analytic solution.

The Improvement of Efficiency in the Numerical Computation of Orbit Trajectories

NASA Technical Reports Server (NTRS)

Dyer, J.; Danchick, R.; Pierce, S.; Haney, R.

1972-01-01

An analysis, system design, programming, and evaluation of results are described for numerical computation of orbit trajectories. Evaluation of generalized methods, interaction of different formulations for satellite motion, transformation of equations of motion and integrator loads, and development of efficient integrators are also considered.

Radiative transfer in a sphere illuminated by a parallel beam - An integral equation approach. [in planetary atmosphere

NASA Technical Reports Server (NTRS)

Shia, R.-L.; Yung, Y. L.

1986-01-01

The problem of multiple scattering of nonpolarized light in a planetary body of arbitrary shape illuminated by a parallel beam is formulated using the integral equation approach. There exists a simple functional whose stationarity condition is equivalent to solving the equation of radiative transfer and whose value at the stationary point is proportional to the differential cross section. The analysis reveals a direct relation between the microscopic symmetry of the phase function for each scattering event and the macroscopic symmetry of the differential cross section for the entire planetary body, and the interconnection of these symmetry relations and the variational principle. The case of a homogeneous sphere containing isotropic scatterers is investigated in detail. It is shown that the solution can be expanded in a multipole series such that the general spherical problem is reduced to solving a set of decoupled integral equations in one dimension. Computations have been performed for a range of parameters of interest, and illustrative examples of applications to planetary problems as provided.

Synthesis, Characterization, and Sensitivity Analysis of Urea Nitrate (UN)

DTIC Science & Technology

2015-04-01

of the line is the rate of the reaction for the corresponding temperature. The equations for the zero order reaction and half- life equation follow...rate law (k is rate constant; [A] is the concentration of UN) Rate = k[A]n . (1) Eq. 2 shows the half-life (t½) equation for a zero order reaction...MGMT 1 GOVT PRINTG OFC (PDF) A MALHOTRA 2 WEAPONS DEV & (PDF) INTEGRATION DIRCTRT AMRDEC RDMR WDN J NEIDERT G DRAKE

Quantum integrability and functional equations

NASA Astrophysics Data System (ADS)

Volin, Dmytro

2010-03-01

In this thesis a general procedure to represent the integral Bethe Ansatz equations in the form of the Reimann-Hilbert problem is given. This allows us to study in simple way integrable spin chains in the thermodynamic limit. Based on the functional equations we give the procedure that allows finding the subleading orders in the solution of various integral equations solved to the leading order by the Wiener-Hopf technics. The integral equations are studied in the context of the AdS/CFT correspondence, where their solution allows verification of the integrability conjecture up to two loops of the strong coupling expansion. In the context of the two-dimensional sigma models we analyze the large-order behavior of the asymptotic perturbative expansion. Obtained experience with the functional representation of the integral equations allowed us also to solve explicitly the crossing equations that appear in the AdS/CFT spectral problem.

Explicit solution of integrated 1 - exp equation for predicting accumulation and decline of concentrations for drugs obeying nonlinear saturation kinetics.

PubMed

Keller, Frieder; Hartmann, Bertram; Czock, David

2009-12-01

To describe nonlinear, saturable pharmacokinetics, the Michaelis-Menten equation is frequently used. However, the Michaelis-Menten equation has no integrated solution for concentrations but only for the time factor. Application of the Lambert W function was proposed recently to obtain an integrated solution of the Michaelis-Menten equation. As an alternative to the Michaelis-Menten equation, a 1 - exp equation has been used to describe saturable kinetics, with the advantage that the integrated 1 - exp equation has an explicit solution for concentrations. We used the integrated 1 - exp equation to predict the accumulation kinetics and the nonlinear concentration decline for a proposed fictive drug. In agreement with the recently proposed method, we found that for the integrated 1 - exp equation no steady state is obtained if the maximum rate of change in concentrations (Vmax) within interval (Tau) is less than the difference between peak and trough concentrations (Vmax x Tau < C peak - C trough).

Analysis of scattering by a linear chain of spherical inclusions in an optical fiber

NASA Astrophysics Data System (ADS)

Chremmos, Ioannis D.; Uzunoglu, Nikolaos K.

2006-12-01

The scattering by a linear chain of spherical dielectric inclusions, embedded along the axis of an optical fiber, is analyzed using a rigorous integral equation formulation, based on the dyadic Green's function theory. The coupled electric field integral equations are solved by applying the Galerkin technique with Mie-type expansion of the field inside the spheres in terms of spherical waves. The analysis extends the previously studied case of a single spherical inhomogeneity inside a fiber to the multisphere-scattering case, by utilizing the classic translational addition theorems for spherical waves in order to analytically extract the direct-intersphere-coupling coefficients. Results for the transmitted and reflected power, on incidence of the fundamental HE11 mode, are presented for several cases.

Integral equation and discontinuous Galerkin methods for the analysis of light-matter interaction

NASA Astrophysics Data System (ADS)

Baczewski, Andrew David

Light-matter interaction is among the most enduring interests of the physical sciences. The understanding and control of this physics is of paramount importance to the design of myriad technologies ranging from stained glass, to molecular sensing and characterization techniques, to quantum computers. The development of complex engineered systems that exploit this physics is predicated at least partially upon in silico design and optimization that properly capture the light-matter coupling. In this thesis, the details of computational frameworks that enable this type of analysis, based upon both Integral Equation and Discontinuous Galerkin formulations will be explored. There will be a primary focus on the development of efficient and accurate software, with results corroborating both. The secondary focus will be on the use of these tools in the analysis of a number of exemplary systems.

Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons.

PubMed

Ratas, Irmantas; Pyragas, Kestutis

2017-10-01

We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of nonsymmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, and chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.

Symmetry breaking in two interacting populations of quadratic integrate-and-fire neurons

NASA Astrophysics Data System (ADS)

Ratas, Irmantas; Pyragas, Kestutis

2017-10-01

We analyze the dynamics of two coupled identical populations of quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The populations are heterogeneous; they include both inherently spiking and excitable neurons. The coupling within and between the populations is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rates and the mean membrane potentials in both populations. The reduced equations are exact in the infinite-size limit. The bifurcation analysis of the equations reveals a rich variety of nonsymmetric patterns, including a splay state, antiphase periodic oscillations, chimera-like states, and chaotic oscillations as well as bistabilities between various states. The validity of the reduced equations is confirmed by direct numerical simulations of the finite-size networks.

A multivariate variational objective analysis-assimilation method. Part 1: Development of the basic model

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Ochs, Harry T., III

1988-01-01

The variational method of undetermined multipliers is used to derive a multivariate model for objective analysis. The model is intended for the assimilation of 3-D fields of rawinsonde height, temperature and wind, and mean level temperature observed by satellite into a dynamically consistent data set. Relative measurement errors are taken into account. The dynamic equations are the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation. The model Euler-Lagrange equations are eleven linear and/or nonlinear partial differential and/or algebraic equations. A cyclical solution sequence is described. Other model features include a nonlinear terrain-following vertical coordinate that eliminates truncation error in the pressure gradient terms of the horizontal momentum equations and easily accommodates satellite observed mean layer temperatures in the middle and upper troposphere. A projection of the pressure gradient onto equivalent pressure surfaces removes most of the adverse impacts of the lower coordinate surface on the variational adjustment.

Direct solution for thermal stresses in a nose cap under an arbitrary axisymmetric temperature distribution

NASA Technical Reports Server (NTRS)

Davis, Randall C.

1988-01-01

The design of a nose cap for a hypersonic vehicle is an iterative process requiring a rapid, easy to use and accurate stress analysis. The objective of this paper is to develop such a stress analysis technique from a direct solution of the thermal stress equations for a spherical shell. The nose cap structure is treated as a thin spherical shell with an axisymmetric temperature distribution. The governing differential equations are solved by expressing the stress solution to the thermoelastic equations in terms of a series of derivatives of the Legendre polynomials. The process of finding the coefficients for the series solution in terms of the temperature distribution is generalized by expressing the temperature along the shell and through the thickness as a polynomial in the spherical angle coordinate. Under this generalization the orthogonality property of the Legendre polynomials leads to a sequence of integrals involving powers of the spherical shell coordinate times the derivative of the Legendre polynomials. The coefficients of the temperature polynomial appear outside of these integrals. Thus, the integrals are evaluated only once and their values tabulated for use with any arbitrary polynomial temperature distribution.

Finite time step and spatial grid effects in δf simulation of warm plasmas

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sturdevant, Benjamin J., E-mail: benjamin.j.sturdevant@gmail.com; Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309; Parker, Scott E.

2016-01-15

This paper introduces a technique for analyzing time integration methods used with the particle weight equations in δf method particle-in-cell (PIC) schemes. The analysis applies to the simulation of warm, uniform, periodic or infinite plasmas in the linear regime and considers the collective behavior similar to the analysis performed by Langdon for full-f PIC schemes [1,2]. We perform both a time integration analysis and spatial grid analysis for a kinetic ion, adiabatic electron model of ion acoustic waves. An implicit time integration scheme is studied in detail for δf simulations using our weight equation analysis and for full-f simulations usingmore » the method of Langdon. It is found that the δf method exhibits a CFL-like stability condition for low temperature ions, which is independent of the parameter characterizing the implicitness of the scheme. The accuracy of the real frequency and damping rate due to the discrete time and spatial schemes is also derived using a perturbative method. The theoretical analysis of numerical error presented here may be useful for the verification of simulations and for providing intuition for the design of new implicit time integration schemes for the δf method, as well as understanding differences between δf and full-f approaches to plasma simulation.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kanna, T.; Sakkaravarthi, K.; Kumar, C. Senthil

In this paper, we have studied the integrability nature of a system of three-coupled Gross-Pitaevskii type nonlinear evolution equations arising in the context of spinor Bose-Einstein condensates by applying the Painleve singularity structure analysis. We show that only for two sets of parametric choices, corresponding to the known integrable cases, the system passes the Painleve test.

VERTICAL INTEGRATION OF THREE-PHASE FLOW EQUATIONS FOR ANALYSIS OF LIGHT HYDROCARBON PLUME MOVEMENT

EPA Science Inventory

A mathematical model is derived for areal flow of water and light hydrocarbon in the presence of gas at atmospheric pressure. Closed-form expressions for the vertically integrated constitutive relations are derived based on a three-phase extension of the Brooks-Corey saturation-...

Equations of motion of slung load systems with results for dual lift

NASA Technical Reports Server (NTRS)

Cicolani, Luigi S.; Kanning, Gerd

1990-01-01

General simulation equations are derived for the rigid body motion of slung load systems. These systems are viewed as consisting of several rigid bodies connected by straight-line cables or links. The suspension can be assumed to be elastic or inelastic, both cases being of interest in simulation and control studies. Equations for the general system are obtained via D'Alembert's principle and the introduction of generalized velocity coordinates. Three forms are obtained. Two of these generalize previous case-specific results for single helicopter systems with elastic or inelastic suspensions. The third is a new formulation for inelastic suspensions. It is derived from the elastic suspension equations by choosing the generalized coordinates so as to separate motion due to cable stretching from motion with invariant cable lengths. The result is computationally more efficient than the conventional formulation, and is readily integrated with the elastic suspension formulation and readily applied to the complex dual lift and multilift systems. Equations are derived for dual lift systems. Three proposed suspension arrangements can be integrated in a single equation set. The equations are given in terms of the natural vectors and matrices of three-dimensional rigid body mechanics and are tractable for both analysis and programming.

On the integration of a class of nonlinear systems of ordinary differential equations

NASA Astrophysics Data System (ADS)

Talyshev, Aleksandr A.

2017-11-01

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

Convection equation modeling: A non-iterative direct matrix solution algorithm for use with SINDA

NASA Technical Reports Server (NTRS)

Schrage, Dean S.

1993-01-01

The determination of the boundary conditions for a component-level analysis, applying discrete finite element and finite difference modeling techniques often requires an analysis of complex coupled phenomenon that cannot be described algebraically. For example, an analysis of the temperature field of a coldplate surface with an integral fluid loop requires a solution to the parabolic heat equation and also requires the boundary conditions that describe the local fluid temperature. However, the local fluid temperature is described by a convection equation that can only be solved with the knowledge of the locally-coupled coldplate temperatures. Generally speaking, it is not computationally efficient, and sometimes, not even possible to perform a direct, coupled phenomenon analysis of the component-level and boundary condition models within a single analysis code. An alternative is to perform a disjoint analysis, but transmit the necessary information between models during the simulation to provide an indirect coupling. For this approach to be effective, the component-level model retains full detail while the boundary condition model is simplified to provide a fast, first-order prediction of the phenomenon in question. Specifically for the present study, the coldplate structure is analyzed with a discrete, numerical model (SINDA) while the fluid loop convection equation is analyzed with a discrete, analytical model (direct matrix solution). This indirect coupling allows a satisfactory prediction of the boundary condition, while not subjugating the overall computational efficiency of the component-level analysis. In the present study a discussion of the complete analysis of the derivation and direct matrix solution algorithm of the convection equation is presented. Discretization is analyzed and discussed to extend of solution accuracy, stability and computation speed. Case studies considering a pulsed and harmonic inlet disturbance to the fluid loop are analyzed to assist in the discussion of numerical dissipation and accuracy. In addition, the issues of code melding or integration with standard class solvers such as SINDA are discussed to advise the user of the potential problems to be encountered.

Nonlinear integral equations for the sausage model

NASA Astrophysics Data System (ADS)

Ahn, Changrim; Balog, Janos; Ravanini, Francesco

2017-08-01

The sausage model, first proposed by Fateev, Onofri, and Zamolodchikov, is a deformation of the O(3) sigma model preserving integrability. The target space is deformed from the sphere to ‘sausage’ shape by a deformation parameter ν. This model is defined by a factorizable S-matrix which is obtained by deforming that of the O(3) sigma model by a parameter λ. Clues for the deformed sigma model are provided by various UV and IR information through the thermodynamic Bethe ansatz (TBA) analysis based on the S-matrix. Application of TBA to the sausage model is, however, limited to the case of 1/λ integer where the coupled integral equations can be truncated to a finite number. In this paper, we propose a finite set of nonlinear integral equations (NLIEs), which are applicable to generic value of λ. Our derivation is based on T-Q relations extracted from the truncated TBA equations. For a consistency check, we compute next-leading order corrections of the vacuum energy and extract the S-matrix information in the IR limit. We also solved the NLIE both analytically and numerically in the UV limit to get the effective central charge and compared with that of the zero-mode dynamics to obtain exact relation between ν and λ. Dedicated to the memory of Petr Petrovich Kulish.

Mathematical Methods for Optical Physics and Engineering

NASA Astrophysics Data System (ADS)

Gbur, Gregory J.

2011-01-01

1. Vector algebra; 2. Vector calculus; 3. Vector calculus in curvilinear coordinate systems; 4. Matrices and linear algebra; 5. Advanced matrix techniques and tensors; 6. Distributions; 7. Infinite series; 8. Fourier series; 9. Complex analysis; 10. Advanced complex analysis; 11. Fourier transforms; 12. Other integral transforms; 13. Discrete transforms; 14. Ordinary differential equations; 15. Partial differential equations; 16. Bessel functions; 17. Legendre functions and spherical harmonics; 18. Orthogonal functions; 19. Green's functions; 20. The calculus of variations; 21. Asymptotic techniques; Appendices; References; Index.

Global solution branches for a nonlocal Allen-Cahn equation

NASA Astrophysics Data System (ADS)

Kuto, Kousuke; Mori, Tatsuki; Tsujikawa, Tohru; Yotsutani, Shoji

2018-05-01

We consider the Neumann problem of a 1D stationary Allen-Cahn equation with nonlocal term. Our previous paper [4] obtained a local branch of asymmetric solutions which bifurcates from a point on the branch of odd-symmetric solutions. This paper derives the global behavior of the branch of asymmetric solutions, and moreover, determines the set of all solutions to the nonlocal Allen-Cahn equation. Our proof is based on a level set analysis for an integral map associated with the nonlocal term.

Dynamic analysis of a system of hinge-connected rigid bodies with nonrigid appendages. [equations of motion

NASA Technical Reports Server (NTRS)

Likins, P. W.

1974-01-01

Equations of motion are derived for use in simulating a spacecraft or other complex electromechanical system amenable to idealization as a set of hinge-connected rigid bodies of tree topology, with rigid axisymmetric rotors and nonrigid appendages attached to each rigid body in the set. In conjunction with a previously published report on finite-element appendage vibration equations, this report provides a complete minimum-dimension formulation suitable for generic programming for digital computer numerical integration.

Time domain convergence properties of Lyapunov stable penalty methods

NASA Technical Reports Server (NTRS)

Kurdila, A. J.; Sunkel, John

1991-01-01

Linear hyperbolic partial differential equations are analyzed using standard techniques to show that a sequence of solutions generated by the Liapunov stable penalty equations approaches the solution of the differential-algebraic equations governing the dynamics of multibody problems arising in linear vibrations. The analysis does not require that the system be conservative and does not impose any specific integration scheme. Variational statements are derived which bound the error in approximation by the norm of the constraint violation obtained in the approximate solutions.

Cavity equations for a positive- or negative-refraction-index material with electric and magnetic nonlinearities.

PubMed

Mártin, Daniel A; Hoyuelos, Miguel

2009-11-01

We study evolution equations for electric and magnetic field amplitudes in a ring cavity with plane mirrors. The cavity is filled with a positive or negative-refraction-index material with third-order effective electric and magnetic nonlinearities. Two coupled nonlinear equations for the electric and magnetic amplitudes are obtained. We prove that the description can be reduced to one Lugiato-Lefever equation with generalized coefficients. A stability analysis of the homogeneous solution, complemented with numerical integration, shows that any combination of the parameters should correspond to one of three characteristic behaviors.

The Dirac equation and the normalization of its solutions in a closed Friedmann- Robertson-Walker universe

NASA Astrophysics Data System (ADS)

Finster, Felix; Reintjes, Moritz

2009-05-01

We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a spacetime normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form. First author supported in part by the Deutsche Forschungsgemeinschaft.

Full nonlinear treatment of the global thermospheric wind system. Part 1: Mathematical method and analysis of forces

NASA Technical Reports Server (NTRS)

Blum, P. W.; Harris, I.

1973-01-01

The equations of horizontal motion of the neutral atmosphere between 120 and 500 km are integrated with the inclusion of all the nonlinear terms of the convective derivative and the viscous forces due to vertical and horizontal velocity gradients. Empirical models of the distribution of neutral and charged particles are assumed to be known. The model of velocities developed is a steady state model. In part 1 the mathematical method used in the integration of the Navier-Stokes equations is described and the various forces are analysed.

Probability density function evolution of power systems subject to stochastic variation of renewable energy

NASA Astrophysics Data System (ADS)

Wei, J. Q.; Cong, Y. C.; Xiao, M. Q.

2018-05-01

As renewable energies are increasingly integrated into power systems, there is increasing interest in stochastic analysis of power systems.Better techniques should be developed to account for the uncertainty caused by penetration of renewables and consequently analyse its impacts on stochastic stability of power systems. In this paper, the Stochastic Differential Equations (SDEs) are used to represent the evolutionary behaviour of the power systems. The stationary Probability Density Function (PDF) solution to SDEs modelling power systems excited by Gaussian white noise is analysed. Subjected to such random excitation, the Joint Probability Density Function (JPDF) solution to the phase angle and angular velocity is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation. To solve this equation, the numerical method is adopted. Special measure is taken such that the generalized FPK equation is satisfied in the average sense of integration with the assumed PDF. Both weak and strong intensities of the stochastic excitations are considered in a single machine infinite bus power system. The numerical analysis has the same result as the one given by the Monte Carlo simulation. Potential studies on stochastic behaviour of multi-machine power systems with random excitations are discussed at the end.

Steady-state and transient analysis of a squeeze film damper bearing for rotor stability

NASA Technical Reports Server (NTRS)

Barrett, L. E.; Gunter, E. J.

1975-01-01

A study of the steady-state and transient response of the squeeze film damper bearing is presented. Both the steady-state and transient equations for the hydrodynamic bearing forces are derived. The bearing equivalent stiffness and damping coefficients are determined by steady-state equations. These coefficients are used to find the bearing configuration which will provide the optimum support characteristics based on a stability analysis of the rotor-bearing system. The transient analysis of rotor-bearing systems is performed by coupling the bearing and journal equations and integrating forward in time. The effects of unbalance, cavitation, and retainer springs are included in the analysis. Methods of determining the stability of a rotor-bearing system under the influence of aerodynamic forces and internal shaft friction are discussed with emphasis on solving the system characteristic frequency equation and on producing stability maps. It is shown that for optimum stability and low force transmissability the squeeze bearing should operate at an eccentricity ratio epsilon 0.4.

Dynamics of a differential-difference integrable (2+1)-dimensional system.

PubMed

Yu, Guo-Fu; Xu, Zong-Wei

2015-06-01

A Kadomtsev-Petviashvili- (KP-) type equation appears in fluid mechanics, plasma physics, and gas dynamics. In this paper, we propose an integrable semidiscrete analog of a coupled (2+1)-dimensional system which is related to the KP equation and the Zakharov equation. N-soliton solutions of the discrete equation are presented. Some interesting examples of soliton resonance related to the two-soliton and three-soliton solutions are investigated. Numerical computations using the integrable semidiscrete equation are performed. It is shown that the integrable semidiscrete equation gives very accurate numerical results in the cases of one-soliton evolution and soliton interactions.

An Obstruction to the Integrability of a Class of Non-linear Wave Equations by 1-Stable Cartan Characteristics

NASA Astrophysics Data System (ADS)

Fackerell, E. D.; Hartley, D.; Tucker, R. W.

We examine in detail the Cauchy problem for a class of non-linear hyperbolic equations in two independent variables. This class is motivated by the analysis of the dynamics of a line of non-linearly coupled particles by Fermi, Pasta, and Ulam and extends the recent investigation of this problem by Gardner and Kamran. We find conditions for the existence of a 1-stable Cartan characteristic of a Pfaffian exterior differential system whose integral curves provide a solution to the Cauchy problem. The same obstruction to involution is exposed in Darboux's method of integration and the two approaches are compared. A class of particular solutions to the obstruction is constructed.

Boundary integral equation analysis for suspension of spheres in Stokes flow

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Veerapaneni, Shravan

2018-06-01

We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.

A theoretical analysis of fluid flow and energy transport in hydrothermal systems

USGS Publications Warehouse

Faust, Charles R.; Mercer, James W.

1977-01-01

A mathematical derivation for fluid flow and energy transport in hydrothermal systems is presented. Specifically, the mathematical model describes the three-dimensional flow of both single- and two-phase, single-component water and the transport of heat in porous media. The derivation begins with the point balance equations for mass, momentum, and energy. These equations are then averaged over a finite volume to obtain the macroscopic balance equations for a porous medium. The macroscopic equations are combined by appropriate constitutive relationships to form two similified partial differential equations posed in terms of fluid pressure and enthalpy. A two-dimensional formulation of the simplified equations is also derived by partial integration in the vertical dimension. (Woodard-USGS)

Determination of elementary first integrals of a generalized Raychaudhuri equation by the Darboux integrability method

NASA Astrophysics Data System (ADS)

Choudhury, A. Ghose; Guha, Partha; Khanra, Barun

2009-10-01

The Darboux integrability method is particularly useful to determine first integrals of nonplanar autonomous systems of ordinary differential equations, whose associated vector fields are polynomials. In particular, we obtain first integrals for a variant of the generalized Raychaudhuri equation, which has appeared in string inspired modern cosmology.

Computational singular perturbation analysis of stochastic chemical systems with stiffness

NASA Astrophysics Data System (ADS)

Wang, Lijin; Han, Xiaoying; Cao, Yanzhao; Najm, Habib N.

2017-04-01

Computational singular perturbation (CSP) is a useful method for analysis, reduction, and time integration of stiff ordinary differential equation systems. It has found dominant utility, in particular, in chemical reaction systems with a large range of time scales at continuum and deterministic level. On the other hand, CSP is not directly applicable to chemical reaction systems at micro or meso-scale, where stochasticity plays an non-negligible role and thus has to be taken into account. In this work we develop a novel stochastic computational singular perturbation (SCSP) analysis and time integration framework, and associated algorithm, that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems, but also analyze the dynamics of the reduced stochastic reaction systems. The algorithm is illustrated by an application to a benchmark stochastic differential equation model, and numerical experiments are carried out to demonstrate the effectiveness of the construction.

Atomic Physics Effects on Convergent, Child-Langmuir Ion Flow between Nearly Transparent Electrodes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Santarius, John F.; Emmert, Gilbert A.

Research during this project at the University of Wisconsin Fusion Technology Institute (UW FTI) on ion and neutral flow through an arbitrary, monotonic potential difference created by nearly transparent electrodes accomplished the following: (1) developed and implemented an integral equation approach for atomic physics effects in helium plasmas; (2) extended the analysis to coupled integral equations that treat atomic and molecular deuterium ions and neutrals; (3) implemented the key deuterium and helium atomic and molecular cross sections; (4) added negative ion production and related cross sections; and (5) benchmarked the code against experimental results. The analysis and codes treat themore » species D0, D20, D+, D2+, D3+, D and, separately at present, He0 and He+. Extensions enhanced the analysis and related computer codes to include He++ ions plus planar and cylindrical geometries.« less

Computation of Sound Propagation by Boundary Element Method

NASA Technical Reports Server (NTRS)

Guo, Yueping

2005-01-01

This report documents the development of a Boundary Element Method (BEM) code for the computation of sound propagation in uniform mean flows. The basic formulation and implementation follow the standard BEM methodology; the convective wave equation and the boundary conditions on the surfaces of the bodies in the flow are formulated into an integral equation and the method of collocation is used to discretize this equation into a matrix equation to be solved numerically. New features discussed here include the formulation of the additional terms due to the effects of the mean flow and the treatment of the numerical singularities in the implementation by the method of collocation. The effects of mean flows introduce terms in the integral equation that contain the gradients of the unknown, which is undesirable if the gradients are treated as additional unknowns, greatly increasing the sizes of the matrix equation, or if numerical differentiation is used to approximate the gradients, introducing numerical error in the computation. It is shown that these terms can be reformulated in terms of the unknown itself, making the integral equation very similar to the case without mean flows and simple for numerical implementation. To avoid asymptotic analysis in the treatment of numerical singularities in the method of collocation, as is conventionally done, we perform the surface integrations in the integral equation by using sub-triangles so that the field point never coincide with the evaluation points on the surfaces. This simplifies the formulation and greatly facilitates the implementation. To validate the method and the code, three canonic problems are studied. They are respectively the sound scattering by a sphere, the sound reflection by a plate in uniform mean flows and the sound propagation over a hump of irregular shape in uniform flows. The first two have analytical solutions and the third is solved by the method of Computational Aeroacoustics (CAA), all of which are used to compare the BEM solutions. The comparisons show very good agreements and validate the accuracy of the BEM approach implemented here.

Autonomous Aerobraking: Thermal Analysis and Response Surface Development

NASA Technical Reports Server (NTRS)

Dec, John A.; Thornblom, Mark N.

2011-01-01

A high-fidelity thermal model of the Mars Reconnaissance Orbiter was developed for use in an autonomous aerobraking simulation study. Response surface equations were derived from the high-fidelity thermal model and integrated into the autonomous aerobraking simulation software. The high-fidelity thermal model was developed using the Thermal Desktop software and used in all phases of the analysis. The use of Thermal Desktop exclusively, represented a change from previously developed aerobraking thermal analysis methodologies. Comparisons were made between the Thermal Desktop solutions and those developed for the previous aerobraking thermal analyses performed on the Mars Reconnaissance Orbiter during aerobraking operations. A variable sensitivity screening study was performed to reduce the number of variables carried in the response surface equations. Thermal analysis and response surface equation development were performed for autonomous aerobraking missions at Mars and Venus.

Two-dimensional integrating matrices on rectangular grids. [solving differential equations associated with rotating structures

NASA Technical Reports Server (NTRS)

Lakin, W. D.

1981-01-01

The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.

SMAUMAT_ITI

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jannetti, C.; Becker, R.

The software is an ABAQUS/Standard UMAT (user defined material behavior subroutine) that implements the constitutive model for shape-memory alloy materials developed by Jannetti et. al. (2003a) using a fully implicit time integration scheme to integrate the constitutive equations. The UMAT is used in conjunction with ABAQUS/Standard to perform a finite-element analysis of SMA materials.

Utilization of integrated Michaelis-Menten equations for enzyme inhibition diagnosis and determination of kinetic constants using Solver supplement of Microsoft Office Excel.

PubMed

Bezerra, Rui M F; Fraga, Irene; Dias, Albino A

2013-01-01

Enzyme kinetic parameters are usually determined from initial rates nevertheless, laboratory instruments only measure substrate or product concentration versus reaction time (progress curves). To overcome this problem we present a methodology which uses integrated models based on Michaelis-Menten equation. The most severe practical limitation of progress curve analysis occurs when the enzyme shows a loss of activity under the chosen assay conditions. To avoid this problem it is possible to work with the same experimental points utilized for initial rates determination. This methodology is illustrated by the use of integrated kinetic equations with the well-known reaction catalyzed by alkaline phosphatase enzyme. In this work nonlinear regression was performed with the Solver supplement (Microsoft Office Excel). It is easy to work with and track graphically the convergence of SSE (sum of square errors). The diagnosis of enzyme inhibition was performed according to Akaike information criterion. Copyright © 2012 Elsevier Ireland Ltd. All rights reserved.

Three-dimensional finite element analysis for high velocity impact. [of projectiles from space debris

NASA Technical Reports Server (NTRS)

Chan, S. T. K.; Lee, C. H.; Brashears, M. R.

1975-01-01

A finite element algorithm for solving unsteady, three-dimensional high velocity impact problems is presented. A computer program was developed based on the Eulerian hydroelasto-viscoplastic formulation and the utilization of the theorem of weak solutions. The equations solved consist of conservation of mass, momentum, and energy, equation of state, and appropriate constitutive equations. The solution technique is a time-dependent finite element analysis utilizing three-dimensional isoparametric elements, in conjunction with a generalized two-step time integration scheme. The developed code was demonstrated by solving one-dimensional as well as three-dimensional impact problems for both the inviscid hydrodynamic model and the hydroelasto-viscoplastic model.

Accurate D-bar Reconstructions of Conductivity Images Based on a Method of Moment with Sinc Basis.

PubMed

Abbasi, Mahdi

2014-01-01

Planar D-bar integral equation is one of the inverse scattering solution methods for complex problems including inverse conductivity considered in applications such as Electrical impedance tomography (EIT). Recently two different methodologies are considered for the numerical solution of D-bar integrals equation, namely product integrals and multigrid. The first one involves high computational burden and the other one suffers from low convergence rate (CR). In this paper, a novel high speed moment method based using the sinc basis is introduced to solve the two-dimensional D-bar integral equation. In this method, all functions within D-bar integral equation are first expanded using the sinc basis functions. Then, the orthogonal properties of their products dissolve the integral operator of the D-bar equation and results a discrete convolution equation. That is, the new moment method leads to the equation solution without direct computation of the D-bar integral. The resulted discrete convolution equation maybe adapted to a suitable structure to be solved using fast Fourier transform. This allows us to reduce the order of computational complexity to as low as O (N (2)log N). Simulation results on solving D-bar equations arising in EIT problem show that the proposed method is accurate with an ultra-linear CR.

Geometry, Heat Equation and Path Integrals on the Poincaré Upper Half-Plane

NASA Astrophysics Data System (ADS)

Kubo, R.

1988-01-01

Geometry, heat equation and Feynman's path integrals are studied on the Poincaré upper half-plane. The fundamental solution to the heat equation partial f/partial t = Delta_{H} f is expressed in terms of a path integral defined on the upper half-plane. It is shown that Kac's statement that Feynman's path integral satisfies the Schrödinger equation is also valid for our case.

Nodal Green’s Function Method Singular Source Term and Burnable Poison Treatment in Hexagonal Geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

A.A. Bingham; R.M. Ferrer; A.M. ougouag

2009-09-01

An accurate and computationally efficient two or three-dimensional neutron diffusion model will be necessary for the development, safety parameters computation, and fuel cycle analysis of a prismatic Very High Temperature Reactor (VHTR) design under Next Generation Nuclear Plant Project (NGNP). For this purpose, an analytical nodal Green’s function solution for the transverse integrated neutron diffusion equation is developed in two and three-dimensional hexagonal geometry. This scheme is incorporated into HEXPEDITE, a code first developed by Fitzpatrick and Ougouag. HEXPEDITE neglects non-physical discontinuity terms that arise in the transverse leakage due to the transverse integration procedure application to hexagonal geometry andmore » cannot account for the effects of burnable poisons across nodal boundaries. The test code being developed for this document accounts for these terms by maintaining an inventory of neutrons by using the nodal balance equation as a constraint of the neutron flux equation. The method developed in this report is intended to restore neutron conservation and increase the accuracy of the code by adding these terms to the transverse integrated flux solution and applying the nodal Green’s function solution to the resulting equation to derive a semi-analytical solution.« less

Viewpoint on ISA TR84.0.02--simplified methods and fault tree analysis.

PubMed

Summers, A E

2000-01-01

ANSI/ISA-S84.01-1996 and IEC 61508 require the establishment of a safety integrity level for any safety instrumented system or safety related system used to mitigate risk. Each stage of design, operation, maintenance, and testing is judged against this safety integrity level. Quantitative techniques can be used to verify whether the safety integrity level is met. ISA-dTR84.0.02 is a technical report under development by ISA, which discusses how to apply quantitative analysis techniques to safety instrumented systems. This paper discusses two of those techniques: (1) Simplified equations and (2) Fault tree analysis.

Squared eigenfunctions for the Sasa-Satsuma equation

NASA Astrophysics Data System (ADS)

Yang, Jianke; Kaup, D. J.

2009-02-01

Squared eigenfunctions are quadratic combinations of Jost functions and adjoint Jost functions which satisfy the linearized equation of an integrable equation. They are needed for various studies related to integrable equations, such as the development of its soliton perturbation theory. In this article, squared eigenfunctions are derived for the Sasa-Satsuma equation whose spectral operator is a 3×3 system, while its linearized operator is a 2×2 system. It is shown that these squared eigenfunctions are sums of two terms, where each term is a product of a Jost function and an adjoint Jost function. The procedure of this derivation consists of two steps: First is to calculate the variations of the potentials via variations of the scattering data by the Riemann-Hilbert method. The second one is to calculate the variations of the scattering data via the variations of the potentials through elementary calculations. While this procedure has been used before on other integrable equations, it is shown here, for the first time, that for a general integrable equation, the functions appearing in these variation relations are precisely the squared eigenfunctions and adjoint squared eigenfunctions satisfying, respectively, the linearized equation and the adjoint linearized equation of the integrable system. This proof clarifies this procedure and provides a unified explanation for previous results of squared eigenfunctions on individual integrable equations. This procedure uses primarily the spectral operator of the Lax pair. Thus two equations in the same integrable hierarchy will share the same squared eigenfunctions (except for a time-dependent factor). In the Appendix, the squared eigenfunctions are presented for the Manakov equations whose spectral operator is closely related to that of the Sasa-Satsuma equation.

Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions

DOE PAGES

Baczewski, Andrew David; Miller, Nicholas C.; Shanker, Balasubramaniam

2012-03-22

Here, the analysis of fields in periodic dielectric structures arise in numerous applications of recent interest, ranging from photonic bandgap structures and plasmonically active nanostructures to metamaterials. To achieve an accurate representation of the fields in these structures using numerical methods, dense spatial discretization is required. This, in turn, affects the cost of analysis, particularly for integral-equation-based methods, for which traditional iterative methods require Ο(Ν 2) operations, Ν being the number of spatial degrees of freedom. In this paper, we introduce a method for the rapid solution of volumetric electric field integral equations used in the analysis of doubly periodicmore » dielectric structures. The crux of our method is the accelerated Cartesian expansion algorithm, which is used to evaluate the requisite potentials in Ο(Ν) cost. Results are provided that corroborate our claims of acceleration without compromising accuracy, as well as the application of our method to a number of compelling photonics applications.« less

Mathematical Methods for Physics and Engineering Third Edition Paperback Set

NASA Astrophysics Data System (ADS)

Riley, Ken F.; Hobson, Mike P.; Bence, Stephen J.

2006-06-01

Prefaces; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics; Index.

Prediction of tautomer ratios by embedded-cluster integral equation theory

NASA Astrophysics Data System (ADS)

Kast, Stefan M.; Heil, Jochen; Güssregen, Stefan; Schmidt, K. Friedemann

2010-04-01

The "embedded cluster reference interaction site model" (EC-RISM) approach combines statistical-mechanical integral equation theory and quantum-chemical calculations for predicting thermodynamic data for chemical reactions in solution. The electronic structure of the solute is determined self-consistently with the structure of the solvent that is described by 3D RISM integral equation theory. The continuous solvent-site distribution is mapped onto a set of discrete background charges ("embedded cluster") that represent an additional contribution to the molecular Hamiltonian. The EC-RISM analysis of the SAMPL2 challenge set of tautomers proceeds in three stages. Firstly, the group of compounds for which quantitative experimental free energy data was provided was taken to determine appropriate levels of quantum-chemical theory for geometry optimization and free energy prediction. Secondly, the resulting workflow was applied to the full set, allowing for chemical interpretations of the results. Thirdly, disclosure of experimental data for parts of the compounds facilitated a detailed analysis of methodical issues and suggestions for future improvements of the model. Without specifically adjusting parameters, the EC-RISM model yields the smallest value of the root mean square error for the first set (0.6 kcal mol-1) as well as for the full set of quantitative reaction data (2.0 kcal mol-1) among the SAMPL2 participants.

Numerical solution of stiff systems of ordinary differential equations with applications to electronic circuits

NASA Technical Reports Server (NTRS)

Rosenbaum, J. S.

1971-01-01

Systems of ordinary differential equations in which the magnitudes of the eigenvalues (or time constants) vary greatly are commonly called stiff. Such systems of equations arise in nuclear reactor kinetics, the flow of chemically reacting gas, dynamics, control theory, circuit analysis and other fields. The research reported develops an A-stable numerical integration technique for solving stiff systems of ordinary differential equations. The method, which is called the generalized trapezoidal rule, is a modification of the trapezoidal rule. However, the method is computationally more efficient than the trapezoidal rule when the solution of the almost-discontinuous segments is being calculated.

Coprimeness-preserving non-integrable extension to the two-dimensional discrete Toda lattice equation

NASA Astrophysics Data System (ADS)

Kamiya, Ryo; Kanki, Masataka; Mase, Takafumi; Tokihiro, Tetsuji

2017-01-01

We introduce a so-called coprimeness-preserving non-integrable extension to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equations defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence.

The dual boundary element formulation for elastoplastic fracture mechanics

NASA Astrophysics Data System (ADS)

Leitao, V.; Aliabadi, M. H.; Rooke, D. P.

1993-08-01

The extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied to one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks, and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elastoplastic behavior is modeled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral, and/or triangular cells. A center-cracked plate and a slant edge-cracked plate subjected to tensile load are analyzed and the results are compared with others available in the literature. J-type integrals are calculated.

Analysis of Large Quasistatic Deformations of Inelastic Solids by a New Stress Based Finite Element Method. Ph.D. Thesis Final Report

NASA Technical Reports Server (NTRS)

Reed, Kenneth W.

1992-01-01

A new hybrid stress finite element algorithm suitable for analyses of large quasistatic deformation of inelastic solids is presented. Principal variables in the formulation are the nominal stress rate and spin. The finite element equations which result are discrete versions of the equations of compatibility and angular momentum balance. Consistent reformulation of the constitutive equation and accurate and stable time integration of the stress are discussed at length. Examples which bring out the feasibility and performance of the algorithm conclude the work.

Computer program for the load and trajectory analysis of two DOF bodies connected by an elastic tether: Users manual

NASA Technical Reports Server (NTRS)

Doyle, G. R., Jr.; Burbick, J. W.

1973-01-01

The derivation of the differential equations of motion of a 3 Degrees of Freedom body joined to a 3 Degrees of Freedom body by an elastic tether. The tether is represented by a spring and dashpot in parallel. A computer program which integrates the equations of motion is also described. Although the derivation of the equations of motions are for a general system, the computer program is written for defining loads in large boosters recovered by parachutes.

Time-temperature effect in adhesively bonded joints

NASA Technical Reports Server (NTRS)

Delale, F.; Erdogan, F.

1981-01-01

The viscoelastic analysis of an adhesively bonded lap joint was reconsidered. The adherends are approximated by essentially Reissner plates and the adhesive is linearly viscoelastic. The hereditary integrals are used to model the adhesive. A linear integral differential equations system for the shear and the tensile stress in the adhesive is applied. The equations have constant coefficients and are solved by using Laplace transforms. It is shown that if the temperature variation in time can be approximated by a piecewise constant function, then the method of Laplace transforms can be used to solve the problem. A numerical example is given for a single lap joint under various loading conditions.

Torsion analysis of cracked circular bars actuated by a piezoelectric coating

NASA Astrophysics Data System (ADS)

Hassani, A. R.; Faal, R. T.

2016-12-01

This study presents a formulation for a bar with circular cross-section, coated by a piezoelectric layer and subjected to Saint-Venant torsion loading. The bar is weakened by a Volterra-type screw dislocation. First, with aid of the finite Fourier transform, the stress fields in the circular bar and the piezoelectric layer are obtained. The problem is then reduced to a set of singular integral equations with a Cauchy-type singularity. Unknown dislocation density is achieved by numerical solution of these integral equations. Numerical results are discussed, to reveal the effect of the piezoelectric layer on the reduction of the mechanical stress intensity factor in the bar.

Scattering by a groove in an impedance plane

NASA Technical Reports Server (NTRS)

Bindiganavale, Sunil; Volakis, John L.

1993-01-01

An analysis of two-dimensional scattering from a narrow groove in an impedance plane is presented. The groove is represented by a impedance surface and the problem reduces to that of scattering from an impedance strip in an otherwise uniform impedance plane. On the basis of this model, appropriate integral equations are constructed using a form of the impedance plane Green's functions involving rapidly convergent integrals. The integral equations are solved by introducing a single basis representation of the equivalent current on the narrow impedance insert. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated. The resulting solution is validated by comparison with results from the standard boundary integral method (BIM) and a high frequency solution. It is found that the presented solution for narrow impedance inserts can be used in conjunction with the high frequency solution for the characterization of impedance inserts of any given width.

Nonlinear Solver Approaches for the Diffusive Wave Approximation to the Shallow Water Equations

NASA Astrophysics Data System (ADS)

Collier, N.; Knepley, M.

2015-12-01

The diffusive wave approximation to the shallow water equations (DSW) is a doubly-degenerate, nonlinear, parabolic partial differential equation used to model overland flows. Despite its challenges, the DSW equation has been extensively used to model the overland flow component of various integrated surface/subsurface models. The equation's complications become increasingly problematic when ponding occurs, a feature which becomes pervasive when solving on large domains with realistic terrain. In this talk I discuss the various forms and regularizations of the DSW equation and highlight their effect on the solvability of the nonlinear system. In addition to this analysis, I present results of a numerical study which tests the applicability of a class of composable nonlinear algebraic solvers recently added to the Portable, Extensible, Toolkit for Scientific Computation (PETSc).

Calculation of transonic flows using an extended integral equation method

NASA Technical Reports Server (NTRS)

Nixon, D.

1976-01-01

An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.

Fredholm-Volterra Integral Equation with a Generalized Singular Kernel and its Numerical Solutions

NASA Astrophysics Data System (ADS)

El-Kalla, I. L.; Al-Bugami, A. M.

2010-11-01

In this paper, the existence and uniqueness of solution of the Fredholm-Volterra integral equation (F-VIE), with a generalized singular kernel, are discussed and proved in the spaceL2(Ω)×C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced to a linear algebraic system (LAS) of equations by using two different methods. These methods are: Toeplitz matrix method and Product Nyström method. A numerical examples are considered when the generalized kernel takes the following forms: Carleman function, logarithmic form, Cauchy kernel, and Hilbert kernel.

Inverse scattering transform for the KPI equation on the background of a one-line soliton*Inverse scattering transform for the KPI equation on the background of a one-line soliton

NASA Astrophysics Data System (ADS)

Fokas, A. S.; Pogrebkov, A. K.

2003-03-01

We study the initial value problem of the Kadomtsev-Petviashvili I (KPI) equation with initial data u(x1,x2,0) = u1(x1)+u2(x1,x2), where u1(x1) is the one-soliton solution of the Korteweg-de Vries equation evaluated at zero time and u2(x1,x2) decays sufficiently rapidly on the (x1,x2)-plane. This involves the analysis of the nonstationary Schrödinger equation (with time replaced by x2) with potential u(x1,x2,0). We introduce an appropriate sectionally analytic eigenfunction in the complex k-plane where k is the spectral parameter. This eigenfunction has the novelty that in addition to the usual jump across the real k-axis, it also has a jump across a segment of the imaginary k-axis. We show that this eigenfunction can be reconstructed through a linear integral equation uniquely defined in terms of appropriate scattering data. In turn, these scattering data are uniquely constructed in terms of u1(x1) and u2(x1,x2). This result implies that the solution of the KPI equation can be obtained through the above linear integral equation where the scattering data have a simple t-dependence.

Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2006-03-01

Preface; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics.

Trajectory-Based Loads for the Ares I-X Test Flight Vehicle

NASA Technical Reports Server (NTRS)

Vause, Roland F.; Starr, Brett R.

2011-01-01

In trajectory-based loads, the structural engineer treats each point on the trajectory as a load case. Distributed aero, inertial, and propulsion forces are developed for the structural model which are equivalent to the integrated values of the trajectory model. Free-body diagrams are then used to solve for the internal forces, or loads, that keep the applied aero, inertial, and propulsion forces in dynamic equilibrium. There are several advantages to using trajectory-based loads. First, consistency is maintained between the integrated equilibrium equations of the trajectory analysis and the distributed equilibrium equations of the structural analysis. Second, the structural loads equations are tied to the uncertainty model for the trajectory systems analysis model. Atmosphere, aero, propulsion, mass property, and controls uncertainty models all feed into the dispersions that are generated for the trajectory systems analysis model. Changes in any of these input models will affect structural loads response. The trajectory systems model manages these inputs as well as the output from the structural model over thousands of dispersed cases. Large structural models with hundreds of thousands of degrees of freedom would execute too slowly to be an efficient part of several thousand system analyses. Trajectory-based loads provide a means for the structures discipline to be included in the integrated systems analysis. Successful applications of trajectory-based loads methods for the Ares I-X vehicle are covered in this paper. Preliminary design loads were based on 2000 trajectories using Monte Carlo dispersions. Range safety loads were tied to 8423 malfunction turn trajectories. In addition, active control system loads were based on 2000 preflight trajectories using Monte Carlo dispersions.

Boundary layers at the interface of two different shear flows

NASA Astrophysics Data System (ADS)

Weidman, Patrick D.; Wang, C. Y.

2018-05-01

We present solutions for the boundary layer between two uniform shear flows flowing in the same direction. In the upper layer, the flow has shear strength a, fluid density ρ1, and kinematic viscosity ν1, while the lower layer has shear strength b, fluid density ρ2, and kinematic viscosity ν2. Similarity transformations reduce the boundary-layer equations to a pair of ordinary differential equations governed by three dimensionless parameters: the shear strength ratio γ = b/a, the density ratio ρ = ρ2/ρ1, and the viscosity ratio ν = ν2/ν1. Further analysis shows that an affine transformation reduces this multi-parameter problem to a single ordinary differential equation which may be efficiently integrated as an initial-value problem. Solutions of the original boundary-value problem are shown to agree with the initial-value integrations, but additional dual and quadruple solutions are found using this method. We argue on physical grounds and through bifurcation analysis that these additional solutions are not tenable. The present problem is applicable to the trailing edge flow over a thin airfoil with camber.

Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation

NASA Astrophysics Data System (ADS)

Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.

2010-05-01

The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.

Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations

PubMed Central

Feischl, Michael; Gantner, Gregor; Praetorius, Dirk

2015-01-01

We consider the Galerkin boundary element method (BEM) for weakly-singular integral equations of the first-kind in 2D. We analyze some residual-type a posteriori error estimator which provides a lower as well as an upper bound for the unknown Galerkin BEM error. The required assumptions are weak and allow for piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. In particular, our analysis gives a first contribution to adaptive BEM in the frame of isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments underline the theoretical findings and show that the proposed adaptive strategy leads to optimal convergence. PMID:26085698

Comparative Analysis of Models of the Earth's Gravity: 3. Accuracy of Predicting EAS Motion

NASA Astrophysics Data System (ADS)

Kuznetsov, E. D.; Berland, V. E.; Wiebe, Yu. S.; Glamazda, D. V.; Kajzer, G. T.; Kolesnikov, V. I.; Khremli, G. P.

2002-05-01

This paper continues a comparative analysis of modern satellite models of the Earth's gravity which we started in [6, 7]. In the cited works, the uniform norms of spherical functions were compared with their gradients for individual harmonics of the geopotential expansion [6] and the potential differences were compared with the gravitational accelerations obtained in various models of the Earth's gravity [7]. In practice, it is important to know how consistently the EAS motion is represented by various geopotential models. Unless otherwise stated, a model version in which the equations of motion are written using the classical Encke scheme and integrated together with the variation equations by the implicit one-step Everhart's algorithm [1] was used. When calculating coordinates and velocities on the integration step (at given instants of time), the approximate Everhart formula was employed.

A study of the viscous and nonadiabatic flow in radial turbines

NASA Technical Reports Server (NTRS)

Khalil, I.; Tabakoff, W.

1981-01-01

A method for analyzing the viscous nonadiabatic flow within turbomachine rotors is presented. The field analysis is based upon the numerical integration of the incompressible Navier-Stokes equations together with the energy equation over the rotors blade-to-blade stream channels. The numerical code used to solve the governing equations employs a nonorthogonal boundary fitted coordinate system that suits the most complicated blade geometries. Effects of turbulence are modeled with two equations; one expressing the development of the turbulence kinetic energy and the other its dissipation rate. The method of analysis is applied to a radial inflow turbine. The solution obtained indicates the severity of the complex interaction mechanism that occurs between different flow regimes (i.e., boundary layers, recirculating eddies, separation zones, etc.). Comparison with nonviscous flow solutions tend to justify strongly the inadequacy of using the latter with standard boundary layer techniques to obtain viscous flow details within turbomachine rotors. Capabilities and limitations of the present method of analysis are discussed.

A new (2+1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mukherjee, Abhik, E-mail: abhik.mukherjee@saha.ac.in; Janaki, M. S., E-mail: ms.janaki@saha.ac.in; Kundu, Anjan, E-mail: anjan.kundu@saha.ac.in

2015-07-15

A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a modulated wavepacket with weak transverse propagation, which has resemblance to nonlinear Schrödinger (NLS) equation and has a connection to Kadomtsev-Petviashvili equation through a constraint relation. Higher soliton solutions of the equation are derived through Hirota bilinearization procedure, and an exact lump solution is calculated exhibiting 2D structure. Some mathematical properties demonstrating the completely integrable nature of this equation are described. Modulational instability using nonlinear frequency correction is derived, and the correspondingmore » growth rate is calculated, which shows the directional asymmetry of the system. The discovery of this novel (2+1) dimensional integrable NLS type equation for a magnetized plasma should pave a new direction of research in the field.« less

Algorithms for the computation of solutions of the Ornstein-Zernike equation.

PubMed

Peplow, A T; Beardmore, R E; Bresme, F

2006-10-01

We introduce a robust and efficient methodology to solve the Ornstein-Zernike integral equation using the pseudoarc length (PAL) continuation method that reformulates the integral equation in an equivalent but nonstandard form. This enables the computation of solutions in regions where the compressibility experiences large changes or where the existence of multiple solutions and so-called branch points prevents Newton's method from converging. We illustrate the use of the algorithm with a difficult problem that arises in the numerical solution of integral equations, namely the evaluation of the so-called no-solution line of the Ornstein-Zernike hypernetted chain (HNC) integral equation for the Lennard-Jones potential. We are able to use the PAL algorithm to solve the integral equation along this line and to connect physical and nonphysical solution branches (both isotherms and isochores) where appropriate. We also show that PAL continuation can compute solutions within the no-solution region that cannot be computed when Newton and Picard methods are applied directly to the integral equation. While many solutions that we find are new, some correspond to states with negative compressibility and consequently are not physical.

Integral equations in the study of polar and ionic interaction site fluids

PubMed Central

Howard, Jesse J.

2011-01-01

In this review article we consider some of the current integral equation approaches and application to model polar liquid mixtures. We consider the use of multidimensional integral equations and in particular progress on the theory and applications of three dimensional integral equations. The IEs we consider may be derived from equilibrium statistical mechanical expressions incorporating a classical Hamiltonian description of the system. We give example including salt solutions, inhomogeneous solutions and systems including proteins and nucleic acids. PMID:22383857

Shock formation in the dispersionless Kadomtsev-Petviashvili equation

NASA Astrophysics Data System (ADS)

Grava, T.; Klein, C.; Eggers, J.

2016-04-01

The dispersionless Kadomtsev-Petviashvili (dKP) equation {{≤ft({{u}t}+u{{u}x}\\right)}x}={{u}yy} is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation {{u}t}+u{{u}x}=0 . We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral.

Regularized Generalized Structured Component Analysis

ERIC Educational Resources Information Center

Hwang, Heungsun

2009-01-01

Generalized structured component analysis (GSCA) has been proposed as a component-based approach to structural equation modeling. In practice, GSCA may suffer from multi-collinearity, i.e., high correlations among exogenous variables. GSCA has yet no remedy for this problem. Thus, a regularized extension of GSCA is proposed that integrates a ridge…

Iterative discrete ordinates solution of the equation for surface-reflected radiance

NASA Astrophysics Data System (ADS)

Radkevich, Alexander

2017-11-01

This paper presents a new method of numerical solution of the integral equation for the radiance reflected from an anisotropic surface. The equation relates the radiance at the surface level with BRDF and solutions of the standard radiative transfer problems for a slab with no reflection on its surfaces. It is also shown that the kernel of the equation satisfies the condition of the existence of a unique solution and the convergence of the successive approximations to that solution. The developed method features two basic steps: discretization on a 2D quadrature, and solving the resulting system of algebraic equations with successive over-relaxation method based on the Gauss-Seidel iterative process. Presented numerical examples show good coincidence between the surface-reflected radiance obtained with DISORT and the proposed method. Analysis of contributions of the direct and diffuse (but not yet reflected) parts of the downward radiance to the total solution is performed. Together, they represent a very good initial guess for the iterative process. This fact ensures fast convergence. The numerical evidence is given that the fastest convergence occurs with the relaxation parameter of 1 (no relaxation). An integral equation for BRDF is derived as inversion of the original equation. The potential of this new equation for BRDF retrievals is analyzed. The approach is found not viable as the BRDF equation appears to be an ill-posed problem, and it requires knowledge the surface-reflected radiance on the entire domain of both Sun and viewing zenith angles.

A spectral boundary integral equation method for the 2-D Helmholtz equation

NASA Technical Reports Server (NTRS)

Hu, Fang Q.

1994-01-01

In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.

The application of the least squares finite element method to Abel's integral equation. [with application to glow discharge problem

NASA Technical Reports Server (NTRS)

Balasubramanian, R.; Norrie, D. H.; De Vries, G.

1979-01-01

Abel's integral equation is the governing equation for certain problems in physics and engineering, such as radiation from distributed sources. The finite element method for the solution of this non-linear equation is presented for problems with cylindrical symmetry and the extension to more general integral equations is indicated. The technique was applied to an axisymmetric glow discharge problem and the results show excellent agreement with previously obtained solutions

On one solution of Volterra integral equations of second kind

NASA Astrophysics Data System (ADS)

Myrhorod, V.; Hvozdeva, I.

2016-10-01

A solution of Volterra integral equations of the second kind with separable and difference kernels based on solutions of corresponding equations linking the kernel and resolvent is suggested. On the basis of a discrete functions class, the equations linking the kernel and resolvent are obtained and the methods of their analytical solutions are proposed. A mathematical model of the gas-turbine engine state modification processes in the form of Volterra integral equation of the second kind with separable kernel is offered.

A parameter study of the two-fluid solar wind

NASA Technical Reports Server (NTRS)

Sandbaek, Ornulf; Leer, Egil; Holzer, Thomas E.

1992-01-01

A two-fluid model of the solar wind was introduced by Sturrock and Hartle (1966) and Hartle and Sturrock (1968). In these studies the proton energy equation was integrated neglecting the heat conductive term. Later several authors solved the equations for the two-fluid solar wind model keeping the proton heat conductive term. Methods where the equations are integrated simultaneously outward and inward from the critical point were used. The equations were also integrated inward from a large heliocentric distance. These methods have been applied to cases with low coronal base electron densities and high base temperatures. In this paper we present a method of integrating the two-fluid solar wind equations using an iteration procedure where the equations are integrated separately and the proton flux is kept constant during the integrations. The technique is applicable for a wide range of coronal base densities and temperatures. The method is used to carry out a parameter study of the two-fluid solar wind.

On the Analysis of Multistep-Out-of-Grid Method for Celestial Mechanics Tasks

NASA Astrophysics Data System (ADS)

Olifer, L.; Choliy, V.

2016-09-01

Occasionally, there is a necessity in high-accurate prediction of celestial body trajectory. The most common way to do that is to solve Kepler's equation analytically or to use Runge-Kutta or Adams integrators to solve equation of motion numerically. For low-orbit satellites, there is a critical need in accounting geopotential and another forces which influence motion. As the result, the right side of equation of motion becomes much bigger, and classical integrators will not be quite effective. On the other hand, there is a multistep-out-of-grid (MOG) method which combines Runge-Kutta and Adams methods. The MOG method is based on using m on-grid values of the solution and n × m off-grid derivative estimations. Such method could provide stable integrators of maximum possible order, O (hm+mn+n-1). The main subject of this research was to implement and analyze the MOG method for solving satellite equation of motion with taking into account Earth geopotential model (ex. EGM2008 (Pavlis at al., 2008)) and with possibility to add other perturbations such as atmospheric drag or solar radiation pressure. Simulations were made for satellites on low orbit and with various eccentricities (from 0.1 to 0.9). Results of the MOG integrator were compared with results of Runge-Kutta and Adams integrators. It was shown that the MOG method has better accuracy than classical ones of the same order and less right-hand value estimations when is working on high orders. That gives it some advantage over "classical" methods.

The staircase method: integrals for periodic reductions of integrable lattice equations

NASA Astrophysics Data System (ADS)

van der Kamp, Peter H.; Quispel, G. R. W.

2010-11-01

We show, in full generality, that the staircase method (Papageorgiou et al 1990 Phys. Lett. A 147 106-14, Quispel et al 1991 Physica A 173 243-66) provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the quotient-difference (QD)-algorithm and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r, then one can introduce q <= 2r variables, which reduce the dimension of the mapping from n to q. These dimension-reducing variables are obtained as joint invariants of k-symmetries of the mappings. Our results support the idea that often the staircase method provides sufficiently many integrals for the periodic reductions of integrable lattice equations to be completely integrable. We also study reductions on other quad-graphs than the regular {\\ Z}^2 lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.

An integrable semi-discrete Degasperis-Procesi equation

NASA Astrophysics Data System (ADS)

Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro

2017-06-01

Based on our previous work on the Degasperis-Procesi equation (Feng et al J. Phys. A: Math. Theor. 46 045205) and the integrable semi-discrete analogue of its short wave limit (Feng et al J. Phys. A: Math. Theor. 48 135203), we derive an integrable semi-discrete Degasperis-Procesi equation by Hirota’s bilinear method. Furthermore, N-soliton solution to the semi-discrete Degasperis-Procesi equation is constructed. It is shown that both the proposed semi-discrete Degasperis-Procesi equation, and its N-soliton solution converge to ones of the original Degasperis-Procesi equation in the continuum limit.

The Kadomtsev{endash}Petviashvili equation as a source of integrable model equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maccari, A.

1996-12-01

A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev{endash}Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev{endash}Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. {copyright} {ital 1996 American Institute of Physics.}

An improved semi-implicit method for structural dynamics analysis

NASA Technical Reports Server (NTRS)

Park, K. C.

1982-01-01

A semi-implicit algorithm is presented for direct time integration of the structural dynamics equations. The algorithm avoids the factoring of the implicit difference solution matrix and mitigates the unacceptable accuracy losses which plagued previous semi-implicit algorithms. This substantial accuracy improvement is achieved by augmenting the solution matrix with two simple diagonal matrices of the order of the integration truncation error.

A finite-element analysis for steady and oscillatory supersonic flows around complex configurations

NASA Technical Reports Server (NTRS)

Morino, L.; Chen, L. T.

1974-01-01

The problem of small perturbation potential supersonic flow around complex configurations is considered. This problem requires the solution of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the small perturbation boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, sigma sub i, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element, and this yields a set of linear algebraic equations. The coefficients of the equation are given by source and doublet integrals over the surface elements, sigma sub i. The results obtained using the above formulation are compared with existing analytical and experimental results.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wang, Y. B.; Zhu, X. W., E-mail: xiaowuzhu1026@znufe.edu.cn; Dai, H. H.

Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings aremore » considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.« less

DOE Office of Scientific and Technical Information (OSTI.GOV)

Stoynov, Y.; Dineva, P.

The stress, magnetic and electric field analysis of multifunctional composites, weakened by impermeable cracks, is of fundamental importance for their structural integrity and reliable service performance. The aim is to study dynamic behavior of a plane of functionally graded magnetoelectroelastic composite with more than one crack. The coupled material properties vary exponentially in an arbitrary direction. The plane is subjected to anti-plane mechanical and in-plane electric and magnetic load. The boundary value problem described by the partial differential equations with variable coefficients is reduced to a non-hypersingular traction boundary integral equation based on the appropriate functional transform and frequency-dependent fundamentalmore » solution derived in a closed form by Radon transform. Software code based on the boundary integral equation method (BIEM) is developed, validated and inserted in numerical simulations. The obtained results show the sensitivity of the dynamic stress, magnetic and electric field concentration in the cracked plane to the type and characteristics of the dynamic load, to the location and cracks disposition, to the wave-crack-crack interactions and to the magnitude and direction of the material gradient.« less

Simplifying Differential Equations for Multiscale Feynman Integrals beyond Multiple Polylogarithms.

PubMed

Adams, Luise; Chaubey, Ekta; Weinzierl, Stefan

2017-04-07

In this Letter we exploit factorization properties of Picard-Fuchs operators to decouple differential equations for multiscale Feynman integrals. The algorithm reduces the differential equations to blocks of the size of the order of the irreducible factors of the Picard-Fuchs operator. As a side product, our method can be used to easily convert the differential equations for Feynman integrals which evaluate to multiple polylogarithms to an ϵ form.

Slackline dynamics and the Helmholtz-Duffing oscillator

NASA Astrophysics Data System (ADS)

Athanasiadis, Panos J.

2018-01-01

Slacklining is a new, rapidly expanding sport, and understanding its physics is paramount for maximizing fun and safety. Yet, compared to other sports, very little has been published so far on slackline dynamics. The equations of motion describing a slackline are fundamentally nonlinear, and assuming linear elasticity, they lead to a form of the Duffing equation. Following this approach, characteristic examples of slackline motion are simulated, including trickline bouncing, leash falls and longline surfing. The time-dependent solutions of the differential equations describing the system are acquired by numerical integration. A simple form of energy dissipation (linear drag) is added in some cases. It is recognized in this study that geometric nonlinearity is a fundamental aspect characterizing the dynamics of slacklines. Sports, and particularly slackline, is an excellent way of engaging young people with physics. A slackline is a simple yet insightful example of a nonlinear oscillator. It is very easy to model in the laboratory, as well as to rig and try on a university campus. For instructive purposes, its behaviour can be explored by numerically integrating the respective equations of motion. A form of the Duffing equation emerges naturally in the analysis and provides a powerful introduction to nonlinear dynamics. The material is suitable for graduate students and undergraduates with a background in classical mechanics and differential equations.

Computational singular perturbation analysis of stochastic chemical systems with stiffness

DOE PAGES

Wang, Lijin; Han, Xiaoying; Cao, Yanzhao; ...

2017-01-25

Computational singular perturbation (CSP) is a useful method for analysis, reduction, and time integration of stiff ordinary differential equation systems. It has found dominant utility, in particular, in chemical reaction systems with a large range of time scales at continuum and deterministic level. On the other hand, CSP is not directly applicable to chemical reaction systems at micro or meso-scale, where stochasticity plays an non-negligible role and thus has to be taken into account. In this work we develop a novel stochastic computational singular perturbation (SCSP) analysis and time integration framework, and associated algorithm, that can be used to notmore » only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems, but also analyze the dynamics of the reduced stochastic reaction systems. Furthermore, the algorithm is illustrated by an application to a benchmark stochastic differential equation model, and numerical experiments are carried out to demonstrate the effectiveness of the construction.« less

Solution of the nonlinear mixed Volterra-Fredholm integral equations by hybrid of block-pulse functions and Bernoulli polynomials.

PubMed

Mashayekhi, S; Razzaghi, M; Tripak, O

2014-01-01

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

PubMed Central

Mashayekhi, S.; Razzaghi, M.; Tripak, O.

2014-01-01

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. PMID:24523638

Two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography

NASA Astrophysics Data System (ADS)

Rahmouni, Lyes; Mitharwal, Rajendra; Andriulli, Francesco P.

2017-11-01

This work presents two new volume integral equations for the Electroencephalography (EEG) forward problem which, differently from the standard integral approaches in the domain, can handle heterogeneities and anisotropies of the head/brain conductivity profiles. The new formulations translate to the quasi-static regime some volume integral equation strategies that have been successfully applied to high frequency electromagnetic scattering problems. This has been obtained by extending, to the volume case, the two classical surface integral formulations used in EEG imaging and by introducing an extra surface equation, in addition to the volume ones, to properly handle boundary conditions. Numerical results corroborate theoretical treatments, showing the competitiveness of our new schemes over existing techniques and qualifying them as a valid alternative to differential equation based methods.

Algebraic Construction of Exact Difference Equations from Symmetry of Equations

NASA Astrophysics Data System (ADS)

Itoh, Toshiaki

2009-09-01

Difference equations or exact numerical integrations, which have general solutions, are treated algebraically. Eliminating the symmetries of the equation, we can construct difference equations (DCE) or numerical integrations equivalent to some ODEs or PDEs that means both have the same solution functions. When arbitrary functions are given, whether we can construct numerical integrations that have solution functions equal to given function or not are treated in this work. Nowadays, Lie's symmetries solver for ODE and PDE has been implemented in many symbolic software. Using this solver we can construct algebraic DCEs or numerical integrations which are correspond to some ODEs or PDEs. In this work, we treated exact correspondence between ODE or PDE and DCE or numerical integration with Gröbner base and Janet base from the view of Lie's symmetries.

Approximation for limit cycles and their isochrons.

PubMed

Demongeot, Jacques; Françoise, Jean-Pierre

2006-12-01

Local analysis of trajectories of dynamical systems near an attractive periodic orbit displays the notion of asymptotic phase and isochrons. These notions are quite useful in applications to biosciences. In this note, we give an expression for the first approximation of equations of isochrons in the setting of perturbations of polynomial Hamiltonian systems. This method can be generalized to perturbations of systems that have a polynomial integral factor (like the Lotka-Volterra equation).

Liouvillian propagators, Riccati equation and differential Galois theory

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo; Suazo, Erwin

2013-11-01

In this paper a Galoisian approach to building propagators through Riccati equations is presented. The main result corresponds to the relationship between the Galois integrability of the linear Schrödinger equation and the virtual solvability of the differential Galois group of its associated characteristic equation. As the main application of this approach we solve Ince’s differential equation through the Hamiltonian algebrization procedure and the Kovacic algorithm to find the propagator for a generalized harmonic oscillator. This propagator has applications which describe the process of degenerate parametric amplification in quantum optics and light propagation in a nonlinear anisotropic waveguide. Toy models of propagators inspired by integrable Riccati equations and integrable characteristic equations are also presented.

Analysis of crack propagation in roller bearings using the boundary integral equation method - A mixed-mode loading problem

NASA Technical Reports Server (NTRS)

Ghosn, L. J.

1988-01-01

Crack propagation in a rotating inner raceway of a high-speed roller bearing is analyzed using the boundary integral method. The model consists of an edge plate under plane strain condition upon which varying Hertzian stress fields are superimposed. A multidomain boundary integral equation using quadratic elements was written to determine the stress intensity factors KI and KII at the crack tip for various roller positions. The multidomain formulation allows the two faces of the crack to be modeled in two different subregions, making it possible to analyze crack closure when the roller is positioned on or close to the crack line. KI and KII stress intensity factors along any direction were computed. These calculations permit determination of crack growth direction along which the average KI times the alternating KI is maximum.

On finite element implementation and computational techniques for constitutive modeling of high temperature composites

NASA Technical Reports Server (NTRS)

Saleeb, A. F.; Chang, T. Y. P.; Wilt, T.; Iskovitz, I.

1989-01-01

The research work performed during the past year on finite element implementation and computational techniques pertaining to high temperature composites is outlined. In the present research, two main issues are addressed: efficient geometric modeling of composite structures and expedient numerical integration techniques dealing with constitutive rate equations. In the first issue, mixed finite elements for modeling laminated plates and shells were examined in terms of numerical accuracy, locking property and computational efficiency. Element applications include (currently available) linearly elastic analysis and future extension to material nonlinearity for damage predictions and large deformations. On the material level, various integration methods to integrate nonlinear constitutive rate equations for finite element implementation were studied. These include explicit, implicit and automatic subincrementing schemes. In all cases, examples are included to illustrate the numerical characteristics of various methods that were considered.

High Energy Laser Beam Propagation in the Atmosphere: The Integral Invariants of the Nonlinear Parabolic Equation and the Method of Moments

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2012-01-01

The method of moments is used to define and derive expressions for laser beam deflection and beam radius broadening for high-energy propagation through the Earth s atmosphere. These expressions are augmented with the integral invariants of the corresponding nonlinear parabolic equation that describes the electric field of high-energy laser beam to propagation to yield universal equations for the aforementioned quantities; the beam deflection is a linear function of the propagation distance whereas the beam broadening is a quadratic function of distance. The coefficients of these expressions are then derived from a thin screen approximation solution of the nonlinear parabolic equation to give corresponding analytical expressions for a target located outside the Earth s atmospheric layer. These equations, which are graphically presented for a host of propagation scenarios, as well as the thin screen model, are easily amenable to the phase expansions of the wave front for the specification and design of adaptive optics algorithms to correct for the inherent phase aberrations. This work finds application in, for example, the analysis of beamed energy propulsion for space-based vehicles.

Equations of Motion for the g-LIMIT Microgravity Vibration Isolation System

NASA Technical Reports Server (NTRS)

Kim, Y. K.; Whorton, M. S.

2001-01-01

A desirable microgravity environment for experimental science payloads may require an active vibration isolation control system. A vibration isolation system named g-LIMIT (GLovebox Integrated Microgravity Isolation Technology) is being developed by NASA Marshall Space Flight Center to support microgravity science experiments using the microgravity science glovebox. In this technical memorandum, the full six-degree-of-freedom nonlinear equations of motion for g-LIMIT are derived. Although the motivation for this model development is control design and analysis of g-LIMIT, the equations are derived for a general configuration and may be used for other isolation systems as well.

Applications of a Property of the Schrödinger Equation to the Modeling of Conservative Discrete Systems

NASA Astrophysics Data System (ADS)

Popa, Alexandru

1998-08-01

Recently we have demonstrated in a mathematical paper the following property: The energy which results from the Schrödinger equation can be rigorously calculated by line integrals of analytical functions, if the Hamilton-Jacobi equation, written for the same system, is satisfied in the space of coordinates by a periodical trajectory. We present now an accurate analysis model of the conservative discrete systems, that is based on this property. The theory is checked for a lot of atomic systems. The experimental data, which are ionization energies, are taken from well known books.

A Constant Percentage Bandwidth Transform for Acoustic Signal Processing

DTIC Science & Technology

1980-01-01

t)eJwt d . 27Th (0) Equation 2.9 is not, however, the most general form for short-time Fourier synthesis, but is in fact a particular case of the...form of the analysis integral. F(Wk? t)eJwkt = f(t) * h( t)’ukt (3.4) Fourier transforming both sides of this equation (and invoking the convolution...properties. In what follows, define 38 f(t) - F(w,t) (3.24) to be an equivalent statement to equation 3.1. 3.6.1 Linearity Property If F1 w,t) and F2 (w

Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations

NASA Astrophysics Data System (ADS)

Eden, Burkhard; Smirnov, Vladimir A.

2016-10-01

We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.

Parametrically excited non-linear multidegree-of-freedom systems with repeated natural frequencies

NASA Astrophysics Data System (ADS)

Tezak, E. G.; Nayfeh, A. H.; Mook, D. T.

1982-12-01

A method for analyzing multidegree-of-freedom systems having a repeated natural frequency subjected to a parametric excitation is presented. Attention is given to the ordering of the various terms (linear and non-linear) in the governing equations. The analysis is based on the method of multiple scales. As a numerical example involving a parametric resonance, panel flutter is discussed in detail in order to illustrate the type of results one can expect to obtain with this analysis. Some of the analytical results are verified by a numerical integration of the governing equations.

Application of the Sumudu Transform to Discrete Dynamic Systems

ERIC Educational Resources Information Center

Asiru, Muniru Aderemi

2003-01-01

The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and…

Computational Algorithms or Identification of Distributed Parameter Systems

DTIC Science & Technology

1993-04-24

delay-differential equations, Volterra integral equations, and partial differential equations with memory terms . In particular we investigated a...tested for estimating parameters in a Volterra integral equation arising from a viscoelastic model of a flexible structure with Boltzmann damping. In...particular, one of the parameters identified was the order of the derivative in Volterra integro-differential equations containing fractional

Effects of Thermal Resistance on One-Dimensional Thermal Analysis of the Epidermal Flexible Electronic Devices Integrated with Human Skin

NASA Astrophysics Data System (ADS)

Li, He; Cui, Yun

2017-12-01

Nowadays, flexible electronic devices are increasingly used in direct contact with human skin to monitor the real-time health of human body. Based on the Fourier heat conduction equation and Pennes bio-heat transfer equation, this paper deduces the analytical solutions of one - dimensional heat transfer for flexible electronic devices integrated with human skin under the condition of a constant power. The influence of contact thermal resistance between devices and skin is considered as well. The corresponding finite element model is established to verify the correctness of analytical solutions. The results show that the finite element analysis agrees well with the analytical solution. With bigger thermal resistance, temperature increase of skin surface will decrease. This result can provide guidance for the design of flexible electronic devices to reduce the negative impact that exceeding temperature leave on human skin.

Application of integral equation theory to analyze stability of electric field in multimode microwave heating cavity

NASA Astrophysics Data System (ADS)

Tang, Zhengming; Hong, Tao; Chen, Fangyuan; Zhu, Huacheng; Huang, Kama

2017-10-01

Microwave heating uniformity is mainly dependent on and affected by electric field. However, little study has paid attention to its stability characteristics in multimode cavity. In this paper, this problem is studied by the theory of Freedholm integral equation. Firstly, Helmholtz equation and the electric dyadic Green's function are used to derive the electric field integral equation. Then, the stability of electric field is demonstrated as the characteristics of solutions to Freedholm integral equation. Finally, the stability characteristics are obtained and verified by finite element calculation. This study not only can provide a comprehensive interpretation of electric field in multimode cavity but also help us make better use of microwave energy.

A semi-discrete Kadomtsev-Petviashvili equation and its coupled integrable system

NASA Astrophysics Data System (ADS)

Li, Chun-Xia; Lafortune, Stéphane; Shen, Shou-Feng

2016-05-01

We establish connections between two cascades of integrable systems generated from the continuum limits of the Hirota-Miwa equation and its remarkable nonlinear counterpart under the Miwa transformation, respectively. Among these equations, we are mainly concerned with the semi-discrete bilinear Kadomtsev-Petviashvili (KP) equation which is seldomly studied in literature. We present both of its Casorati and Grammian determinant solutions. Through the Pfaffianization procedure proposed by Hirota and Ohta, we are able to derive the coupled integrable system for the semi-discrete KP equation.

An iwatsubo-based solution for labyrinth seals - comparison with experimental results

NASA Technical Reports Server (NTRS)

Childs, D. W.; Scharrer, J. K.

1984-01-01

The basic equations are derived for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. Linearized zeroth and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equation. The first-order equations are solved by a separation of variables solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to published test results.

Space tug economic analysis study. Volume 2: Tug concepts analysis. Part 1: Overall approach and data generation

NASA Technical Reports Server (NTRS)

1972-01-01

An economic analysis of space tug operations is presented. The subjects discussed are: (1) data base for orbit injection stages, (2) data base for reusable space tug, (3) performance equations, (4) data integration and interpretation, (5) tug performance and mission model accomodation, (6) total program cost, (7) payload analysis, (8) computer software, and (9) comparison of tug concepts.

A Mapping from the Human Factors Analysis and Classification System (DOD-HFACS) to the Domains of Human Systems Integration (HSI)

DTIC Science & Technology

2009-11-01

Equation Chapter 1 Section 1 A MAPPING FROM THE HUMAN FACTORS ANALYSIS AND CLASSIFICATION SYSTEM (DOD...OMB control number. 1. REPORT DATE NOV 2009 2. REPORT TYPE 3. DATES COVERED 4. TITLE AND SUBTITLE A Mapping from the Human Factors Analysis ...7 The Human Factors Analysis and Classification System .................................................. 7 Mapping of DoD

Whitham modulation theory for (2  +  1)-dimensional equations of Kadomtsev–Petviashvili type

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Biondini, Gino; Rumanov, Igor

2018-05-01

Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev–Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the two-dimensional Benjamin–Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich–Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.

Two Optical Atmospheric Remote Sensing Techniques and AN Associated Analytic Solution to a Class of Integral Equations

NASA Astrophysics Data System (ADS)

Manning, Robert Michael

This work concerns itself with the analysis of two optical remote sensing methods to be used to obtain parameters of the turbulent atmosphere pertinent to stochastic electromagnetic wave propagation studies, and the well -posed solution to a class of integral equations that are central to the development of these remote sensing methods. A remote sensing technique is theoretically developed whereby the temporal frequency spectrum of the scintillations of a stellar source or a point source within the atmosphere, observed through a variable radius aperture, is related to the space-time spectrum of atmospheric scintillation. The key to this spectral remote sensing method is the spatial filtering performed by a finite aperture. The entire method is developed without resorting to a priori information such as results from stochastic wave propagation theory. Once the space-time spectrum of the scintillations is obtained, an application of known results of atmospheric wave propagation theory and simple geometric considerations are shown to yield such important information such as the spectrum of atmospheric turbulence, the cross-wind velocity, and the path profile of the atmospheric refractive index structure parameter. A method is also developed to independently verify the Taylor frozen flow hypothesis. The success of the spectral remote sensing method relies on the solution to a Fredholm integral equation of the first kind. An entire class of such equations, that are peculiar to inverse diffraction problems, is studied and a well-posed solution (in the sense of Hadamard) is obtained and probed. Conditions of applicability are derived and shown not to limit the useful operating range of the spectral remote sensing method. The general integral equation solution obtained is then applied to another remote sensing problem having to do with the characterization of the particle size distribution to atmospheric aerosols and hydrometeors. By measuring the diffraction pattern in the focal plane of a lens created by the passage of a laser beam through a distribution of particles, it is shown that the particle-size distribution of the particles can be obtained. An intermediate result of the analysis also gives the total volume concentration of the particles.

First integrals of the axisymmetric shape equation of lipid membranes

NASA Astrophysics Data System (ADS)

Zhang, Yi-Heng; McDargh, Zachary; Tu, Zhan-Chun

2018-03-01

The shape equation of lipid membranes is a fourth-order partial differential equation. Under the axisymmetric condition, this equation was transformed into a second-order ordinary differential equation (ODE) by Zheng and Liu (Phys. Rev. E 48 2856 (1993)). Here we try to further reduce this second-order ODE to a first-order ODE. First, we invert the usual process of variational calculus, that is, we construct a Lagrangian for which the ODE is the corresponding Euler–Lagrange equation. Then, we seek symmetries of this Lagrangian according to the Noether theorem. Under a certain restriction on Lie groups of the shape equation, we find that the first integral only exists when the shape equation is identical to the Willmore equation, in which case the symmetry leading to the first integral is scale invariance. We also obtain the mechanical interpretation of the first integral by using the membrane stress tensor. Project supported by the National Natural Science Foundation of China (Grant No. 11274046) and the National Science Foundation of the United States (Grant No. 1515007).

ALESEP. Part 2: A computer program for the analysis of leading edge separation bubbles on infinite swept wings

NASA Technical Reports Server (NTRS)

Davis, R. L.

1986-01-01

A program called ALESEP is presented for the analysis of the inviscid-viscous interaction which occurs due to the presence of a closed laminar-transitional separation bubble on an airfoil or infinite swept wing. The ALESEP code provides an iterative solution of the boundary layer equations expressed in an inverse formulation coupled to a Cauchy integral representation of the inviscid flow. This interaction analysis is treated as a local perturbation to a known solution obtained from a global airfoil analysis; hence, part of the required input to the ALESEP code are the reference displacement thickness and tangential velocity distributions. Special windward differencing may be used in the reversed flow regions of the separation bubble to accurately account for the flow direction in the discretization of the streamwise convection of momentum. The ALESEP code contains a forced transition model based on a streamwise intermittency function, a natural transition model based on a solution of the integral form of the turbulent kinetic energy equation, and an empirical natural transition model.

Optical solitons and modulation instability analysis with (3 + 1)-dimensional nonlinear Shrödinger equation

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Aliyu, Aliyu Isa; Yusuf, Abdullahi; Baleanu, Dumitru

2017-12-01

This paper addresses the (3 + 1)-dimensional nonlinear Shrödinger equation (NLSE) that serves as the model to study the propagation of optical solitons through nonlinear optical fibers. Two integration schemes are employed to study the equation. These are the complex envelope function ansatz and the solitary wave ansatz with Jaccobi elliptic function methods, we present the exact dark, bright and dark-bright or combined optical solitons to the model. The intensity as well as the nonlinear phase shift of the solitons are reported. The modulation instability aspects are discussed using the concept of linear stability analysis. The MI gain is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Design and application of squeeze film dampers for turbomachinery stabilization

NASA Technical Reports Server (NTRS)

Gunter, E. J.; Barrett, L. E.; Allaire, P. E.

1975-01-01

The steady-state transient response of the squeeze film damper bearing was investigated. Both the steady-state and transient equations for the hydrodynamic bearing forces are derived; the steady-state equations were used to determine the damper equivalent stiffness and damping coefficients. These coefficients are used to find the damper configuration which will provide the optimum support characteristics based on a stability analysis of the rotor-bearing system. The effects of end seals and cavitated fluid film are included. The transient analysis of rotor-bearing systems was conducted by coupling the damping and rotor equations and integrating forward in time. The effects of unbalance, cavitation, and retainer springs are included. Methods of determining the stability of a rotor-bearing system under the influence of aerodynamic forces and internal shaft friction are discussed.

Shape reanalysis and sensitivities utilizing preconditioned iterative boundary solvers

NASA Technical Reports Server (NTRS)

Guru Prasad, K.; Kane, J. H.

1992-01-01

The computational advantages associated with the utilization of preconditined iterative equation solvers are quantified for the reanalysis of perturbed shapes using continuum structural boundary element analysis (BEA). Both single- and multi-zone three-dimensional problems are examined. Significant reductions in computer time are obtained by making use of previously computed solution vectors and preconditioners in subsequent analyses. The effectiveness of this technique is demonstrated for the computation of shape response sensitivities required in shape optimization. Computer times and accuracies achieved using the preconditioned iterative solvers are compared with those obtained via direct solvers and implicit differentiation of the boundary integral equations. It is concluded that this approach employing preconditioned iterative equation solvers in reanalysis and sensitivity analysis can be competitive with if not superior to those involving direct solvers.

Three-dimensional analysis of chevron-notched specimens by boundary integral method

NASA Technical Reports Server (NTRS)

Mendelson, A.; Ghosn, L.

1983-01-01

The chevron-notched short bar and short rod specimens was analyzed by the boundary integral equations method. This method makes use of boundary surface elements in obtaining the solution. The boundary integral models were composed of linear triangular and rectangular surface segments. Results were obtained for two specimens with width to thickness ratios of 1.45 and 2.00 and for different crack length to width ratios ranging from 0.4 to 0.7. Crack opening displacement and stress intensity factors determined from displacement calculations along the crack front and compliance calculations were compared with experimental values and with finite element analysis.

Macroscopic self-oscillations and aging transition in a network of synaptically coupled quadratic integrate-and-fire neurons.

PubMed

Ratas, Irmantas; Pyragas, Kestutis

2016-09-01

We analyze the dynamics of a large network of coupled quadratic integrate-and-fire neurons, which represent the canonical model for class I neurons near the spiking threshold. The network is heterogeneous in that it includes both inherently spiking and excitable neurons. The coupling is global via synapses that take into account the finite width of synaptic pulses. Using a recently developed reduction method based on the Lorentzian ansatz, we derive a closed system of equations for the neuron's firing rate and the mean membrane potential, which are exact in the infinite-size limit. The bifurcation analysis of the reduced equations reveals a rich scenario of asymptotic behavior, the most interesting of which is the macroscopic limit-cycle oscillations. It is shown that the finite width of synaptic pulses is a necessary condition for the existence of such oscillations. The robustness of the oscillations against aging damage, which transforms spiking neurons into nonspiking neurons, is analyzed. The validity of the reduced equations is confirmed by comparing their solutions with the solutions of microscopic equations for the finite-size networks.

G-DYN Multibody Dynamics Engine

NASA Technical Reports Server (NTRS)

Acikmese, Behcet; Blackmore, James C.; Broderick, Daniel

2011-01-01

G-DYN is a multi-body dynamic simulation software engine that automatically assembles and integrates equations of motion for arbitrarily connected multibody dynamic systems. The algorithm behind G-DYN is based on a primal-dual formulation of the dynamics that captures the position and velocity vectors (primal variables) of each body and the interaction forces (dual variables) between bodies, which are particularly useful for control and estimation analysis and synthesis. It also takes full advantage of the spare matrix structure resulting from the system dynamics to numerically integrate the equations of motion efficiently. Furthermore, the dynamic model for each body can easily be replaced without re-deriving the overall equations of motion, and the assembly of the equations of motion is done automatically. G-DYN proved an essential software tool in the simulation of spacecraft systems used for small celestial body surface sampling, specifically in simulating touch-and-go (TAG) maneuvers of a robotic sampling system from a comet and asteroid. It is used extensively in validating mission concepts for small body sample return, such as Comet Odyssey and Galahad New Frontiers proposals.

Dual boundary element formulation for elastoplastic fracture mechanics

NASA Astrophysics Data System (ADS)

Leitao, V.; Aliabadi, M. H.; Rooke, D. P.

1995-01-01

In this paper the extension of the dual boundary element method (DBEM) to the analysis of elastoplastic fracture mechanics (EPFM) problems is presented. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. In order to avoid collocation at crack tips, crack kinks and crack-edge corners, both crack surfaces are discretized with discontinuous quadratic boundary elements. The elasto-plastic behavior is modelled through the use of an approximation for the plastic component of the strain tensor on the region expected to yield. This region is discretized with internal quadratic, quadrilateral and/or triangular cells. This formulation was implemented for two-dimensional domains only, although there is no theoretical or numerical limitation to its application to three-dimensional ones. A center-cracked plate and a slant edge-cracked plate subjected to tensile load are analysed and the results are compared with others available in the literature. J-type integrals are calculated.

Variational formulation for dissipative continua and an incremental J-integral

NASA Astrophysics Data System (ADS)

Rahaman, Md. Masiur; Dhas, Bensingh; Roy, D.; Reddy, J. N.

2018-01-01

Our aim is to rationally formulate a proper variational principle for dissipative (viscoplastic) solids in the presence of inertia forces. As a first step, a consistent linearization of the governing nonlinear partial differential equations (PDEs) is carried out. An additional set of complementary (adjoint) equations is then formed to recover an underlying variational structure for the augmented system of linearized balance laws. This makes it possible to introduce an incremental Lagrangian such that the linearized PDEs, including the complementary equations, become the Euler-Lagrange equations. Continuous groups of symmetries of the linearized PDEs are computed and an analysis is undertaken to identify the variational groups of symmetries of the linearized dissipative system. Application of Noether's theorem leads to the conservation laws (conserved currents) of motion corresponding to the variational symmetries. As a specific outcome, we exploit translational symmetries of the functional in the material space and recover, via Noether's theorem, an incremental J-integral for viscoplastic solids in the presence of inertia forces. Numerical demonstrations are provided through a two-dimensional plane strain numerical simulation of a compact tension specimen of annealed mild steel under dynamic loading.

Constraint treatment techniques and parallel algorithms for multibody dynamic analysis. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Chiou, Jin-Chern

1990-01-01

Computational procedures for kinematic and dynamic analysis of three-dimensional multibody dynamic (MBD) systems are developed from the differential-algebraic equations (DAE's) viewpoint. Constraint violations during the time integration process are minimized and penalty constraint stabilization techniques and partitioning schemes are developed. The governing equations of motion, a two-stage staggered explicit-implicit numerical algorithm, are treated which takes advantage of a partitioned solution procedure. A robust and parallelizable integration algorithm is developed. This algorithm uses a two-stage staggered central difference algorithm to integrate the translational coordinates and the angular velocities. The angular orientations of bodies in MBD systems are then obtained by using an implicit algorithm via the kinematic relationship between Euler parameters and angular velocities. It is shown that the combination of the present solution procedures yields a computationally more accurate solution. To speed up the computational procedures, parallel implementation of the present constraint treatment techniques, the two-stage staggered explicit-implicit numerical algorithm was efficiently carried out. The DAE's and the constraint treatment techniques were transformed into arrowhead matrices to which Schur complement form was derived. By fully exploiting the sparse matrix structural analysis techniques, a parallel preconditioned conjugate gradient numerical algorithm is used to solve the systems equations written in Schur complement form. A software testbed was designed and implemented in both sequential and parallel computers. This testbed was used to demonstrate the robustness and efficiency of the constraint treatment techniques, the accuracy of the two-stage staggered explicit-implicit numerical algorithm, and the speed up of the Schur-complement-based parallel preconditioned conjugate gradient algorithm on a parallel computer.

Derivation of the Schrodinger Equation from the Hamilton-Jacobi Equation in Feynman's Path Integral Formulation of Quantum Mechanics

ERIC Educational Resources Information Center

Field, J. H.

2011-01-01

It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…

Path integral analysis of Jarzynski's equality: Analytical results

NASA Astrophysics Data System (ADS)

Minh, David D. L.; Adib, Artur B.

2009-02-01

We apply path integrals to study nonequilibrium work theorems in the context of Brownian dynamics, deriving in particular the equations of motion governing the most typical and most dominant trajectories. For the analytically soluble cases of a moving harmonic potential and a harmonic oscillator with a time-dependent natural frequency, we find such trajectories, evaluate the work-weighted propagators, and validate Jarzynski’s equality.

Exponential Methods for the Time Integration of Schroedinger Equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cano, B.; Gonzalez-Pachon, A.

2010-09-30

We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.

Path-integral methods for analyzing the effects of fluctuations in stochastic hybrid neural networks.

PubMed

Bressloff, Paul C

2015-01-01

We consider applications of path-integral methods to the analysis of a stochastic hybrid model representing a network of synaptically coupled spiking neuronal populations. The state of each local population is described in terms of two stochastic variables, a continuous synaptic variable and a discrete activity variable. The synaptic variables evolve according to piecewise-deterministic dynamics describing, at the population level, synapses driven by spiking activity. The dynamical equations for the synaptic currents are only valid between jumps in spiking activity, and the latter are described by a jump Markov process whose transition rates depend on the synaptic variables. We assume a separation of time scales between fast spiking dynamics with time constant [Formula: see text] and slower synaptic dynamics with time constant τ. This naturally introduces a small positive parameter [Formula: see text], which can be used to develop various asymptotic expansions of the corresponding path-integral representation of the stochastic dynamics. First, we derive a variational principle for maximum-likelihood paths of escape from a metastable state (large deviations in the small noise limit [Formula: see text]). We then show how the path integral provides an efficient method for obtaining a diffusion approximation of the hybrid system for small ϵ. The resulting Langevin equation can be used to analyze the effects of fluctuations within the basin of attraction of a metastable state, that is, ignoring the effects of large deviations. We illustrate this by using the Langevin approximation to analyze the effects of intrinsic noise on pattern formation in a spatially structured hybrid network. In particular, we show how noise enlarges the parameter regime over which patterns occur, in an analogous fashion to PDEs. Finally, we carry out a [Formula: see text]-loop expansion of the path integral, and use this to derive corrections to voltage-based mean-field equations, analogous to the modified activity-based equations generated from a neural master equation.

Weierstrass traveling wave solutions for dissipative Benjamin, Bona, and Mahony (BBM) equation

NASA Astrophysics Data System (ADS)

Mancas, Stefan C.; Spradlin, Greg; Khanal, Harihar

2013-08-01

In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified Benjamin, Bona, and Mahony (BBM) equation by viscosity. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant makes the equation integrable in terms of Weierstrass ℘ functions. We will use a general formalism based on Ince's transformation to write the general solution of dissipative BBM in terms of ℘ functions, from which all the other known solutions can be obtained via simplifying assumptions. Using ODE (ordinary differential equations) analysis we show that the traveling wave speed is a bifurcation parameter that makes transition between different classes of waves.

Approximate analysis for repeated eigenvalue problems with applications to controls-structure integrated design

NASA Technical Reports Server (NTRS)

Kenny, Sean P.; Hou, Gene J. W.

1994-01-01

A method for eigenvalue and eigenvector approximate analysis for the case of repeated eigenvalues with distinct first derivatives is presented. The approximate analysis method developed involves a reparameterization of the multivariable structural eigenvalue problem in terms of a single positive-valued parameter. The resulting equations yield first-order approximations to changes in the eigenvalues and the eigenvectors associated with the repeated eigenvalue problem. This work also presents a numerical technique that facilitates the definition of an eigenvector derivative for the case of repeated eigenvalues with repeated eigenvalue derivatives (of all orders). Examples are given which demonstrate the application of such equations for sensitivity and approximate analysis. Emphasis is placed on the application of sensitivity analysis to large-scale structural and controls-structures optimization problems.

The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2

NASA Technical Reports Server (NTRS)

Poole, Eugene L.; Overman, Andrea L.

1988-01-01

Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution.

A Solution Space for a System of Null-State Partial Differential Equations: Part 2

NASA Astrophysics Data System (ADS)

Flores, Steven M.; Kleban, Peter

2015-01-01

This article is the second of four that completely and rigorously characterize a solution space for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE). The system comprises 2 N null-state equations and three conformal Ward identities which govern CFT correlation functions of 2 N one-leg boundary operators. In the first article (Flores and Kleban, Commun Math Phys, arXiv:1212.2301, 2012), we use methods of analysis and linear algebra to prove that dim , with C N the Nth Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. The purpose of this article is to provide that proof. The lemma states that if every interval among ( x 2, x 3), ( x 3, x 4),…,( x 2 N-1, x 2 N ) is a two-leg interval of (defined in Flores and Kleban, Commun Math Phys, arXiv:1212.2301, 2012), then F vanishes. Proving this lemma by contradiction, we show that the existence of such a nonzero function implies the existence of a non-vanishing CFT two-point function involving primary operators with different conformal weights, an impossibility. This proof (which is rigorous in spite of our occasional reference to CFT) involves two different types of estimates, those that give the asymptotic behavior of F as the length of one interval vanishes, and those that give this behavior as the lengths of two intervals vanish simultaneously. We derive these estimates by using Green functions to rewrite certain null-state PDEs as integral equations, combining other null-state PDEs to obtain Schauder interior estimates, and then repeatedly integrating the integral equations with these estimates until we obtain optimal bounds. Estimates in which two interval lengths vanish simultaneously divide into two cases: two adjacent intervals and two non-adjacent intervals. The analysis of the latter case is similar to that for one vanishing interval length. In contrast, the analysis of the former case is more complicated, involving a Green function that contains the Jacobi heat kernel as its essential ingredient.

Compactness and robustness: Applications in the solution of integral equations for chemical kinetics and electromagnetic scattering

NASA Astrophysics Data System (ADS)

Zhou, Yajun

This thesis employs the topological concept of compactness to deduce robust solutions to two integral equations arising from chemistry and physics: the inverse Laplace problem in chemical kinetics and the vector wave scattering problem in dielectric optics. The inverse Laplace problem occurs in the quantitative understanding of biological processes that exhibit complex kinetic behavior: different subpopulations of transition events from the "reactant" state to the "product" state follow distinct reaction rate constants, which results in a weighted superposition of exponential decay modes. Reconstruction of the rate constant distribution from kinetic data is often critical for mechanistic understandings of chemical reactions related to biological macromolecules. We devise a "phase function approach" to recover the probability distribution of rate constants from decay data in the time domain. The robustness (numerical stability) of this reconstruction algorithm builds upon the continuity of the transformations connecting the relevant function spaces that are compact metric spaces. The robust "phase function approach" not only is useful for the analysis of heterogeneous subpopulations of exponential decays within a single transition step, but also is generalizable to the kinetic analysis of complex chemical reactions that involve multiple intermediate steps. A quantitative characterization of the light scattering is central to many meteoro-logical, optical, and medical applications. We give a rigorous treatment to electromagnetic scattering on arbitrarily shaped dielectric media via the Born equation: an integral equation with a strongly singular convolution kernel that corresponds to a non-compact Green operator. By constructing a quadratic polynomial of the Green operator that cancels out the kernel singularity and satisfies the compactness criterion, we reveal the universality of a real resonance mode in dielectric optics. Meanwhile, exploiting the properties of compact operators, we outline the geometric and physical conditions that guarantee a robust solution to the light scattering problem, and devise an asymptotic solution to the Born equation of electromagnetic scattering for arbitrarily shaped dielectric in a non-perturbative manner.

Introduction to Generalized Functions with Applications in Aerodynamics and Aeroacoustics

NASA Technical Reports Server (NTRS)

Farassat, F.

1994-01-01

Generalized functions have many applications in science and engineering. One useful aspect is that discontinuous functions can be handled as easily as continuous or differentiable functions and provide a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists. We define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, are derived with the generalized function theory. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic flow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics include the derivation of the Kirchhoff formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Discrete integration of continuous Kalman filtering equations for time invariant second-order structural systems

NASA Technical Reports Server (NTRS)

Park, K. C.; Belvin, W. Keith

1990-01-01

A general form for the first-order representation of the continuous second-order linear structural-dynamics equations is introduced to derive a corresponding form of first-order continuous Kalman filtering equations. Time integration of the resulting equations is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete Kalman filtering equations involving only symmetric sparse N x N solution matrices.

Integrability and structural stability of solutions to the Ginzburg-Landau equation

NASA Technical Reports Server (NTRS)

Keefe, Laurence R.

1986-01-01

The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).

Symbolic computation of recurrence equations for the Chebyshev series solution of linear ODE's. [ordinary differential equations

NASA Technical Reports Server (NTRS)

Geddes, K. O.

1977-01-01

If a linear ordinary differential equation with polynomial coefficients is converted into integrated form then the formal substitution of a Chebyshev series leads to recurrence equations defining the Chebyshev coefficients of the solution function. An explicit formula is presented for the polynomial coefficients of the integrated form in terms of the polynomial coefficients of the differential form. The symmetries arising from multiplication and integration of Chebyshev polynomials are exploited in deriving a general recurrence equation from which can be derived all of the linear equations defining the Chebyshev coefficients. Procedures for deriving the general recurrence equation are specified in a precise algorithmic notation suitable for translation into any of the languages for symbolic computation. The method is algebraic and it can therefore be applied to differential equations containing indeterminates.

Shear and compression buckling analysis for anisotropic panels with centrally located elliptical cutouts

NASA Technical Reports Server (NTRS)

Britt, V. O.

1993-01-01

An approximate analysis for buckling of biaxial- and shear-loaded anisotropic panels with centrally located elliptical cutouts is presented in the present paper. The analysis is composed of two parts, a prebuckling analysis and a buckling analysis. The prebuckling solution is determined using Lekhnitskii's complex variable equations of plane elastostatics combined with a Laurent series approximation and a boundary collocation method. The buckling solution is obtained using the principle of minimum potential energy. A by-product of the minimum potential energy equation is an integral equation which is solved using Gaussian quadrature. Comparisons with documented experimental results and finite element analyses indicate that the approximate analysis accurately predicts the buckling loads of square biaxial- and shear-loaded panels having elliptical cutouts with major axes up to sixty percent of the panel width. Results of a parametric study are presented for shear- and compression-loaded rectangular anisotropic panels with elliptical cutouts. The effects of panel aspect ratio, cutout shape, cutout size, cutout orientation, laminate anisotropy, and combined loading on the buckling load are examined.

The optimal modified variational iteration method for the Lane-Emden equations with Neumann and Robin boundary conditions

NASA Astrophysics Data System (ADS)

Singh, Randhir; Das, Nilima; Kumar, Jitendra

2017-06-01

An effective analytical technique is proposed for the solution of the Lane-Emden equations. The proposed technique is based on the variational iteration method (VIM) and the convergence control parameter h . In order to avoid solving a sequence of nonlinear algebraic or complicated integrals for the derivation of unknown constant, the boundary conditions are used before designing the recursive scheme for solution. The series solutions are found which converges rapidly to the exact solution. Convergence analysis and error bounds are discussed. Accuracy, applicability of the method is examined by solving three singular problems: i) nonlinear Poisson-Boltzmann equation, ii) distribution of heat sources in the human head, iii) second-kind Lane-Emden equation.

Closed Form Equations for the Preliminary Design of a Heat-Pipe-Cooled Leading Edge

NASA Technical Reports Server (NTRS)

Glass, David E.

1998-01-01

A set of closed form equations for the preliminary evaluation and design of a heat-pipe-cooled leading edge is presented. The set of equations can provide a leading-edge designer with a quick evaluation of the feasibility of using heat-pipe cooling. The heat pipes can be embedded in a metallic or composite structure. The maximum heat flux, total integrated heat load, and thermal properties of the structure and heat-pipe container are required input. The heat-pipe operating temperature, maximum surface temperature, heat-pipe length, and heat pipe-spacing can be estimated. Results using the design equations compared well with those from a 3-D finite element analysis for both a large and small radius leading edge.

A fully coupled variable properties thermohydraulic model for a cryogenic hydrostatic journal bearing

NASA Technical Reports Server (NTRS)

Braun, M. J.; Wheeler, R. L., III; Hendricks, R. C.

1986-01-01

The goal set forth here is to continue the work started by Braun et al. (1984-1985) and present an integrated analysis of the behavior of the two row, 20 staggered pockets, hydrostatic cryogenic bearing used by the turbopumps of the Space Shuttle main engine. The variable properties Reynolds equation is fully coupled with the two-dimensional fluid film energy equation. The three-dimensional equations of the shaft and bushing model the boundary conditions of the fluid film energy equation. The effects of shaft eccentricity, angular velocity, and inertia pressure drops at pocket edge are incorporated in the model. Their effects on the bearing fluid properties, load carrying capacity, mass flow, pressure, velocity, and temperature form the ultimate object of this paper.

Computer-Aided Design Of Turbine Blades And Vanes

NASA Technical Reports Server (NTRS)

Hsu, Wayne Q.

1988-01-01

Quasi-three-dimensional method for determining aerothermodynamic configuration of turbine uses computer-interactive analysis and design and computer-interactive graphics. Design procedure executed rapidly so designer easily repeats it to arrive at best performance, size, structural integrity, and engine life. Sequence of events in aerothermodynamic analysis and design starts with engine-balance equations and ends with boundary-layer analysis and viscous-flow calculations. Analysis-and-design procedure interactive and iterative throughout.

Effective quadrature formula in solving linear integro-differential equations of order two

NASA Astrophysics Data System (ADS)

Eshkuvatov, Z. K.; Kammuji, M.; Long, N. M. A. Nik; Yunus, Arif A. M.

2017-08-01

In this note, we solve general form of Fredholm-Volterra integro-differential equations (IDEs) of order 2 with boundary condition approximately and show that proposed method is effective and reliable. Initially, IDEs is reduced into integral equation of the third kind by using standard integration techniques and identity between multiple and single integrals then truncated Legendre series are used to estimate the unknown function. For the kernel integrals, we have applied Gauss-Legendre quadrature formula and collocation points are chosen as the roots of the Legendre polynomials. Finally, reduce the integral equations of the third kind into the system of algebraic equations and Gaussian elimination method is applied to get approximate solutions. Numerical examples and comparisons with other methods reveal that the proposed method is very effective and dominated others in many cases. General theory of existence of the solution is also discussed.

A finite-element analysis for steady and oscillatory subsonic flow around complex configurations

NASA Technical Reports Server (NTRS)

Chen, L. T.; Suciu, E. O.; Morino, L.

1974-01-01

The problem of potential subsonic flow around complex configurations is considered. The solution is given of an integral equation relating the values of the potential on the surface of the body to the values of the normal derivative, which is known from the boundary conditions. The surface of the body is divided into small (hyperboloidal quadrilateral) surface elements, which are described in terms of the Cartesian components of the four corner points. The values of the potential (and its normal derivative) within each element is assumed to be constant and equal to its value at the centroid of the element. The coefficients of the equation are given by source and doublet integrals over the surface elements. Closed form evaluations of the integrals are presented. The results obtained with the above formulation are compared with existing analytical and experimental results.

An integrable family of Monge-Ampère equations and their multi-Hamiltonian structure

NASA Astrophysics Data System (ADS)

Nutku, Y.; Sarioǧlu, Ö.

1993-02-01

We have identified a completely integrable family of Monge-Ampère equations through an examination of their Hamiltonian structure. Starting with a variational formulation of the Monge-Ampère equations we have constructed the first Hamiltonian operator through an application of Dirac's theory of constraints. The completely integrable class of Monge-Ampère equations are then obtained by solving the Jacobi identities for a sufficiently general form of the second Hamiltonian operator that is compatible with the first.

Extremely Fast Numerical Integration of Ocean Surface Wave Dynamics

DTIC Science & Technology

2007-09-30

sub-processor must be added as shown in the blue box of Fig. 1. We first consider the Kadomtsev - Petviashvili (KP) equation ηt + coηx +αηηx + βη ...analytic integration of the so-called “soliton equations ,” I have discovered how the GFT can be used to solved higher order equations for which study...analytical study and extremely fast numerical integration of the extended nonlinear Schroedinger equation for fully three dimensional wave motion

Continuous properties of the data-to-solution map for a generalized μ-Camassa-Holm integrable equation

NASA Astrophysics Data System (ADS)

Yu, Shengqi

2018-05-01

This work studies a generalized μ-type integrable equation with both quadratic and cubic nonlinearities; the μ-Camassa-Holm and modified μ-Camassa-Holm equations are members of this family of equations. It has been shown that the Cauchy problem for this generalized μ-Camassa-Holm integrable equation is locally well-posed for initial data u0 ∈ Hs, s > 5/2. In this work, we further investigate the continuity properties to this equation. It is proved in this work that the data-to-solution map of the proposed equation is not uniformly continuous. It is also found that the solution map is Hölder continuous in the Hr-topology when 0 ≤ r < s with Hölder exponent α depending on both s and r.

A family of wave equations with some remarkable properties.

PubMed

da Silva, Priscila Leal; Freire, Igor Leite; Sampaio, Júlio Cesar Santos

2018-02-01

We consider a family of homogeneous nonlinear dispersive equations with two arbitrary parameters. Conservation laws are established from the point symmetries and imply that the whole family admits square integrable solutions. Recursion operators are found for two members of the family investigated. For one of them, a Lax pair is also obtained, proving its complete integrability. From the Lax pair, we construct a Miura-type transformation relating the original equation to the Korteweg-de Vries (KdV) equation. This transformation, on the other hand, enables us to obtain solutions of the equation from the kernel of a Schrödinger operator with potential parametrized by the solutions of the KdV equation. In particular, this allows us to exhibit a kink solution to the completely integrable equation from the 1-soliton solution of the KdV equation. Finally, peakon-type solutions are also found for a certain choice of the parameters, although for this particular case the equation is reduced to a homogeneous second-order nonlinear evolution equation.

A Generalized Eulerian-Lagrangian Analysis, with Application to Liquid Flows with Vapor Bubbles

NASA Technical Reports Server (NTRS)

Dejong, Frederik J.; Meyyappan, Meyya

1993-01-01

Under a NASA MSFC SBIR Phase 2 effort an analysis has been developed for liquid flows with vapor bubbles such as those in liquid rocket engine components. The analysis is based on a combined Eulerian-Lagrangian technique, in which Eulerian conservation equations are solved for the liquid phase, while Lagrangian equations of motion are integrated in computational coordinates for the vapor phase. The novel aspect of the Lagrangian analysis developed under this effort is that it combines features of the so-called particle distribution approach with those of the so-called particle trajectory approach and can, in fact, be considered as a generalization of both of those traditional methods. The result of this generalization is a reduction in CPU time and memory requirements. Particle time step (stability) limitations have been eliminated by semi-implicit integration of the particle equations of motion (and, for certain applications, the particle temperature equation), although practical limitations remain in effect for reasons of accuracy. The analysis has been applied to the simulation of cavitating flow through a single-bladed section of a labyrinth seal. Models for the simulation of bubble formation and growth have been included, as well as models for bubble drag and heat transfer. The results indicate that bubble formation is more or less 'explosive'. for a given flow field, the number density of bubble nucleation sites is very sensitive to the vapor properties and the surface tension. The bubble motion, on the other hand, is much less sensitive to the properties, but is affected strongly by the local pressure gradients in the flow field. In situations where either the material properties or the flow field are not known with sufficient accuracy, parametric studies can be carried out rapidly to assess the effect of the important variables. Future work will include application of the analysis to cavitation in inducer flow fields.

On a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation

NASA Astrophysics Data System (ADS)

Zhao, Hai-qiong

2017-02-01

In this paper, a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation is investigated. The underlying integrable structures like the Lax pair, the infinite number of conservation laws, the Darboux-Bäcklund transformation, and the solutions are presented in the explicit form. The theory of the semi-discrete equation including integrable properties yields the corresponding theory of the continuous counterpart in the continuous limit. Finally, numerical experiments are provided to demonstrate the effectiveness of the developed integrable semi-discretization algorithms.

Calculation of Moment Matrix Elements for Bilinear Quadrilaterals and Higher-Order Basis Functions

DTIC Science & Technology

2016-01-06

methods are known as boundary integral equation (BIE) methods and the present study falls into this category. The numerical solution of the BIE is...iterated integrals. The inner integral involves the product of the free-space Green’s function for the Helmholtz equation multiplied by an appropriate...Website: http://www.wipl-d.com/ 5. Y. Zhang and T. K. Sarkar, Parallel Solution of Integral Equation -Based EM Problems in the Frequency Domain. New

Performance analysis of three dimensional integral equation computations on a massively parallel computer. M.S. Thesis

NASA Technical Reports Server (NTRS)

Logan, Terry G.

1994-01-01

The purpose of this study is to investigate the performance of the integral equation computations using numerical source field-panel method in a massively parallel processing (MPP) environment. A comparative study of computational performance of the MPP CM-5 computer and conventional Cray-YMP supercomputer for a three-dimensional flow problem is made. A serial FORTRAN code is converted into a parallel CM-FORTRAN code. Some performance results are obtained on CM-5 with 32, 62, 128 nodes along with those on Cray-YMP with a single processor. The comparison of the performance indicates that the parallel CM-FORTRAN code near or out-performs the equivalent serial FORTRAN code for some cases.

A numerical solution for two-dimensional Fredholm integral equations of the second kind with kernels of the logarithmic potential form

NASA Technical Reports Server (NTRS)

Gabrielsen, R. E.; Uenal, A.

1981-01-01

Two dimensional Fredholm integral equations with logarithmic potential kernels are numerically solved. The explicit consequence of these solutions to their true solutions is demonstrated. The results are based on a previous work in which numerical solutions were obtained for Fredholm integral equations of the second kind with continuous kernels.

Properties of the two-dimensional heterogeneous Lennard-Jones dimers: An integral equation study

PubMed Central

Urbic, Tomaz

2016-01-01

Structural and thermodynamic properties of a planar heterogeneous soft dumbbell fluid are examined using Monte Carlo simulations and integral equation theory. Lennard-Jones particles of different sizes are the building blocks of the dimers. The site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions and the thermodynamic properties. Obtained results are compared to Monte Carlo simulation data. The critical parameters for selected types of dimers were also estimated and the influence of the Lennard-Jones parameters was studied. We have also tested the correctness of the site-site integral equation theory using different closures. PMID:27875894

Collision properties of overtaking supersolitons with small amplitudes

NASA Astrophysics Data System (ADS)

Olivier, C. P.; Verheest, F.; Hereman, W. A.

2018-03-01

The collision properties of overtaking small-amplitude supersolitons are investigated for the fluid model of a plasma consisting of cold ions and two-temperature Boltzmann electrons. A reductive perturbation analysis is performed for compositional parameters near the supercritical composition. A generalized Korteweg-de Vries equation with a quartic nonlinearity is derived, referred to as the modified Gardner equation. Criteria for the existence of small-amplitude supersolitons are derived. The modified Gardner equation is shown to be not completely integrable, implying that supersoliton collisions are inelastic, as confirmed by numerical simulations. These simulations also show that supersolitons may reduce to regular solitons as a result of overtaking collisions.

Evolution of spherical cavitation bubbles: Parametric and closed-form solutions

NASA Astrophysics Data System (ADS)

Mancas, Stefan C.; Rosu, Haret C.

2016-02-01

We present an analysis of the Rayleigh-Plesset equation for a three dimensional vacuous bubble in water. In the simplest case when the effects of surface tension are neglected, the known parametric solutions for the radius and time evolution of the bubble in terms of a hypergeometric function are briefly reviewed. By including the surface tension, we show the connection between the Rayleigh-Plesset equation and Abel's equation, and obtain the parametric rational Weierstrass periodic solutions following the Abel route. In the same Abel approach, we also provide a discussion of the nonintegrable case of nonzero viscosity for which we perform a numerical integration.

Universal and integrable nonlinear evolution systems of equations in 2+1 dimensions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maccari, A.

1997-08-01

Integrable systems of nonlinear partial differential equations (PDEs) are obtained from integrable equations in 2+1 dimensions, by means of a reduction method of broad applicability based on Fourier expansion and spatio{endash}temporal rescalings, which is asymptotically exact in the limit of weak nonlinearity. The integrability by the spectral transform is explicitly demonstrated, because the corresponding Lax pairs have been derived, applying the same reduction method to the Lax pair of the initial equation. These systems of nonlinear PDEs are likely to be of applicative relevance and have a {open_quotes}universal{close_quotes} character, inasmuch as they may be derived from a very large classmore » of nonlinear evolution equations with a linear dispersive part. {copyright} {ital 1997 American Institute of Physics.}« less

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

NASA Astrophysics Data System (ADS)

Galenko, Peter K.; Alexandrov, Dmitri V.; Titova, Ekaterina A.

2018-01-01

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay-Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals. This article is part of the theme issue `From atomistic interfaces to dendritic patterns'.

Finite elements: Theory and application

NASA Technical Reports Server (NTRS)

Dwoyer, D. L. (Editor); Hussaini, M. Y. (Editor); Voigt, R. G. (Editor)

1988-01-01

Recent advances in FEM techniques and applications are discussed in reviews and reports presented at the ICASE/LaRC workshop held in Hampton, VA in July 1986. Topics addressed include FEM approaches for partial differential equations, mixed FEMs, singular FEMs, FEMs for hyperbolic systems, iterative methods for elliptic finite-element equations on general meshes, mathematical aspects of FEMS for incompressible viscous flows, and gradient weighted moving finite elements in two dimensions. Consideration is given to adaptive flux-corrected FEM transport techniques for CFD, mixed and singular finite elements and the field BEM, p and h-p versions of the FEM, transient analysis methods in computational dynamics, and FEMs for integrated flow/thermal/structural analysis.

Recent developments in rotary-wing aerodynamic theory

NASA Technical Reports Server (NTRS)

Johnson, W.

1986-01-01

Current progress in the computational analysis of rotary-wing flowfields is surveyed, and some typical results are presented in graphs. Topics examined include potential theory, rotating coordinate systems, lifting-surface theory (moving singularity, fixed wing, and rotary wing), panel methods (surface singularity representations, integral equations, and compressible flows), transonic theory (the small-disturbance equation), wake analysis (hovering rotor-wake models and transonic blade-vortex interaction), limitations on computational aerodynamics, and viscous-flow methods (dynamic-stall theories and lifting-line theory). It is suggested that the present algorithms and advanced computers make it possible to begin working toward the ultimate goal of turbulent Navier-Stokes calculations for an entire rotorcraft.

Analysis and synthesis of distributed-lumped-active networks by digital computer

NASA Technical Reports Server (NTRS)

1973-01-01

The use of digital computational techniques in the analysis and synthesis of DLA (distributed lumped active) networks is considered. This class of networks consists of three distinct types of elements, namely, distributed elements (modeled by partial differential equations), lumped elements (modeled by algebraic relations and ordinary differential equations), and active elements (modeled by algebraic relations). Such a characterization is applicable to a broad class of circuits, especially including those usually referred to as linear integrated circuits, since the fabrication techniques for such circuits readily produce elements which may be modeled as distributed, as well as the more conventional lumped and active ones.

Dispersive analysis of the scalar form factor of the nucleon

NASA Astrophysics Data System (ADS)

Hoferichter, M.; Ditsche, C.; Kubis, B.; Meißner, U.-G.

2012-06-01

Based on the recently proposed Roy-Steiner equations for pion-nucleon ( πN) scattering [1], we derive a system of coupled integral equations for the π π to overline N N and overline K K to overline N N S-waves. These equations take the form of a two-channel Muskhelishvili-Omnès problem, whose solution in the presence of a finite matching point is discussed. We use these results to update the dispersive analysis of the scalar form factor of the nucleon fully including overline K K intermediate states. In particular, we determine the correction {Δ_{σ }} = σ ( {2M_{π }^2} ) - {σ_{{π N}}} , which is needed for the extraction of the pion-nucleon σ term from πN scattering, as a function of pion-nucleon subthreshold parameters and the πN coupling constant.

Least-Squares, Continuous Sensitivity Analysis for Nonlinear Fluid-Structure Interaction

DTIC Science & Technology

2009-08-20

Tangential stress optimization convergence to uniform value  1.797  as a function of eccentric anomaly   E and Objective function value as a...up to the domain dimension, domainn . Equation (3.7) expands as truncation error round-off error decreasing step size FD e rr or 54...force, and E is Young’s modulus. Equations (3.31) and (3.32) may be directly integrated to yield the stress and displacement solutions, which, for no

Solvent dynamics and electron transfer reactions

NASA Astrophysics Data System (ADS)

Rasaiah, Jayendran C.; Zhu, Jianjun

1994-02-01

Recent experimental and theoretical studies of the influence of solvent dynamics on electron transfer (ET) reactions are discussed. It is seen that the survival probabilities of the reactants and products can be obtained as the solution to an integral equation using experimental or simulation data on the solvation dynamics. The theory developed for ET between thermally equilibrated reactants in solution, in which the ligand vibrations were treated classically, is extended to include quantum effects on the inner-shell ligand vibration and electron transfer from a nonequilibrium initial state prepared, for example, by laser excitation. This leads to a slight modification of the integral equation which is easily solved on a personal computer to provide results that can be directly compared with experiment. Analytic approximations to the solutions of the integral equation, ranging from a single exponential to multiexponential time dependence of the survival probabilities are discussed. The rate constant for the single exponential decay of the reactants interpolates between the thermal equilibrium rate constant kie (that is independent of solvent dynamics) and a diffusion controlled rate constant kid (determined by solvent dynamics) and also between the wide (A=0) and narrow (A=1) window limits dominated by inner-sphere ligand vibration and outer-sphere solvent reorganization respectively. The explicit dependence of the integral equation solutions on solvation dynamics S(t), the free energy of reaction ΔG0, the total reorganization energy λ and its partitioning between ligand vibration λq and solvent polarization fluctuations λ0, and the nature of the initial state should be useful in the analysis and design of ET experiments in different solvents.

Green function of the double-fractional Fokker-Planck equation: path integral and stochastic differential equations.

PubMed

Kleinert, H; Zatloukal, V

2013-11-01

The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.

Localization of the eigenvalues of linear integral equations with applications to linear ordinary differential equations.

NASA Technical Reports Server (NTRS)

Sloss, J. M.; Kranzler, S. K.

1972-01-01

The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.

Inverse scattering transform analysis of rogue waves using local periodization procedure

NASA Astrophysics Data System (ADS)

Randoux, Stéphane; Suret, Pierre; El, Gennady

2016-07-01

The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra.

Inverse scattering transform analysis of rogue waves using local periodization procedure

PubMed Central

Randoux, Stéphane; Suret, Pierre; El, Gennady

2016-01-01

The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays a prominent role in the modeling and understanding of the wave phenomena relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence, and the specific question of the formation of rogue waves (RWs) has been recently extensively studied in this context. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping RW events of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the RW classification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of RWs from standard breathers and their collisions to more general nonlinear modes characterized by their nonlinear spectra. PMID:27385164

Volume integral equation for electromagnetic scattering: Rigorous derivation and analysis for a set of multilayered particles with piecewise-smooth boundaries in a passive host medium

NASA Astrophysics Data System (ADS)

Yurkin, Maxim A.; Mishchenko, Michael I.

2018-04-01

We present a general derivation of the frequency-domain volume integral equation (VIE) for the electric field inside a nonmagnetic scattering object from the differential Maxwell equations, transmission boundary conditions, radiation condition at infinity, and locally-finite-energy condition. The derivation applies to an arbitrary spatially finite group of particles made of isotropic materials and embedded in a passive host medium, including those with edges, corners, and intersecting internal interfaces. This is a substantially more general type of scatterer than in all previous derivations. We explicitly treat the strong singularity of the integral kernel, but keep the entire discussion accessible to the applied scattering community. We also consider the known results on the existence and uniqueness of VIE solution and conjecture a general sufficient condition for that. Finally, we discuss an alternative way of deriving the VIE for an arbitrary object by means of a continuous transformation of the everywhere smooth refractive-index function into a discontinuous one. Overall, the paper examines and pushes forward the state-of-the-art understanding of various analytical aspects of the VIE.

Exact solutions for the static bending of Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model

NASA Astrophysics Data System (ADS)

Wang, Y. B.; Zhu, X. W.; Dai, H. H.

2016-08-01

Though widely used in modelling nano- and micro- structures, Eringen's differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

Finite-volume spectra of the Lee-Yang model

NASA Astrophysics Data System (ADS)

Bajnok, Zoltan; el Deeb, Omar; Pearce, Paul A.

2015-04-01

We consider the non-unitary Lee-Yang minimal model in three different finite geometries: (i) on the interval with integrable boundary conditions labelled by the Kac labels ( r, s) = (1 , 1) , (1 , 2), (ii) on the circle with periodic boundary conditions and (iii) on the periodic circle including an integrable purely transmitting defect. We apply φ 1,3 integrable perturbations on the boundary and on the defect and describe the flow of the spectrum. Adding a Φ1,3 integrable perturbation to move off-criticality in the bulk, we determine the finite size spectrum of the massive scattering theory in the three geometries via Thermodynamic Bethe Ansatz (TBA) equations. We derive these integral equations for all excitations by solving, in the continuum scaling limit, the TBA functional equations satisfied by the transfer matrices of the associated A 4 RSOS lattice model of Forrester and Baxter in Regime III. The excitations are classified in terms of ( m, n) systems. The excited state TBA equations agree with the previously conjectured equations in the boundary and periodic cases. In the defect case, new TBA equations confirm previously conjectured transmission factors.

Nonzero solutions of nonlinear integral equations modeling infectious disease

DOE Office of Scientific and Technical Information (OSTI.GOV)

Williams, L.R.; Leggett, R.W.

1982-01-01

Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.

Multistep integration formulas for the numerical integration of the satellite problem

NASA Technical Reports Server (NTRS)

Lundberg, J. B.; Tapley, B. D.

1981-01-01

The use of two Class 2/fixed mesh/fixed order/multistep integration packages of the PECE type for the numerical integration of the second order, nonlinear, ordinary differential equation of the satellite orbit problem. These two methods are referred to as the general and the second sum formulations. The derivation of the basic equations which characterize each formulation and the role of the basic equations in the PECE algorithm are discussed. Possible starting procedures are examined which may be used to supply the initial set of values required by the fixed mesh/multistep integrators. The results of the general and second sum integrators are compared to the results of various fixed step and variable step integrators.

Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations.

PubMed

Fu, Wei; Nijhoff, Frank W

2017-07-01

A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.

Asymptotic/numerical analysis of supersonic propeller noise

NASA Technical Reports Server (NTRS)

Myers, M. K.; Wydeven, R.

1989-01-01

An asymptotic analysis based on the Mach surface structure of the field of a supersonic helical source distribution is applied to predict thickness and loading noise radiated by high speed propeller blades. The theory utilizes an integral representation of the Ffowcs-Williams Hawkings equation in a fully linearized form. The asymptotic results are used for chordwise strips of the blade, while required spanwise integrations are performed numerically. The form of the analysis enables predicted waveforms to be interpreted in terms of Mach surface propagation. A computer code developed to implement the theory is described and found to yield results in close agreement with more exact computations.

Stability and Interaction of Coherent Structure in Supersonic Reactive Wakes

NASA Technical Reports Server (NTRS)

Menon, Suresh

1983-01-01

A theoretical formulation and analysis is presented for a study of the stability and interaction of coherent structure in reacting free shear layers. The physical problem under investigation is a premixed hydrogen-oxygen reacting shear layer in the wake of a thin flat plate. The coherent structure is modeled as a periodic disturbance and its stability is determined by the application of linearized hydrodynamic stability theory which results in a generalized eigenvalue problem for reactive flows. Detailed stability analysis of the reactive wake for neutral, symmetrical and antisymmetrical disturbance is presented. Reactive stability criteria is shown to be quite different from classical non-reactive stability. The interaction between the mean flow, coherent structure and fine-scale turbulence is theoretically formulated using the von-Kaman integral technique. Both time-averaging and conditional phase averaging are necessary to separate the three types of motion. The resulting integro-differential equations can then be solved subject to initial conditions with appropriate shape functions. In the laminar flow transition region of interest, the spatial interaction between the mean motion and coherent structure is calculated for both non-reactive and reactive conditions and compared with experimental data wherever available. The fine-scale turbulent motion determined by the application of integral analysis to the fluctuation equations. Since at present this turbulence model is still untested, turbulence is modeled in the interaction problem by a simple algebraic eddy viscosity model. The applicability of the integral turbulence model formulated here is studied parametrically by integrating these equations for the simple case of self-similar mean motion with assumed shape functions. The effect of the motion of the coherent structure is studied and very good agreement is obtained with previous experimental and theoretical works for non-reactive flow. For the reactive case, lack of experimental data made direct comparison difficult. It was determined that the growth rate of the disturbance amplitude is lower for reactive case. The results indicate that the reactive flow stability is in qualitative agreement with experimental observation.

Transient Effects in Planar Solidification of Dilute Binary Alloys

NASA Technical Reports Server (NTRS)

Mazuruk, Konstantin; Volz, Martin P.

2008-01-01

The initial transient during planar solidification of dilute binary alloys is studied in the framework of the boundary integral method that leads to the non-linear Volterra integral governing equation. An analytical solution of this equation is obtained for the case of a constant growth rate which constitutes the well-known Tiller's formula for the solute transient. The more physically relevant, constant ramping down temperature case has been studied both numerically and analytically. In particular, an asymptotic analytical solution is obtained for the initial transient behavior. A numerical technique to solve the non-linear Volterra equation is developed and the solution is obtained for a family of the governing parameters. For the rapid solidification condition, growth rate spikes have been observed even for the infinite kinetics model. When recirculating fluid flow is included into the analysis, the spike feature is dramatically diminished. Finally, we have investigated planar solidification with a fluctuating temperature field as a possible mechanism for frequently observed solute trapping bands.

Refraction of dispersive shock waves

NASA Astrophysics Data System (ADS)

El, G. A.; Khodorovskii, V. V.; Leszczyszyn, A. M.

2012-09-01

We study a dispersive counterpart of the classical gas dynamics problem of the interaction of a shock wave with a counter-propagating simple rarefaction wave, often referred to as the shock wave refraction. The refraction of a one-dimensional dispersive shock wave (DSW) due to its head-on collision with the centred rarefaction wave (RW) is considered in the framework of the defocusing nonlinear Schrödinger (NLS) equation. For the integrable cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain main physical parameters of the DSW refraction. The key features of the DSW-RW interaction predicted by our modulation theory analysis are confirmed by direct numerical solutions of the full dispersive problem.

Stability of the Superconducting d-Wave Pairing Toward the Intersite Coulomb Repulsion in CuO_2 Plane

NASA Astrophysics Data System (ADS)

Val'kov, V. V.; Dzebisashvili, D. M.; Korovushkin, M. M.; Barabanov, A. F.

2018-06-01

Taking into account the real crystalline structure of the CuO_2 plane and the strong spin-fermion coupling, we study the influence of the intersite Coulomb repulsion between holes on the Cooper instability of the spin-polaron quasiparticles in cuprate superconductors. The analysis shows that only the superconducting d-wave pairing is implemented in the whole region of doping, whereas the solutions of the self-consistent equations for the s-wave pairing are absent. It is shown that intersite Coulomb interaction V_1 between the holes located at the nearest oxygen ions does not affect the d-wave pairing, because its Fourier transform V_q vanishes in the kernel of the corresponding integral equation. The intersite Coulomb interaction V_2 of quasiparticles located at the next-nearest oxygen ions does not vanish in the integral equations, however, but it is also shown that the d-wave pairing is robust toward this interaction for physically reasonable values of V_2.

Stability of the Superconducting d-Wave Pairing Toward the Intersite Coulomb Repulsion in CuO_2 Plane

NASA Astrophysics Data System (ADS)

Val'kov, V. V.; Dzebisashvili, D. M.; Korovushkin, M. M.; Barabanov, A. F.

2018-03-01

Taking into account the real crystalline structure of the CuO_2 plane and the strong spin-fermion coupling, we study the influence of the intersite Coulomb repulsion between holes on the Cooper instability of the spin-polaron quasiparticles in cuprate superconductors. The analysis shows that only the superconducting d-wave pairing is implemented in the whole region of doping, whereas the solutions of the self-consistent equations for the s-wave pairing are absent. It is shown that intersite Coulomb interaction V_1 between the holes located at the nearest oxygen ions does not affect the d-wave pairing, because its Fourier transform V_q vanishes in the kernel of the corresponding integral equation. The intersite Coulomb interaction V_2 of quasiparticles located at the next-nearest oxygen ions does not vanish in the integral equations, however, but it is also shown that the d-wave pairing is robust toward this interaction for physically reasonable values of V_2.

Integrability of the Kruskal--Zabusky Discrete Equation by Multiscale Expansion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Levi, Decio; Scimiterna, Christian

2010-03-08

In 1965 Kruskal and Zabusky in a very famous article in Physical Review Letters introduced the notion of 'soliton' to describe the interaction of solitary waves solutions of the Korteweg de Vries equation (KdV). To do so they introduced a discrete approximation to the KdV, the Kruskal-Zabusky equation (KZ). Here we analyze the KZ equation using the multiscale expansion and show that the equation is only A{sub 2} integrable.

Penalized differential pathway analysis of integrative oncogenomics studies.

PubMed

van Wieringen, Wessel N; van de Wiel, Mark A

2014-04-01

Through integration of genomic data from multiple sources, we may obtain a more accurate and complete picture of the molecular mechanisms underlying tumorigenesis. We discuss the integration of DNA copy number and mRNA gene expression data from an observational integrative genomics study involving cancer patients. The two molecular levels involved are linked through the central dogma of molecular biology. DNA copy number aberrations abound in the cancer cell. Here we investigate how these aberrations affect gene expression levels within a pathway using observational integrative genomics data of cancer patients. In particular, we aim to identify differential edges between regulatory networks of two groups involving these molecular levels. Motivated by the rate equations, the regulatory mechanism between DNA copy number aberrations and gene expression levels within a pathway is modeled by a simultaneous-equations model, for the one- and two-group case. The latter facilitates the identification of differential interactions between the two groups. Model parameters are estimated by penalized least squares using the lasso (L1) penalty to obtain a sparse pathway topology. Simulations show that the inclusion of DNA copy number data benefits the discovery of gene-gene interactions. In addition, the simulations reveal that cis-effects tend to be over-estimated in a univariate (single gene) analysis. In the application to real data from integrative oncogenomic studies we show that inclusion of prior information on the regulatory network architecture benefits the reproducibility of all edges. Furthermore, analyses of the TP53 and TGFb signaling pathways between ER+ and ER- samples from an integrative genomics breast cancer study identify reproducible differential regulatory patterns that corroborate with existing literature.

Integrability and Poisson Structures of Three Dimensional Dynamical Systems and Equations of Hydrodynamic Type

NASA Astrophysics Data System (ADS)

Gumral, Hasan

Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. We show that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a non-trivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of 3-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sl_2 structure is a quadratic unfolding of an integrable 1-form in 3 + 1 dimensions. We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation and a continuum limit of Toda lattice. We present further infinite sequences of conserved quantities for shallow water equations and show that their generalizations by Kodama admit bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.

Variational methods for direct/inverse problems of atmospheric dynamics and chemistry

NASA Astrophysics Data System (ADS)

Penenko, Vladimir; Penenko, Alexey; Tsvetova, Elena

2013-04-01

We present a variational approach for solving direct and inverse problems of atmospheric hydrodynamics and chemistry. It is important that the accurate matching of numerical schemes has to be provided in the chain of objects: direct/adjoint problems - sensitivity relations - inverse problems, including assimilation of all available measurement data. To solve the problems we have developed a new enhanced set of cost-effective algorithms. The matched description of the multi-scale processes is provided by a specific choice of the variational principle functionals for the whole set of integrated models. Then all functionals of variational principle are approximated in space and time by splitting and decomposition methods. Such approach allows us to separately consider, for example, the space-time problems of atmospheric chemistry in the frames of decomposition schemes for the integral identity sum analogs of the variational principle at each time step and in each of 3D finite-volumes. To enhance the realization efficiency, the set of chemical reactions is divided on the subsets related to the operators of production and destruction. Then the idea of the Euler's integrating factors is applied in the frames of the local adjoint problem technique [1]-[3]. The analytical solutions of such adjoint problems play the role of integrating factors for differential equations describing atmospheric chemistry. With their help, the system of differential equations is transformed to the equivalent system of integral equations. As a result we avoid the construction and inversion of preconditioning operators containing the Jacobi matrixes which arise in traditional implicit schemes for ODE solution. This is the main advantage of our schemes. At the same time step but on the different stages of the "global" splitting scheme, the system of atmospheric dynamic equations is solved. For convection - diffusion equations for all state functions in the integrated models we have developed the monotone and stable discrete-analytical numerical schemes [1]-[3] conserving the positivity of the chemical substance concentrations and possessing the properties of energy and mass balance that are postulated in the general variational principle for integrated models. All algorithms for solution of transport, diffusion and transformation problems are direct (without iterations). The work is partially supported by the Programs No 4 of Presidium RAS and No 3 of Mathematical Department of RAS, by RFBR project 11-01-00187 and Integrating projects of SD RAS No 8 and 35. Our studies are in the line with the goals of COST Action ES1004. References Penenko V., Tsvetova E. Discrete-analytical methods for the implementation of variational principles in environmental applications// Journal of computational and applied mathematics, 2009, v. 226, 319-330. Penenko A.V. Discrete-analytic schemes for solving an inverse coefficient heat conduction problem in a layered medium with gradient methods// Numerical Analysis and Applications, 2012, V. 5, pp 326-341. V. Penenko, E. Tsvetova. Variational methods for constructing the monotone approximations for atmospheric chemistry models //Numerical Analysis and Applications, 2013 (in press).

Time-dependent integral equations of neutron transport for calculating the kinetics of nuclear reactors by the Monte Carlo method

DOE Office of Scientific and Technical Information (OSTI.GOV)

Davidenko, V. D., E-mail: Davidenko-VD@nrcki.ru; Zinchenko, A. S., E-mail: zin-sn@mail.ru; Harchenko, I. K.

2016-12-15

Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.

On the solution of integral equations with a generalized cauchy kernal

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1986-01-01

A certain class of singular integral equations that may arise from the mixed boundary value problems in nonhonogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernal has strong singularities of the form (t-x)(-2), x(n-2) (t+x)(n), (n is = or 2, 0 x, t b). The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.

On the solution of integral equations with strongly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1986-01-01

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

On the solution of integral equations with strong ly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1985-01-01

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

On the solution of integral equations with strongly singular kernels

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1987-01-01

Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Urbic, Tomaz, E-mail: tomaz.urbic@fkkt.uni-lj.si; Dias, Cristiano L.

The thermodynamic and structural properties of the planar soft-sites dumbbell fluid are examined by Monte Carlo simulations and integral equation theory. The dimers are built of two Lennard-Jones segments. Site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions for a range of elongations and densities and the results are compared with Monte Carlo simulations. The critical parameters for selected types of dimers were also estimated. We analyze the influence of the bond length on critical point as well as tested correctness of site-site integral equation theory with different closures. The integral equations canmore » be used to predict the phase diagram of dimers whose molecular parameters are known.« less

Numerical techniques in radiative heat transfer for general, scattering, plane-parallel media

NASA Technical Reports Server (NTRS)

Sharma, A.; Cogley, A. C.

1982-01-01

The study of radiative heat transfer with scattering usually leads to the solution of singular Fredholm integral equations. The present paper presents an accurate and efficient numerical method to solve certain integral equations that govern radiative equilibrium problems in plane-parallel geometry for both grey and nongrey, anisotropically scattering media. In particular, the nongrey problem is represented by a spectral integral of a system of nonlinear integral equations in space, which has not been solved previously. The numerical technique is constructed to handle this unique nongrey governing equation as well as the difficulties caused by singular kernels. Example problems are solved and the method's accuracy and computational speed are analyzed.

Generalization of Boundary-Layer Momentum-Integral Equations to Three-Dimensional Flows Including Those of Rotating System

NASA Technical Reports Server (NTRS)

Mager, Arthur

1952-01-01

The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.

Evaluation of atomic pressure in the multiple time-step integration algorithm.

PubMed

Andoh, Yoshimichi; Yoshii, Noriyuki; Yamada, Atsushi; Okazaki, Susumu

2017-04-15

In molecular dynamics (MD) calculations, reduction in calculation time per MD loop is essential. A multiple time-step (MTS) integration algorithm, the RESPA (Tuckerman and Berne, J. Chem. Phys. 1992, 97, 1990-2001), enables reductions in calculation time by decreasing the frequency of time-consuming long-range interaction calculations. However, the RESPA MTS algorithm involves uncertainties in evaluating the atomic interaction-based pressure (i.e., atomic pressure) of systems with and without holonomic constraints. It is not clear which intermediate forces and constraint forces in the MTS integration procedure should be used to calculate the atomic pressure. In this article, we propose a series of equations to evaluate the atomic pressure in the RESPA MTS integration procedure on the basis of its equivalence to the Velocity-Verlet integration procedure with a single time step (STS). The equations guarantee time-reversibility even for the system with holonomic constrants. Furthermore, we generalize the equations to both (i) arbitrary number of inner time steps and (ii) arbitrary number of force components (RESPA levels). The atomic pressure calculated by our equations with the MTS integration shows excellent agreement with the reference value with the STS, whereas pressures calculated using the conventional ad hoc equations deviated from it. Our equations can be extended straightforwardly to the MTS integration algorithm for the isothermal NVT and isothermal-isobaric NPT ensembles. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

Computational flow development for unsteady viscous flows: Foundation of the numerical method

NASA Technical Reports Server (NTRS)

Bratanow, T.; Spehert, T.

1978-01-01

A procedure is presented for effective consideration of viscous effects in computational development of high Reynolds number flows. The procedure is based on the interpretation of the Navier-Stokes equations as vorticity transport equations. The physics of the flow was represented in a form suitable for numerical analysis. Lighthill's concept for flow development for computational purposes was adapted. The vorticity transport equations were cast in a form convenient for computation. A statement for these equations was written using the method of weighted residuals and applying the Galerkin criterion. An integral representation of the induced velocity was applied on the basis of the Biot-Savart law. Distribution of new vorticity, produced at wing surfaces over small computational time intervals, was assumed to be confined to a thin region around the wing surfaces.

Symmetry Analysis of Gauge-Invariant Field Equations via a Generalized Harrison-Estabrook Formalism.

NASA Astrophysics Data System (ADS)

Papachristou, Costas J.

The Harrison-Estabrook formalism for the study of invariance groups of partial differential equations is generalized and extended to equations that define, through their solutions, sections on vector bundles of various kinds. Applications include the Dirac, Yang-Mills, and self-dual Yang-Mills (SDYM) equations. The latter case exhibits interesting connections between the internal symmetries of SDYM and the existence of integrability characteristics such as a linear ("inverse scattering") system and Backlund transformations (BT's). By "verticalizing" the generators of coordinate point transformations of SDYM, nine nonlocal, generalized (as opposed to local, point) symmetries are constructed. The observation is made that the prolongations of these symmetries are parametric BT's for SDYM. It is thus concluded that the entire point group of SDYM contributes, upon verticalization, BT's to the system.

Some Exact Solutions of a Nonintegrable Toda-type Equation

NASA Astrophysics Data System (ADS)

Kim, Chanju

2018-05-01

We study a Toda-type equation with two scalar fields which is not integrable and construct two families of exact solutions which are expressed in terms of rational functions. The equation appears in U(1) Chern-Simons theories coupled to two nonrelativistic matter fields with opposite charges. One family of solutions is a trivial embedding of Liouville-type solutions. The other family is obtained by transforming the equation into the Taubes vortex equation on the hyperbolic space. Though the Taubes equation is not integrable, a trivial vacuum solution provides nontrivial solutions to the original Toda-type equation.

Painlevé equations, elliptic integrals and elementary functions

NASA Astrophysics Data System (ADS)

Żołądek, Henryk; Filipuk, Galina

2015-02-01

The six Painlevé equations can be written in the Hamiltonian form, with time dependent Hamilton functions. We present a rather new approach to this result, leading to rational Hamilton functions. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems with two degrees of freedom. We realize the Bäcklund transformations of the Painlevé equations as symplectic birational transformations in C4 and we interpret the cases with classical solutions as the cases of partial integrability of the extended Hamiltonian systems. We prove that the extended Hamiltonian systems do not have any additional algebraic first integral besides the known special cases of the third and fifth Painlevé equations. We also show that the original Painlevé equations admit the first integrals expressed in terms of the elementary functions only in the special cases mentioned above. In the proofs we use equations in variations with respect to a parameter and Liouville's theory of elementary functions.

External validation of a forest inventory and analysis volume equation and comparisons with estimates from multiple stem-profile models

Treesearch

Christopher M. Oswalt; Adam M. Saunders

2009-01-01

Sound estimation procedures are desideratum for generating credible population estimates to evaluate the status and trends in resource conditions. As such, volume estimation is an integral component of the U.S. Department of Agriculture, Forest Service, Forest Inventory and Analysis (FIA) program's reporting. In effect, reliable volume estimation procedures are...

A Comparison of Experimental and Theoretical Results for Labyrinth Gas Seals. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Scharrer, Joseph Kirk

1987-01-01

The basic equations are derived for a two control volume model for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent and isoenergetic. The wall friction factors are determined using the Blasius formula. Jet flow theory is used for the calculation of the recirculation velocity in the cavity. Linearized zeroth and first order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth order pressure distribution is found by satisfying the leakage equation. The circumferential velocity distribution is determined by satisfying the momentum equations. The first order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to experimental test results.

A Super mKdV Equation: Bosonization, Painlevé Property and Exact Solutions

NASA Astrophysics Data System (ADS)

Ren, Bo; Lou, Sen-Yue

2018-04-01

The symmetry of the fermionic field is obtained by means of the Lax pair of the mKdV equation. A new super mKdV equation is constructed by virtue of the symmetry of the fermionic form. The super mKdV system is changed to a system of coupled bosonic equations with the bosonization approach. The bosonized SmKdV (BSmKdV) equation admits Painlevé property by the standard singularity analysis. The traveling wave solutions of the BSmKdV system are presented by the mapping and deformation method. We also provide other ideas to construct new super integrable systems. Supported by the National Natural Science Foundation of China under Grant Nos. 11775146, 11435005, and 11472177, Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No. ZF1213 and K. C. Wong Magna Fund in Ningbo University

A nodal domain theorem for integrable billiards in two dimensions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Samajdar, Rhine; Jain, Sudhir R., E-mail: srjain@barc.gov.in

Eigenfunctions of integrable planar billiards are studied — in particular, the number of nodal domains, ν of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrödinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, ν satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of mmodkn, given a particular k, for a set of quantum numbers, m,n. Further, we observe that the patterns in a familymore » are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations. - Highlights: • We find that the number of nodal domains of eigenfunctions of integrable, planar billiards satisfy a class of difference equations. • The eigenfunctions labelled by quantum numbers (m,n) can be classified in terms of mmodkn. • A theorem is presented, realising algebraic representations of geometrical patterns exhibited by the domains. • This work presents a connection between integrable systems and difference equations.« less

Soliton solutions, stability analysis and conservation laws for the brusselator reaction diffusion model with time- and constant-dependent coefficients

NASA Astrophysics Data System (ADS)

Inc, Mustafa; Yusuf, Abdullahi; Isa Aliyu, Aliyu; Hashemi, M. S.

2018-05-01

This paper studies the brusselator reaction diffusion model (BRDM) with time- and constant-dependent coefficients. The soliton solutions for BRDM with time-dependent coefficients are obtained via first integral (FIM), ansatz, and sine-Gordon expansion (SGEM) methods. Moreover, it is well known that stability analysis (SA), symmetry analysis and conservation laws (CLs) give several information for modelling a system of differential equations (SDE). This is because they can be used for investigating the internal properties, existence, uniqueness and integrability of different SDE. For this reason, we investigate the SA via linear stability technique, symmetry analysis and CLs for BRDM with constant-dependent coefficients in order to extract more physics and information on the governing equation. The constraint conditions for the existence of the solutions are also examined. The new solutions obtained in this paper can be useful for describing the concentrations of diffusion problems of the BRDM. It is shown that the examined dependent coefficients are some of the factors that are affecting the diffusion rate. So, the present paper provides much motivational information in comparison to the existing results in the literature.

Integrated Force Method Solution to Indeterminate Structural Mechanics Problems

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Hopkins, Dale A.; Halford, Gary R.

2004-01-01

Strength of materials problems have been classified into determinate and indeterminate problems. Determinate analysis primarily based on the equilibrium concept is well understood. Solutions of indeterminate problems required additional compatibility conditions, and its comprehension was not exclusive. A solution to indeterminate problem is generated by manipulating the equilibrium concept, either by rewriting in the displacement variables or through the cutting and closing gap technique of the redundant force method. Compatibility improvisation has made analysis cumbersome. The authors have researched and understood the compatibility theory. Solutions can be generated with equal emphasis on the equilibrium and compatibility concepts. This technique is called the Integrated Force Method (IFM). Forces are the primary unknowns of IFM. Displacements are back-calculated from forces. IFM equations are manipulated to obtain the Dual Integrated Force Method (IFMD). Displacement is the primary variable of IFMD and force is back-calculated. The subject is introduced through response variables: force, deformation, displacement; and underlying concepts: equilibrium equation, force deformation relation, deformation displacement relation, and compatibility condition. Mechanical load, temperature variation, and support settling are equally emphasized. The basic theory is discussed. A set of examples illustrate the new concepts. IFM and IFMD based finite element methods are introduced for simple problems.

Active flow control insight gained from a modified integral boundary layer equation

NASA Astrophysics Data System (ADS)

Seifert, Avraham

2016-11-01

Active Flow Control (AFC) can alter the development of boundary layers with applications (e.g., reducing drag by separation delay or separating the boundary layers and enhancing vortex shedding to increase drag). Historically, significant effects of steady AFC methods were observed. Unsteady actuation is significantly more efficient than steady. Full-scale AFC tests were conducted with varying levels of success. While clearly relevant to industry, AFC implementation relies on expert knowledge with proven intuition and or costly and lengthy computational efforts. This situation hinders the use of AFC while simple, quick and reliable design method is absent. An updated form of the unsteady integral boundary layer (UIBL) equations, that include AFC terms (unsteady wall transpiration and body forces) can be used to assist in AFC analysis and design. With these equations and given a family of suitable velocity profiles, the momentum thickness can be calculated and matched with an outer, potential flow solution in 2D and 3D manner to create an AFC design tool, parallel to proven tools for airfoil design. Limiting cases of the UIBL equation can be used to analyze candidate AFC concepts in terms of their capability to modify the boundary layers development and system performance.

Numerical simulation of rarefied gas flow through a slit

NASA Technical Reports Server (NTRS)

Keith, Theo G., Jr.; Jeng, Duen-Ren; De Witt, Kenneth J.; Chung, Chan-Hong

1990-01-01

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas from one reservoir to another through a two-dimensional slit. The cases considered are for hard vacuum downstream pressure, finite pressure ratios, and isobaric pressure with thermal diffusion, which are not well established in spite of the simplicity of the flow field. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, three kinds of collision sampling techniques, the time counter (TC) method, the null collision (NC) method, and the no time counter (NTC) method, are used.

FDDO and DSMC analyses of rarefied gas flow through 2D nozzles

NASA Technical Reports Server (NTRS)

Chung, Chan-Hong; De Witt, Kenneth J.; Jeng, Duen-Ren; Penko, Paul F.

1992-01-01

Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.

Iso-chemical potential trajectories in the P-T plane for He II

NASA Technical Reports Server (NTRS)

Maytal, B.; Nissen, J. A.; Van Sciver, S. W.

1990-01-01

Trajectories of constant chemical potential in the P-T plane serve as an integral formulation of London's equation. The trajectories are useful for analysis and synthesis of fountain effect pump performance. A family of trajectories is generated from available numerical codes.

Discrete homotopy analysis for optimal trading execution with nonlinear transient market impact

NASA Astrophysics Data System (ADS)

Curato, Gianbiagio; Gatheral, Jim; Lillo, Fabrizio

2016-10-01

Optimal execution in financial markets is the problem of how to trade a large quantity of shares incrementally in time in order to minimize the expected cost. In this paper, we study the problem of the optimal execution in the presence of nonlinear transient market impact. Mathematically such problem is equivalent to solve a strongly nonlinear integral equation, which in our model is a weakly singular Urysohn equation of the first kind. We propose an approach based on Homotopy Analysis Method (HAM), whereby a well behaved initial trading strategy is continuously deformed to lower the expected execution cost. Specifically, we propose a discrete version of the HAM, i.e. the DHAM approach, in order to use the method when the integrals to compute have no closed form solution. We find that the optimal solution is front loaded for concave instantaneous impact even when the investor is risk neutral. More important we find that the expected cost of the DHAM strategy is significantly smaller than the cost of conventional strategies.

Solving Ordinary Differential Equations

NASA Technical Reports Server (NTRS)

Krogh, F. T.

1987-01-01

Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.

FAST TRACK COMMUNICATION: On the Liouvillian solution of second-order linear differential equations and algebraic invariant curves

NASA Astrophysics Data System (ADS)

Man, Yiu-Kwong

2010-10-01

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle-Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

On randomized algorithms for numerical solution of applied Fredholm integral equations of the second kind

NASA Astrophysics Data System (ADS)

Voytishek, Anton V.; Shipilov, Nikolay M.

2017-11-01

In this paper, the systematization of numerical (implemented on a computer) randomized functional algorithms for approximation of a solution of Fredholm integral equation of the second kind is carried out. Wherein, three types of such algorithms are distinguished: the projection, the mesh and the projection-mesh methods. The possibilities for usage of these algorithms for solution of practically important problems is investigated in detail. The disadvantages of the mesh algorithms, related to the necessity of calculation values of the kernels of integral equations in fixed points, are identified. On practice, these kernels have integrated singularities, and calculation of their values is impossible. Thus, for applied problems, related to solving Fredholm integral equation of the second kind, it is expedient to use not mesh, but the projection and the projection-mesh randomized algorithms.

Initial-boundary value problems associated with the Ablowitz-Ladik system

NASA Astrophysics Data System (ADS)

Xia, Baoqiang; Fokas, A. S.

2018-02-01

We employ the Ablowitz-Ladik system as an illustrative example in order to demonstrate how to analyze initial-boundary value problems for integrable nonlinear differential-difference equations via the unified transform (Fokas method). In particular, we express the solutions of the integrable discrete nonlinear Schrödinger and integrable discrete modified Korteweg-de Vries equations in terms of the solutions of appropriate matrix Riemann-Hilbert problems. We also discuss in detail, for both the above discrete integrable equations, the associated global relations and the process of eliminating of the unknown boundary values.

Steady/unsteady aerodynamic analysis of wings at subsonic, sonic and supersonic Mach numbers using a 3D panel method

NASA Astrophysics Data System (ADS)

Cho, Jeonghyun; Han, Cheolheui; Cho, Leesang; Cho, Jinsoo

2003-08-01

This paper treats the kernel function of an integral equation that relates a known or prescribed upwash distribution to an unknown lift distribution for a finite wing. The pressure kernel functions of the singular integral equation are summarized for all speed range in the Laplace transform domain. The sonic kernel function has been reduced to a form, which can be conveniently evaluated as a finite limit from both the subsonic and supersonic sides when the Mach number tends to one. Several examples are solved including rectangular wings, swept wings, a supersonic transport wing and a harmonically oscillating wing. Present results are given with other numerical data, showing continuous results through the unit Mach number. Computed results are in good agreement with other numerical results.

Analysis of a Generally Oriented Crack in a Functionally Graded Strip Sandwiched Between Two Homogeneous Half Planes

NASA Technical Reports Server (NTRS)

Shbeeb, N.; Binienda, W. K.; Kreider, K.

1999-01-01

The driving forces for a generally oriented crack embedded in a Functionally Graded strip sandwiched between two half planes are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various nonhomogeneity ratios, crack lengths. crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.

Simulation electromagnetic scattering on bodies through integral equation and neural networks methods

NASA Astrophysics Data System (ADS)

Lvovich, I. Ya; Preobrazhenskiy, A. P.; Choporov, O. N.

2018-05-01

The paper deals with the issue of electromagnetic scattering on a perfectly conducting diffractive body of a complex shape. Performance calculation of the body scattering is carried out through the integral equation method. Fredholm equation of the second time was used for calculating electric current density. While solving the integral equation through the moments method, the authors have properly described the core singularity. The authors determined piecewise constant functions as basic functions. The chosen equation was solved through the moments method. Within the Kirchhoff integral approach it is possible to define the scattered electromagnetic field, in some way related to obtained electrical currents. The observation angles sector belongs to the area of the front hemisphere of the diffractive body. To improve characteristics of the diffractive body, the authors used a neural network. All the neurons contained a logsigmoid activation function and weighted sums as discriminant functions. The paper presents the matrix of weighting factors of the connectionist model, as well as the results of the optimized dimensions of the diffractive body. The paper also presents some basic steps in calculation technique of the diffractive bodies, based on the combination of integral equation and neural networks methods.

On the solution of integral equations with a generalized cauchy kernel

NASA Technical Reports Server (NTRS)

Kaya, A. C.; Erdogan, F.

1986-01-01

In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form (t-x) sup-2, x sup n-2 (t+x) sup n, (n or = 2, 0x,tb). The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.

An exterior Poisson solver using fast direct methods and boundary integral equations with applications to nonlinear potential flow

NASA Technical Reports Server (NTRS)

Young, D. P.; Woo, A. C.; Bussoletti, J. E.; Johnson, F. T.

1986-01-01

A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.

A remark on fractional differential equation involving I-function

NASA Astrophysics Data System (ADS)

Mishra, Jyoti

2018-02-01

The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.

MEGNO-analysis of the orbital evolution of GEO objects. (Russian Title: MEGNO-анализ орбитальной эволюции объектов зоны ГЕО )

NASA Astrophysics Data System (ADS)

Aleksandrova, A. G.; Chuvashov, I. N.; Bordovitsyna, T. V.

2011-07-01

The results of investigations of the instability of orbits in the GEO are presentеd. Average parameter MEGNO as main indicator of chaotic state has been used. The parameter is computed by combined numerical integration of equations of the motion, equations in variation and equations of MEGNO parameters. The results have been obtained using software package "Numerical model of the systems artificial satellite motion", implemented on the cluster "Skiff Cyberia".

Thermodynamic energy balance equations for Space Shuttle Orbiter gas compartment during ascent and re-entry

NASA Technical Reports Server (NTRS)

Ting, P. C.

1982-01-01

Thermodynamic energy balance equations are derived and applied to midsection Orbiter-payload atmospheric thermal math models (TMMs) to predict Orbiter component, element, compartment, internal insolation and structure temperatures in support of NASA/JSC mission planning, postflight thermal analysis and payload thermal integration planning. The equations are extended and applied to the forward section, midsection, and aft section of the TMMs for five Orbiter mission phases: prelaunch on pad with purge, lift-off to ascent, re-entry to touchdown, post landing without purge, and post-landing with purge. Predicted results from the 390 node/DFI atmospheric TMM are in good agreement with STS-1 flight measurement data.

Alien calculus and a Schwinger-Dyson equation: two-point function with a nonperturbative mass scale

NASA Astrophysics Data System (ADS)

Bellon, Marc P.; Clavier, Pierre J.

2018-02-01

Starting from the Schwinger-Dyson equation and the renormalization group equation for the massless Wess-Zumino model, we compute the dominant nonperturbative contributions to the anomalous dimension of the theory, which are related by alien calculus to singularities of the Borel transform on integer points. The sum of these dominant contributions has an analytic expression. When applied to the two-point function, this analysis gives a tame evolution in the deep euclidean domain at this approximation level, making doubtful the arguments on the triviality of the quantum field theory with positive β -function. On the other side, we have a singularity of the propagator for timelike momenta of the order of the renormalization group invariant scale of the theory, which has a nonperturbative relationship with the renormalization point of the theory. All these results do not seem to have an interpretation in terms of semiclassical analysis of a Feynman path integral.

DEAN: A program for dynamic engine analysis

NASA Technical Reports Server (NTRS)

Sadler, G. G.; Melcher, K. J.

1985-01-01

The Dynamic Engine Analysis program, DEAN, is a FORTRAN code implemented on the IBM/370 mainframe at NASA Lewis Research Center for digital simulation of turbofan engine dynamics. DEAN is an interactive program which allows the user to simulate engine subsystems as well as a full engine systems with relative ease. The nonlinear first order ordinary differential equations which define the engine model may be solved by one of four integration schemes, a second order Runge-Kutta, a fourth order Runge-Kutta, an Adams Predictor-Corrector, or Gear's method for still systems. The numerical data generated by the model equations are displayed at specified intervals between which the user may choose to modify various parameters affecting the model equations and transient execution. Following the transient run, versatile graphics capabilities allow close examination of the data. DEAN's modeling procedure and capabilities are demonstrated by generating a model of simple compressor rig.

Numerical analysis for the stick-slip vibration of a transversely moving beam in contact with a frictional wall

NASA Astrophysics Data System (ADS)

Won, Hong-In; Chung, Jintai

2018-04-01

This paper presents a numerical analysis for the stick-slip vibration of a transversely moving beam, considering both stick-slip transition and friction force discontinuity. The dynamic state of the beam was separated into the stick state and the slip state, and boundary conditions were defined for both. By applying the finite element method, two matrix-vector equations were derived: one for stick state and the other for slip state. However, the equations have different degrees of freedom depending on whether the end of a beam sticks or slips, so we encountered difficulties in time integration. To overcome the difficulties, we proposed a new numerical technique to alternatively use the matrix-vector equations with different matrix sizes. In addition, to eliminate spurious high-frequency responses, we applied the generalized-α time integration method with appropriate value of high-frequency numerical dissipation. Finally, the dynamic responses of stick-slip vibration were analyzed in time and frequency domains: the dynamic behavior of the beam was explained to facilitate understanding of the stick-slip motion, and frequency characteristics of the stick-slip vibration were investigated in relation to the natural frequencies of the beam. The effects of the axial load and the moving speed upon the dynamic response were also examined.

Photodiode design study. Final report, May--December 1977

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lamorte, M.F.

1977-12-01

The purpose of this work was to apply the analytical method developed for single junction and multijunction solar cells, Contract No. F33615-76-C-1283, to photodiodes and avalanche photodiodes. It was anticipated that this analytical method will advance the state-of-the-art because of the following: (1) the analysis considers the total photodetector multilayer structure rather than just the depleted region; (2) a model of the complete band structure is analyzed; (3) application of the integral form of the continuity equation is used; (4) structures that reduce dark current and/or increase the ratio of photocurrent to dark current are obtained; and (5) structures thatmore » increase spectral response in the depleted region and reduce response in other regions of the diode are obtained. The integral form of the continuity equation developed for solar cells is the steady-state or time-independent form. The contract specified that the time-independent equation would only be employed to determine applicability to photodetectors. The GaAsSb photodiode under development at Rockwell International, Thousand Oaks, California was used to determine the applicability to photodetectors. The diode structure is composed of four layers grown on a substrate. The analysis presents calculations of spectral response. This parameter is used in this study to optimize the structure.« less

Davidenko’s Method for the Solution of Nonlinear Operator Equations.

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, NUMERICAL INTEGRATION), OPERATORS(MATHEMATICS), BANACH SPACE , MAPPING (TRANSFORMATIONS), NUMERICAL METHODS AND PROCEDURES, INTEGRALS, SET THEORY, CONVERGENCE, MATRICES(MATHEMATICS)

Integrable Equations in Multi-Dimensions (2+1) are Bi-Hamiltonian Systems,

DTIC Science & Technology

1987-02-01

equation [18]. It should be noted that the 80 equation has more similarities [19] with the Kadomtsev - Petviashvili (KP...Cimento, 39B, 1 (1977). [31] P. Caudrey, Discrete and Periodic Spectral Transforms Related to the Kadomtsev - Petviashvili Equation , preprint U.M.I.S.T. (1985). II ’AI D p-I 4, - -- - -- - - -w 4 ...TOM NONLINEAR STUDIES IDTIC I IELEC )// MAR 2 51988 I / \\ / Integrable Equations in Multi- dimensions (2+1) are Bi-Hamiltonian Systems by A.S.

The non-autonomous YdKN equation and generalized symmetries of Boll equations

NASA Astrophysics Data System (ADS)

Gubbiotti, G.; Scimiterna, C.; Levi, D.

2017-05-01

In this paper, we study the integrability of a class of nonlinear non-autonomous quad graph equations compatible around the cube introduced by Boll in the framework of the generalized Adler, Bobenko, and Suris (ABS) classification. We show that all these equations possess three-point generalized symmetries which are subcases of either the Yamilov discretization of the Krichever-Novikov equation or of its non-autonomous extension. We also prove that all those symmetries are integrable as they pass the algebraic entropy test.

Development and application of a local linearization algorithm for the integration of quaternion rate equations in real-time flight simulation problems

NASA Technical Reports Server (NTRS)

Barker, L. E., Jr.; Bowles, R. L.; Williams, L. H.

1973-01-01

High angular rates encountered in real-time flight simulation problems may require a more stable and accurate integration method than the classical methods normally used. A study was made to develop a general local linearization procedure of integrating dynamic system equations when using a digital computer in real-time. The procedure is specifically applied to the integration of the quaternion rate equations. For this application, results are compared to a classical second-order method. The local linearization approach is shown to have desirable stability characteristics and gives significant improvement in accuracy over the classical second-order integration methods.

Solving Simple Kinetics without Integrals

ERIC Educational Resources Information Center

de la Pen~a, Lisandro Herna´ndez

2016-01-01

The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…

Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions.

PubMed

Ankiewicz, Adrian; Wang, Yan; Wabnitz, Stefan; Akhmediev, Nail

2014-01-01

We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota and the Lakshmanan-Porsezian-Daniel equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue wave solutions of the corresponding equations.

Minimal string theories and integrable hierarchies

NASA Astrophysics Data System (ADS)

Iyer, Ramakrishnan

Well-defined, non-perturbative formulations of the physics of string theories in specific minimal or superminimal model backgrounds can be obtained by solving matrix models in the double scaling limit. They provide us with the first examples of completely solvable string theories. Despite being relatively simple compared to higher dimensional critical string theories, they furnish non-perturbative descriptions of interesting physical phenomena such as geometrical transitions between D-branes and fluxes, tachyon condensation and holography. The physics of these theories in the minimal model backgrounds is succinctly encoded in a non-linear differential equation known as the string equation, along with an associated hierarchy of integrable partial differential equations (PDEs). The bosonic string in (2,2m-1) conformal minimal model backgrounds and the type 0A string in (2,4 m) superconformal minimal model backgrounds have the Korteweg-de Vries system, while type 0B in (2,4m) backgrounds has the Zakharov-Shabat system. The integrable PDE hierarchy governs flows between backgrounds with different m. In this thesis, we explore this interesting connection between minimal string theories and integrable hierarchies further. We uncover the remarkable role that an infinite hierarchy of non-linear differential equations plays in organizing and connecting certain minimal string theories non-perturbatively. We are able to embed the type 0A and 0B (A,A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We find that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A,D) minimal string backgrounds. We explain how these and several other string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painleve IV equation plays a key role in organizing the string theory physics, joining its siblings, Painleve I and II, whose roles have previously been identified in this minimal string context. We then present evidence that the conjectured type II theories have smooth non-perturbative solutions, connecting two perturbative asymptotic regimes, in a 't Hooft limit. Our technique also demonstrates evidence for new minimal string theories that are not apparent in a perturbative analysis.

Analysis of wave propagation and wavefront sensing in target-in-the-loop beam control systems

NASA Astrophysics Data System (ADS)

Vorontsov, Mikhail A.; Kolosov, Valeri V.

2004-10-01

Target-in-the-loop (TIL) wave propagation geometry represents perhaps the most challenging case for adaptive optics applications that are related with maximization of irradiance power density on extended remotely located surfaces in the presence of dynamically changing refractive index inhomogeneities in the propagation medium. We introduce a TIL propagation model that uses a combination of the parabolic equation describing outgoing wave propagation, and the equation describing evolution of the mutual intensity function (MIF) for the backscattered (returned) wave. The resulting evolution equation for the MIF is further simplified by the use of the smooth refractive index approximation. This approximation enables derivation of the transport equation for the returned wave brightness function, analyzed here using method characteristics (brightness function trajectories). The equations for the brightness function trajectories (ray equations) can be efficiently integrated numerically. We also consider wavefront sensors that perform sensing of speckle-averaged characteristics of the wavefront phase (TIL sensors). Analysis of the wavefront phase reconstructed from Shack-Hartmann TIL sensor measurements shows that an extended target introduces a phase modulation (target-induced phase) that cannot be easily separated from the atmospheric turbulence-related phase aberrations. We also show that wavefront sensing results depend on the extended target shape, surface roughness, and the outgoing beam intensity distribution on the target surface.

Transport properties of partially ionized and unmagnetized plasmas.

PubMed

Magin, Thierry E; Degrez, Gérard

2004-10-01

This work is a comprehensive and theoretical study of transport phenomena in partially ionized and unmagnetized plasmas by means of kinetic theory. The pros and cons of different models encountered in the literature are presented. A dimensional analysis of the Boltzmann equation deals with the disparity of mass between electrons and heavy particles and yields the epochal relaxation concept. First, electrons and heavy particles exhibit distinct kinetic time scales and may have different translational temperatures. The hydrodynamic velocity is assumed to be identical for both types of species. Second, at the hydrodynamic time scale the energy exchanged between electrons and heavy particles tends to equalize both temperatures. Global and species macroscopic fluid conservation equations are given. New constrained integral equations are derived from a modified Chapman-Enskog perturbative method. Adequate bracket integrals are introduced to treat thermal nonequilibrium. A symmetric mathematical formalism is preferred for physical and numerical standpoints. A Laguerre-Sonine polynomial expansion allows for systems of transport to be derived. Momentum, mass, and energy fluxes are associated to shear viscosity, diffusion coefficients, thermal diffusion coefficients, and thermal conductivities. A Goldstein expansion of the perturbation function provides explicit expressions of the thermal diffusion ratios and measurable thermal conductivities. Thermal diffusion terms already found in the Russian literature ensure the exact mass conservation. A generalized Stefan-Maxwell equation is derived following the method of Kolesnikov and Tirskiy. The bracket integral reduction in terms of transport collision integrals is presented in Appendix for the thermal nonequilibrium case. A simple Eucken correction is proposed to deal with the internal degrees of freedom of atoms and polyatomic molecules, neglecting inelastic collisions. The authors believe that the final expressions are readily usable for practical applications in fluid dynamics.

Transonic flow of steam with non-equilibrium and homogenous condensation

NASA Astrophysics Data System (ADS)

Virk, Akashdeep Singh; Rusak, Zvi

2017-11-01

A small-disturbance model for studying the physical behavior of a steady transonic flow of steam with non-equilibrium and homogeneous condensation around a thin airfoil is derived. The steam thermodynamic behavior is described by van der Waals equation of state. The water condensation rate is calculated according to classical nucleation and droplet growth models. The current study is based on an asymptotic analysis of the fluid flow and condensation equations and boundary conditions in terms of the small thickness of the airfoil, small angle of attack, closeness of upstream flow Mach number to unity and small amount of condensate. The asymptotic analysis gives the similarity parameters that govern the problem. The flow field may be described by a non-homogeneous transonic small-disturbance equation coupled with a set of four ordinary differential equations for the calculation of the condensate mass fraction. An iterative numerical scheme which combines Murman & Cole's (1971) method with Simpson's integration rule is applied to solve the coupled system of equations. The model is used to study the effects of energy release from condensation on the aerodynamic performance of airfoils operating at high pressures and temperatures and near the vapor-liquid saturation conditions.

Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations.

PubMed

Cooper, F; Hyman, J M; Khare, A

2001-08-01

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

Theoretical study of the incompressible Navier-Stokes equations by the least-squares method

NASA Technical Reports Server (NTRS)

Jiang, Bo-Nan; Loh, Ching Y.; Povinelli, Louis A.

1994-01-01

Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.

Integral equation approach to time-dependent kinematic dynamos in finite domains

NASA Astrophysics Data System (ADS)

Xu, Mingtian; Stefani, Frank; Gerbeth, Gunter

2004-11-01

The homogeneous dynamo effect is at the root of cosmic magnetic field generation. With only a very few exceptions, the numerical treatment of homogeneous dynamos is carried out in the framework of the differential equation approach. The present paper tries to facilitate the use of integral equations in dynamo research. Apart from the pedagogical value to illustrate dynamo action within the well-known picture of the Biot-Savart law, the integral equation approach has a number of practical advantages. The first advantage is its proven numerical robustness and stability. The second and perhaps most important advantage is its applicability to dynamos in arbitrary geometries. The third advantage is its intimate connection to inverse problems relevant not only for dynamos but also for technical applications of magnetohydrodynamics. The paper provides the first general formulation and application of the integral equation approach to time-dependent kinematic dynamos, with stationary dynamo sources, in finite domains. The time dependence is restricted to the magnetic field, whereas the velocity or corresponding mean-field sources of dynamo action are supposed to be stationary. For the spherically symmetric α2 dynamo model it is shown how the general formulation is reduced to a coupled system of two radial integral equations for the defining scalars of the poloidal and toroidal field components. The integral equation formulation for spherical dynamos with general stationary velocity fields is also derived. Two numerical examples—the α2 dynamo model with radially varying α and the Bullard-Gellman model—illustrate the equivalence of the approach with the usual differential equation method. The main advantage of the method is exemplified by the treatment of an α2 dynamo in rectangular domains.

Cartan symmetries and global dynamical systems analysis in a higher-order modified teleparallel theory

NASA Astrophysics Data System (ADS)

Karpathopoulos, L.; Basilakos, S.; Leon, G.; Paliathanasis, A.; Tsamparlis, M.

2018-07-01

In a higher-order modified teleparallel theory cosmological we present analytical cosmological solutions. In particular we determine forms of the unknown potential which drives the scalar field such that the field equations form a Liouville integrable system. For the determination of the conservation laws we apply the Cartan symmetries. Furthermore, inspired from our solutions, a toy model is studied and it is shown that it can describe the Supernova data, while at the same time introduces dark matter components in the Hubble function. When the extra matter source is a stiff fluid then we show how analytical solutions for Bianchi I universes can be constructed from our analysis. Finally, we perform a global dynamical analysis of the field equations by using variables different from that of the Hubble-normalization.

Almost analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2017-11-01

We present an almost analytical new approach to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of solving this matrix eigenvalue problem purely numerically, which may suffer from the computational inaccuracy for big data, first, we consider a pair of integral and differential equations, which are related to the so-called prolate spheroidal wave functions (PSWF). For the PSWF differential equation, the pair of the eigenvectors (PSWF) and eigenvalues can be obtained from a relatively small number of analytical Legendre functions. Then, the eigenvalues in the PSWF integral equation are expressed in terms of functional values of the PSWF and the eigenvalues of the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data; ordinary irregular waves and rogue waves. We found that the present almost analytical method is better than the conventional data-independent Fourier representation and, also, the conventional direct numerical K-L representation in terms of both accuracy and computational cost. This work was supported by the National Research Foundation of Korea (NRF). (NRF-2017R1D1A1B03028299).

An analysis of the flow field near the fuel injection location in a gas core reactor.

NASA Technical Reports Server (NTRS)

Weinstein, H.; Murty, B. G. K.; Porter, R. W.

1971-01-01

An analytical study is presented which shows the effects of large energy release and the concurrent high acceleration of inner stream fluid on the coaxial flow field in a gas core reactor. The governing equations include the assumptions of only radial radiative transport of energy represented as an energy diffusion term in the Euler equations. The method of integral relations is used to obtain the numerical solution. Results show that the rapidly accelerating, heat generating inner stream actually shrinks in radius as it expands axially.

High altitude chemically reacting gas particle mixtures. Volume 1: A theoretical analysis and development of the numerical solution. [rocket nozzle and orbital plume flow fields

NASA Technical Reports Server (NTRS)

Smith, S. D.

1984-01-01

The overall contractual effort and the theory and numerical solution for the Reacting and Multi-Phase (RAMP2) computer code are described. The code can be used to model the dominant phenomena which affect the prediction of liquid and solid rocket nozzle and orbital plume flow fields. Fundamental equations for steady flow of reacting gas-particle mixtures, method of characteristics, mesh point construction, and numerical integration of the conservation equations are considered herein.

On an example of a system of differential equations that are integrated in Abelian functions

NASA Astrophysics Data System (ADS)

Malykh, M. D.; Sevastianov, L. A.

2017-12-01

The short review of the theory of Abelian functions and its applications in mechanics and analytical theory of differential equations is given. We think that Abelian functions are the natural generalization of commonly used functions because if the general solution of the 2nd order differential equation depends algebraically on the constants of integration, then integrating this equation does not lead out of the realm of commonly used functions complemented by the Abelian functions (Painlevé theorem). We present a relatively simple example of a dynamical system that is integrated in Abelian integrals by “pairing” two copies of a hyperelliptic curve. Unfortunately, initially simple formulas unfold into very long ones. Apparently the theory of Abelian functions hasn’t been finished in the last century because without computer algebra systems it was impossible to complete the calculations to the end. All calculations presented in our report are performed in Sage.

GENERIC Integrators: Structure Preserving Time Integration for Thermodynamic Systems

NASA Astrophysics Data System (ADS)

Öttinger, Hans Christian

2018-04-01

Thermodynamically admissible evolution equations for non-equilibrium systems are known to possess a distinct mathematical structure. Within the GENERIC (general equation for the non-equilibrium reversible-irreversible coupling) framework of non-equilibrium thermodynamics, which is based on continuous time evolution, we investigate the possibility of preserving all the structural elements in time-discretized equations. Our approach, which follows Moser's [1] construction of symplectic integrators for Hamiltonian systems, is illustrated for the damped harmonic oscillator. Alternative approaches are sketched.

The application of the integral equation theory to study the hydrophobic interaction

PubMed Central

Mohorič, Tomaž; Urbic, Tomaz; Hribar-Lee, Barbara

2014-01-01

The Wertheim's integral equation theory was tested against newly obtained Monte Carlo computer simulations to describe the potential of mean force between two hydrophobic particles. An excellent agreement was obtained between the theoretical and simulation results. Further, the Wertheim's integral equation theory with polymer Percus-Yevick closure qualitatively correctly (with respect to the experimental data) describes the solvation structure under conditions where the simulation results are difficult to obtain with good enough accuracy. PMID:24437891

Algorithms For Integrating Nonlinear Differential Equations

NASA Technical Reports Server (NTRS)

Freed, A. D.; Walker, K. P.

1994-01-01

Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.

Solving differential equations for Feynman integrals by expansions near singular points

NASA Astrophysics Data System (ADS)

Lee, Roman N.; Smirnov, Alexander V.; Smirnov, Vladimir A.

2018-03-01

We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales, i.e. non-trivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer code constructed with the help of our algorithm for a simple example of four-loop generalized sunset integrals with three equal non-zero masses and two zero masses. Our code gives values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter ɛ.

Heavy use of equations impedes communication among biologists.

PubMed

Fawcett, Tim W; Higginson, Andrew D

2012-07-17

Most research in biology is empirical, yet empirical studies rely fundamentally on theoretical work for generating testable predictions and interpreting observations. Despite this interdependence, many empirical studies build largely on other empirical studies with little direct reference to relevant theory, suggesting a failure of communication that may hinder scientific progress. To investigate the extent of this problem, we analyzed how the use of mathematical equations affects the scientific impact of studies in ecology and evolution. The density of equations in an article has a significant negative impact on citation rates, with papers receiving 28% fewer citations overall for each additional equation per page in the main text. Long, equation-dense papers tend to be more frequently cited by other theoretical papers, but this increase is outweighed by a sharp drop in citations from nontheoretical papers (35% fewer citations for each additional equation per page in the main text). In contrast, equations presented in an accompanying appendix do not lessen a paper's impact. Our analysis suggests possible strategies for enhancing the presentation of mathematical models to facilitate progress in disciplines that rely on the tight integration of theoretical and empirical work.

Contact interaction of thin-walled elements with an elastic layer and an infinite circular cylinder under torsion

NASA Astrophysics Data System (ADS)

Kanetsyan, E. G.; Mkrtchyan, M. S.; Mkhitaryan, S. M.

2018-04-01

We consider a class of contact torsion problems on interaction of thin-walled elements shaped as an elastic thin washer – a flat circular plate of small height – with an elastic layer, in particular, with a half-space, and on interaction of thin cylindrical shells with a solid elastic cylinder, infinite in both directions. The governing equations of the physical models of elastic thin washers and thin circular cylindrical shells under torsion are derived from the exact equations of mathematical theory of elasticity using the Hankel and Fourier transforms. Within the framework of the accepted physical models, the solution of the contact problem between an elastic washer and an elastic layer is reduced to solving the Fredholm integral equation of the first kind with a kernel representable as a sum of the Weber–Sonin integral and some integral regular kernel, while solving the contact problem between a cylindrical shell and solid cylinder is reduced to a singular integral equation (SIE). An effective method for solving the governing integral equations of these problems are specified.

Comparison of numerical techniques for integration of stiff ordinary differential equations arising in combustion chemistry

NASA Technical Reports Server (NTRS)

Radhakrishnan, K.

1984-01-01

The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.

Note on the eigensolution of a homogeneous equation with semi-infinite domain

NASA Technical Reports Server (NTRS)

Wadia, A. R.

1980-01-01

The 'variation-iteration' method using Green's functions to find the eigenvalues and the corresponding eigenfunctions of a homogeneous Fredholm integral equation is employed for the stability analysis of fluid hydromechanics problems with a semiinfinite (infinite) domain of application. The objective of the study is to develop a suitable numerical approach to the solution of such equations in order to better understand the full set of equations for 'real-world' flow models. The study involves a search for a suitable value of the length of the domain which is a fair finite approximation to infinity, which makes the eigensolution an approximation dependent on the length of the interval chosen. In the examples investigated y = 1 = a seems to be the best approximation of infinity; for y greater than unity this method fails due to the polynomial nature of Green's functions.

An introduction to generalized functions with some applications in aerodynamics and aeroacoustics

NASA Technical Reports Server (NTRS)

Farassat, F.

1994-01-01

In this paper, we start with the definition of generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support. The concept of generalization differentiation is introduced next. This is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some of the results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived using the generalized function theory. It is shown that the divergence theorem remains valid for discontinuous vector fields provided that the derivatives are all viewed as generalized derivatives. This implies that all conservation laws of fluid mechanics are valid as they stand for discontinuous fields with all derivatives treated as generalized deriatives. Once these derivatives are written as ordinary derivatives and jumps in the field parameters across discontinuities, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. By using a generalized function theory, this derivative becomes trivial. Other applications of the generalized function theory in aerodynamics discussed in this paper are derivation of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of finite part of divergent integrals, derivation of Oswatiitsch integral equation of transonic flow, and analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchoff formula for moving surfaces,the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams-Hawkings equation.

Differential equations for loop integrals in Baikov representation

NASA Astrophysics Data System (ADS)

Bosma, Jorrit; Larsen, Kasper J.; Zhang, Yang

2018-05-01

We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.

Monograph - The Numerical Integration of Ordinary Differential Equations.

ERIC Educational Resources Information Center

Hull, T. E.

The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…

An Analytical Comparison of the Acoustic Analogy and Kirchhoff Formulation for Moving Surfaces

NASA Technical Reports Server (NTRS)

Brentner, Kenneth S.; Farassat, F.

1997-01-01

The Lighthill acoustic analogy, as embodied in the Ffowcs Williams-Hawkings (FW-H) equation, is compared with the Kirchhoff formulation for moving surfaces. A comparison of the two governing equations reveals that the main Kirchhoff advantage (namely nonlinear flow effects are included in the surface integration) is also available to the FW-H method if the integration surface used in the FW-H equation is not assumed impenetrable. The FW-H equation is analytically superior for aeroacoustics because it is based upon the conservation laws of fluid mechanics rather than the wave equation. This means that the FW-H equation is valid even if the integration surface is in the nonlinear region. This is demonstrated numerically in the paper. The Kirchhoff approach can lead to substantial errors if the integration surface is not positioned in the linear region. These errors may be hard to identify. Finally, new metrics based on the Sobolev norm are introduced which may be used to compare input data for both quadrupole noise calculations and Kirchhoff noise predictions.

Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations

NASA Technical Reports Server (NTRS)

Sidi, A.; Israeli, M.

1986-01-01

High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.

A Corresponding Lie Algebra of a Reductive homogeneous Group and Its Applications

NASA Astrophysics Data System (ADS)

Zhang, Yu-Feng; Wu, Li-Xin; Rui, Wen-Juan

2015-05-01

With the help of a Lie algebra of a reductive homogeneous space G/K, where G is a Lie group and K is a resulting isotropy group, we introduce a Lax pair for which an expanding (2+1)-dimensional integrable hierarchy is obtained by applying the binormial-residue representation (BRR) method, whose Hamiltonian structure is derived from the trace identity for deducing (2+1)-dimensional integrable hierarchies, which was proposed by Tu, et al. We further consider some reductions of the expanding integrable hierarchy obtained in the paper. The first reduction is just right the (2+1)-dimensional AKNS hierarchy, the second-type reduction reveals an integrable coupling of the (2+1)-dimensional AKNS equation (also called the Davey-Stewartson hierarchy), a kind of (2+1)-dimensional Schrödinger equation, which was once reobtained by Tu, Feng and Zhang. It is interesting that a new (2+1)-dimensional integrable nonlinear coupled equation is generated from the reduction of the part of the (2+1)-dimensional integrable coupling, which is further reduced to the standard (2+1)-dimensional diffusion equation along with a parameter. In addition, the well-known (1+1)-dimensional AKNS hierarchy, the (1+1)-dimensional nonlinear Schrödinger equation are all special cases of the (2+1)-dimensional expanding integrable hierarchy. Finally, we discuss a few discrete difference equations of the diffusion equation whose stabilities are analyzed by making use of the von Neumann condition and the Fourier method. Some numerical solutions of a special stationary initial value problem of the (2+1)-dimensional diffusion equation are obtained and the resulting convergence and estimation formula are investigated. Supported by the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), the National Natural Science Foundation of China under Grant No. 11371361, the Fundamental Research Funds for the Central Universities (2013XK03), and the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016

Solving the hypersingular boundary integral equation in three-dimensional acoustics using a regularization relationship.

PubMed

Yan, Zai You; Hung, Kin Chew; Zheng, Hui

2003-05-01

Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.

On solvability of boundary value problems for hyperbolic fourth-order equations with nonlocal boundary conditions of integral type

NASA Astrophysics Data System (ADS)

Popov, Nikolay S.

2017-11-01

Solvability of some initial-boundary value problems for linear hyperbolic equations of the fourth order is studied. A condition on the lateral boundary in these problems relates the values of a solution or the conormal derivative of a solution to the values of some integral operator applied to a solution. Nonlocal boundary-value problems for one-dimensional hyperbolic second-order equations with integral conditions on the lateral boundary were considered in the articles by A.I. Kozhanov. Higher-dimensional hyperbolic equations of higher order with integral conditions on the lateral boundary were not studied earlier. The existence and uniqueness theorems of regular solutions are proven. The method of regularization and the method of continuation in a parameter are employed to establish solvability.

Reply to "Comment on 'Defocusing complex short-pulse equation and its multi-dark-soliton solution' ".

PubMed

Feng, Bao-Feng; Ling, Liming; Zhu, Zuonong

2017-08-01

Our paper [Phys. Rev. E 93, 052227 (2016)PREHBM2470-004510.1103/PhysRevE.93.052227], proposing an integrable model for the propagation of ultrashort pulses, has recently received a Comment by Youssoufa et al. [Phys. Rev. E 96, 026201 (2017)10.1103/PhysRevE.96.026201] about a possible flaw in its derivation. We point out that their claim is incorrect since we have stated explicitly that a term is neglected to derive our model equation in our paper. Furthermore, the integrable model is validated by comparing with the normalized Maxwell equation and other known integrable models. Moreover, we show that a similar approximation has to be performed in deriving the same integrable equation as explained in the Comment.

A new aerodynamic integral equation based on an acoustic formula in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1984-01-01

An aerodynamic integral equation for bodies moving at transonic and supersonic speeds is presented. Based on a time-dependent acoustic formula for calculating the noise emanating from the outer portion of a propeller blade travelling at high speed (the Ffowcs Williams-Hawking formulation), the loading terms and a conventional thickness source terms are retained. Two surface and three line integrals are employed to solve an equation for the loading noise. The near-field term is regularized using the collapsing sphere approach to obtain semiconvergence on the blade surface. A singular integral equation is thereby derived for the unknown surface pressure, and is amenable to numerical solutions using Galerkin or collocation methods. The technique is useful for studying the nonuniform inflow to the propeller.

Integrable equations of the infinite nonlinear Schrödinger equation hierarchy with time variable coefficients.

PubMed

Kedziora, D J; Ankiewicz, A; Chowdury, A; Akhmediev, N

2015-10-01

We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.

Classical integrable defects as quasi Bäcklund transformations

NASA Astrophysics Data System (ADS)

Doikou, Anastasia

2016-10-01

We consider the algebraic setting of classical defects in discrete and continuous integrable theories. We derive the ;equations of motion; on the defect point via the space-like and time-like description. We then exploit the structural similarity of these equations with the discrete and continuous Bäcklund transformations. And although these equations are similar they are not exactly the same to the Bäcklund transformations. We also consider specific examples of integrable models to demonstrate our construction, i.e. the Toda chain and the sine-Gordon model. The equations of the time (space) evolution of the defect (discontinuity) degrees of freedom for these models are explicitly derived.

Analytical expressions for the correlation function of a hard sphere dimer fluid

NASA Astrophysics Data System (ADS)

Kim, Soonho; Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.

Analytical expression for the correlation function of a hard sphere chain fluid

NASA Astrophysics Data System (ADS)

Chang, Jaeeon; Kim, Hwayong

A closed form expression is given for the correlation function of flexible hard sphere chain fluid. A set of integral equations obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with the polymer Percus-Yevick ideal chain approximation is considered. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of individual correlation functions are obtained. By inverse Laplace transformation the inter- and intramolecular radial distribution functions (RDFs) are obtained in closed forms up to 3D(D is segment diameter). These analytical expressions for the RDFs would be useful in developing the perturbation theory of chain fluids.

Entrainment of bed material by Earth-surface mass flows: review and reformulation of depth-integrated theory

USGS Publications Warehouse

Iverson, Richard M.; Chaojun Ouyang,

2015-01-01

Earth-surface mass flows such as debris flows, rock avalanches, and dam-break floods can grow greatly in size and destructive potential by entraining bed material they encounter. Increasing use of depth-integrated mass- and momentum-conservation equations to model these erosive flows motivates a review of the underlying theory. Our review indicates that many existing models apply depth-integrated conservation principles incorrectly, leading to spurious inferences about the role of mass and momentum exchanges at flow-bed boundaries. Model discrepancies can be rectified by analyzing conservation of mass and momentum in a two-layer system consisting of a moving upper layer and static lower layer. Our analysis shows that erosion or deposition rates at the interface between layers must in general satisfy three jump conditions. These conditions impose constraints on valid erosion formulas, and they help determine the correct forms of depth-integrated conservation equations. Two of the three jump conditions are closely analogous to Rankine-Hugoniot conditions that describe the behavior of shocks in compressible gasses, and the third jump condition describes shear traction discontinuities that necessarily exist across eroding boundaries. Grain-fluid mixtures commonly behave as compressible materials as they undergo entrainment, because changes in bulk density occur as the mixtures mobilize and merge with an overriding flow. If no bulk density change occurs, then only the shear-traction jump condition applies. Even for this special case, however, accurate formulation of depth-integrated momentum equations requires a clear distinction between boundary shear tractions that exist in the presence or absence of bed erosion.

Simulation of RF-fields in a fusion device

DOE Office of Scientific and Technical Information (OSTI.GOV)

De Witte, Dieter; Bogaert, Ignace; De Zutter, Daniel

2009-11-26

In this paper the problem of scattering off a fusion plasma is approached from the point of view of integral equations. Using the volume equivalence principle an integral equation is derived which describes the electromagnetic fields in a plasma. The equation is discretized with MoM using conforming basis functions. This reduces the problem to solving a dense matrix equation. This can be done iteratively. Each iteration can be sped up using FFTs.

On the conservation laws and solutions of a (2+1) dimensional KdV-mKdV equation of mathematical physics

NASA Astrophysics Data System (ADS)

Motsepa, Tanki; Masood Khalique, Chaudry

2018-05-01

In this paper, we study a (2+1) dimensional KdV-mKdV equation, which models many physical phenomena of mathematical physics. This equation has two integral terms in it. By an appropriate substitution, we convert this equation into two partial differential equations, which do not have integral terms and are equivalent to the original equation. We then work with the system of two equations and obtain its exact travelling wave solutions in form of Jacobi elliptic functions. Furthermore, we employ the multiplier method to construct conservation laws for the system. Finally, we revert the results obtained into the original variables of the (2+1) dimensional KdV-mKdV equation.

New integrable model of propagation of the few-cycle pulses in an anisotropic microdispersed medium

NASA Astrophysics Data System (ADS)

Sazonov, S. V.; Ustinov, N. V.

2018-03-01

We investigate the propagation of the few-cycle electromagnetic pulses in the anisotropic microdispersed medium. The effects of the anisotropy and spatial dispersion of the medium are created by the two sorts of the two-level atoms. The system of the material equations describing an evolution of the states of the atoms and the wave equations for the ordinary and extraordinary components of the pulses is derived. By applying the approximation of the sudden excitation to exclude the material variables, we reduce this system to the single nonlinear wave equation that generalizes the modified sine-Gordon equation and the Rabelo-Fokas equation. It is shown that this equation is integrable by means of the inverse scattering transformation method if an additional restriction on the parameters is imposed. The multisoliton solutions of this integrable generalization are constructed and investigated.

Interpreting the Coulomb-field approximation for generalized-Born electrostatics using boundary-integral equation theory.

PubMed

Bardhan, Jaydeep P

2008-10-14

The importance of molecular electrostatic interactions in aqueous solution has motivated extensive research into physical models and numerical methods for their estimation. The computational costs associated with simulations that include many explicit water molecules have driven the development of implicit-solvent models, with generalized-Born (GB) models among the most popular of these. In this paper, we analyze a boundary-integral equation interpretation for the Coulomb-field approximation (CFA), which plays a central role in most GB models. This interpretation offers new insights into the nature of the CFA, which traditionally has been assessed using only a single point charge in the solute. The boundary-integral interpretation of the CFA allows the use of multiple point charges, or even continuous charge distributions, leading naturally to methods that eliminate the interpolation inaccuracies associated with the Still equation. This approach, which we call boundary-integral-based electrostatic estimation by the CFA (BIBEE/CFA), is most accurate when the molecular charge distribution generates a smooth normal displacement field at the solute-solvent boundary, and CFA-based GB methods perform similarly. Conversely, both methods are least accurate for charge distributions that give rise to rapidly varying or highly localized normal displacement fields. Supporting this analysis are comparisons of the reaction-potential matrices calculated using GB methods and boundary-element-method (BEM) simulations. An approximation similar to BIBEE/CFA exhibits complementary behavior, with superior accuracy for charge distributions that generate rapidly varying normal fields and poorer accuracy for distributions that produce smooth fields. This approximation, BIBEE by preconditioning (BIBEE/P), essentially generates initial guesses for preconditioned Krylov-subspace iterative BEMs. Thus, iterative refinement of the BIBEE/P results recovers the BEM solution; excellent agreement is obtained in only a few iterations. The boundary-integral-equation framework may also provide a means to derive rigorous results explaining how the empirical correction terms in many modern GB models significantly improve accuracy despite their simple analytical forms.

Singular boundary value problem for the integrodifferential equation in an insurance model with stochastic premiums: Analysis and numerical solution

NASA Astrophysics Data System (ADS)

Belkina, T. A.; Konyukhova, N. B.; Kurochkin, S. V.

2012-10-01

A singular boundary value problem for a second-order linear integrodifferential equation with Volterra and non-Volterra integral operators is formulated and analyzed. The equation is defined on ℝ+, has a weak singularity at zero and a strong singularity at infinity, and depends on several positive parameters. Under natural constraints on the coefficients of the equation, existence and uniqueness theorems for this problem with given limit boundary conditions at singular points are proved, asymptotic representations of the solution are given, and an algorithm for its numerical determination is described. Numerical computations are performed and their interpretation is given. The problem arises in the study of the survival probability of an insurance company over infinite time (as a function of its initial surplus) in a dynamic insurance model that is a modification of the classical Cramer-Lundberg model with a stochastic process rate of premium under a certain investment strategy in the financial market. A comparative analysis of the results with those produced by the model with deterministic premiums is given.

The ε-form of the differential equations for Feynman integrals in the elliptic case

NASA Astrophysics Data System (ADS)

Adams, Luise; Weinzierl, Stefan

2018-06-01

Feynman integrals are easily solved if their system of differential equations is in ε-form. In this letter we show by the explicit example of the kite integral family that an ε-form can even be achieved, if the Feynman integrals do not evaluate to multiple polylogarithms. The ε-form is obtained by a (non-algebraic) change of basis for the master integrals.

Magnetic field homogeneity perturbations in finite Halbach dipole magnets.

PubMed

Turek, Krzysztof; Liszkowski, Piotr

2014-01-01

Halbach hollow cylinder dipole magnets of a low or relatively low aspect ratio attract considerable attention due to their applications, among others, in compact NMR and MRI systems for investigating small objects. However, a complete mathematical framework for the analysis of magnetic fields in these magnets has been developed only for their infinitely long precursors. In such a case the analysis is reduced to two-dimensions (2D). The paper details the analysis of the 3D magnetic field in the Halbach dipole cylinders of a finite length. The analysis is based on three equations in which the components of the magnetic flux density Bx, By and Bz are expanded to infinite power series of the radial coordinate r. The zeroth term in the series corresponds to a homogeneous magnetic field Bc, which is perturbed by the higher order terms due to a finite magnet length. This set of equations is supplemented with an equation for the field profile B(z) along the magnet axis, presented for the first time. It is demonstrated that the geometrical factors in the coefficients of particular powers of r, defined by intricate integrals are the coefficients of the Taylor expansion of the homogeneity profile (B(z)-Bc)/Bc. As a consequence, the components of B can be easily calculated with an arbitrary accuracy. In order to describe perturbations of the field due to segmentation, two additional equations are borrowed from the 2D theory. It is shown that the 2D approach to the perturbations generated by the segmentation can be applied to the 3D Halbach structures unless r is not too close to the inner radius of the cylinder ri. The mathematical framework presented in the paper was verified with great precision by computations of B by a highly accurate integration of the magnetostatic Coulomb law and utilized to analyze the inhomogeneity of the magnetic field in the magnet with the accuracy better than 1 ppm. Copyright © 2013 Elsevier Inc. All rights reserved.

Distribution theory for Schrödinger’s integral equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lange, Rutger-Jan, E-mail: rutger-jan.lange@cantab.net

2015-12-15

Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between bothmore » forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.« less

A boundary integral approach in primitive variables for free surface flows

NASA Astrophysics Data System (ADS)

Casciola, C.; Piva, R.

The boundary integral formulation, very efficient for free surface potential flows, was considered for its possible extension to rotational flows either inviscid or viscous. We first analyze a general formulation for unsteady Navier-Stokes equations in primitive variables, which reduces to a representation for the Euler equations in the limiting case of Reynolds infinity. A first simplified model for rotational flows, obtained by decoupling kinematics and dynamics, reduces the integral equations to a known kinematical form whose mathematical and numerical properties have been studied. The dynamics equations to complete the model are obtained for the free surface and the wake. A simple and efficient scheme for the study of the non linear evolution of the wave system and its interaction with the body wake is presented. A steady state version for the calculation of the wave resistance is also reported. A second model was proposed for the simulation of rotational separated regions, by coupling the integral equations in velocity with an integral equation for the vorticity at the body boundary. The same procedure may be extended to include the diffusion of the vorticity in the flowfield. The vortex shedding from a cylindrical body in unsteady motion is discussed, as a first application of the model.

Chordwise and compressibility corrections to slender-wing theory

NASA Technical Reports Server (NTRS)

Lomax, Harvard; Sluder, Loma

1952-01-01

Corrections to slender-wing theory are obtained by assuming a spanwise distribution of loading and determining the chordwise variation which satisfies the appropriate integral equation. Such integral equations are set up in terms of the given vertical induced velocity on the center line or, depending on the type of wing plan form, its average value across the span at a given chord station. The chordwise distribution is then obtained by solving these integral equations. Results are shown for flat-plate rectangular, and triangular wings.

The ATOMFT integrator - Using Taylor series to solve ordinary differential equations

NASA Technical Reports Server (NTRS)

Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.

1988-01-01

This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.

Matrix algorithms for solving (in)homogeneous bound state equations

PubMed Central

Blank, M.; Krassnigg, A.

2011-01-01

In the functional approach to quantum chromodynamics, the properties of hadronic bound states are accessible via covariant integral equations, e.g. the Bethe–Salpeter equation for mesons. In particular, one has to deal with linear, homogeneous integral equations which, in sophisticated model setups, use numerical representations of the solutions of other integral equations as part of their input. Analogously, inhomogeneous equations can be constructed to obtain off-shell information in addition to bound-state masses and other properties obtained from the covariant analogue to a wave function of the bound state. These can be solved very efficiently using well-known matrix algorithms for eigenvalues (in the homogeneous case) and the solution of linear systems (in the inhomogeneous case). We demonstrate this by solving the homogeneous and inhomogeneous Bethe–Salpeter equations and find, e.g. that for the calculation of the mass spectrum it is as efficient or even advantageous to use the inhomogeneous equation as compared to the homogeneous. This is valuable insight, in particular for the study of baryons in a three-quark setup and more involved systems. PMID:21760640

Analytical Theory of the Destruction Terms in Dissipation Rate Transport Equations

NASA Technical Reports Server (NTRS)

Rubinstein, Robert; Zhou, Ye

1996-01-01

Modeled dissipation rate transport equations are often derived by invoking various hypotheses to close correlations in the corresponding exact equations. D. C. Leslie suggested that these models might be derived instead from Kraichnan's wavenumber space integrals for inertial range transport power. This suggestion is applied to the destruction terms in the dissipation rate equations for incompressible turbulence, buoyant turbulence, rotating incompressible turbulence, and rotating buoyant turbulence. Model constants like C(epsilon 2) are expressed as integrals; convergence of these integrals implies the absence of Reynolds number dependence in the corresponding destruction term. The dependence of C(epsilon 2) on rotation rate emerges naturally; sensitization of the modeled dissipation rate equation to rotation is not required. A buoyancy related effect which is absent in the exact transport equation for temperature variance dissipation, but which sometimes improves computational predictions, also arises naturally. Both the presence of this effect and the appropriate time scale in the modeled transport equation depend on whether Bolgiano or Kolmogorov inertial range scaling applies. A simple application of these methods leads to a preliminary, dissipation rate equation for rotating buoyant turbulence.

Efficiently and easily integrating differential equations with JiTCODE, JiTCDDE, and JiTCSDE

NASA Astrophysics Data System (ADS)

Ansmann, Gerrit

2018-04-01

We present a family of Python modules for the numerical integration of ordinary, delay, or stochastic differential equations. The key features are that the user enters the derivative symbolically and it is just-in-time-compiled, allowing the user to efficiently integrate differential equations from a higher-level interpreted language. The presented modules are particularly suited for large systems of differential equations such as those used to describe dynamics on complex networks. Through the selected method of input, the presented modules also allow almost complete automatization of the process of estimating regular as well as transversal Lyapunov exponents for ordinary and delay differential equations. We conceptually discuss the modules' design, analyze their performance, and demonstrate their capabilities by application to timely problems.

Asymptotic integration algorithms for first-order ODEs with application to viscoplasticity

NASA Technical Reports Server (NTRS)

Freed, Alan D.; Yao, Minwu; Walker, Kevin P.

1992-01-01

When constructing an algorithm for the numerical integration of a differential equation, one must first convert the known ordinary differential equation (ODE), which is defined at a point, into an ordinary difference equation (O(delta)E), which is defined over an interval. Asymptotic, generalized, midpoint, and trapezoidal, O(delta)E algorithms are derived for a nonlinear first order ODE written in the form of a linear ODE. The asymptotic forward (typically underdamped) and backward (typically overdamped) integrators bound these midpoint and trapezoidal integrators, which tend to cancel out unwanted numerical damping by averaging, in some sense, the forward and backward integrations. Viscoplasticity presents itself as a system of nonlinear, coupled first-ordered ODE's that are mathematically stiff, and therefore, difficult to numerically integrate. They are an excellent application for the asymptotic integrators. Considering a general viscoplastic structure, it is demonstrated that one can either integrate the viscoplastic stresses or their associated eigenstrains.

Semi-analytical Karhunen-Loeve representation of irregular waves based on the prolate spheroidal wave functions

NASA Astrophysics Data System (ADS)

Lee, Gibbeum; Cho, Yeunwoo

2018-01-01

A new semi-analytical approach is presented to solving the matrix eigenvalue problem or the integral equation in Karhunen-Loeve (K-L) representation of random data such as irregular ocean waves. Instead of direct numerical approach to this matrix eigenvalue problem, which may suffer from the computational inaccuracy for big data, a pair of integral and differential equations are considered, which are related to the so-called prolate spheroidal wave functions (PSWF). First, the PSWF is expressed as a summation of a small number of the analytical Legendre functions. After substituting them into the PSWF differential equation, a much smaller size matrix eigenvalue problem is obtained than the direct numerical K-L matrix eigenvalue problem. By solving this with a minimal numerical effort, the PSWF and the associated eigenvalue of the PSWF differential equation are obtained. Then, the eigenvalue of the PSWF integral equation is analytically expressed by the functional values of the PSWF and the eigenvalues obtained in the PSWF differential equation. Finally, the analytically expressed PSWFs and the eigenvalues in the PWSF integral equation are used to form the kernel matrix in the K-L integral equation for the representation of exemplary wave data such as ordinary irregular waves. It is found that, with the same accuracy, the required memory size of the present method is smaller than that of the direct numerical K-L representation and the computation time of the present method is shorter than that of the semi-analytical method based on the sinusoidal functions.

Elastic interactions of a fatigue crack with a micro-defect by the mixed boundary integral equation method

NASA Technical Reports Server (NTRS)

Lua, Yuan J.; Liu, Wing K.; Belytschko, Ted

1993-01-01

In this paper, the mixed boundary integral equation method is developed to study the elastic interactions of a fatigue crack and a micro-defect such as a void, a rigid inclusion or a transformation inclusion. The method of pseudo-tractions is employed to study the effect of a transformation inclusion. An enriched element which incorporates the mixed-mode stress intensity factors is applied to characterize the singularity at a moving crack tip. In order to evaluate the accuracy of the numerical procedure, the analysis of a crack emanating from a circular hole in a finite plate is performed and the results are compared with the available numerical solution. The effects of various micro-defects on the crack path and fatigue life are investigated. The results agree with the experimental observations.

Dynamical analysis of an n‑H‑T cosmological quintessence real gas model with a general equation of state

NASA Astrophysics Data System (ADS)

Ivanov, Rossen I.; Prodanov, Emil M.

2018-01-01

The cosmological dynamics of a quintessence model based on real gas with general equation of state is presented within the framework of a three-dimensional dynamical system describing the time evolution of the number density, the Hubble parameter and the temperature. Two global first integrals are found and examples for gas with virial expansion and van der Waals gas are presented. The van der Waals system is completely integrable. In addition to the unbounded trajectories, stemming from the presence of the conserved quantities, stable periodic solutions (closed orbits) also exist under certain conditions and these represent models of a cyclic Universe. The cyclic solutions exhibit regions characterized by inflation and deflation, while the open trajectories are characterized by inflation in a “fly-by” near an unstable critical point.

The Boundary Integral Equation Method for Porous Media Flow

NASA Astrophysics Data System (ADS)

Anderson, Mary P.

Just as groundwater hydrologists are breathing sighs of relief after the exertions of learning the finite element method, a new technique has reared its nodes—the boundary integral equation method (BIEM) or the boundary equation method (BEM), as it is sometimes called. As Liggett and Liu put it in the preface to The Boundary Integral Equation Method for Porous Media Flow, “Lately, the Boundary Integral Equation Method (BIEM) has emerged as a contender in the computation Derby.” In fact, in July 1984, the 6th International Conference on Boundary Element Methods in Engineering will be held aboard the Queen Elizabeth II, en route from Southampton to New York. These conferences are sponsored by the Department of Civil Engineering at Southampton College (UK), whose members are proponents of BIEM. The conferences have featured papers on applications of BIEM to all aspects of engineering, including flow through porous media. Published proceedings are available, as are textbooks on application of BIEM to engineering problems. There is even a 10-minute film on the subject.

Inverse scattering for an exterior Dirichlet program

NASA Technical Reports Server (NTRS)

Hariharan, S. I.

1981-01-01

Scattering due to a metallic cylinder which is in the field of a wire carrying a periodic current is considered. The location and shape of the cylinder is obtained with a far field measurement in between the wire and the cylinder. The same analysis is applicable in acoustics in the situation that the cylinder is a soft wall body and the wire is a line source. The associated direct problem in this situation is an exterior Dirichlet problem for the Helmholtz equation in two dimensions. An improved low frequency estimate for the solution of this problem using integral equation methods is presented. The far field measurements are related to the solutions of boundary integral equations in the low frequency situation. These solutions are expressed in terms of mapping function which maps the exterior of the unknown curve onto the exterior of a unit disk. The coefficients of the Laurent expansion of the conformal transformations are related to the far field coefficients. The first far field coefficient leads to the calculation of the distance between the source and the cylinder.

[Clinical exercise testing and the Fick equation: strategic thinking for optimizing diagnosis].

PubMed

Perrault, H; Richard, R

2012-04-01

This article examines the expected exercise-induced changes in the components of the oxygen transport system as described by the Fick equation with a view to enable a critical analysis of a standard incremental exercise test to identify normal and abnormal patterns of responses and generate hypotheses as to potential physiological and/or pathophysiological causes. The text reviews basic physiological principals and provides useful reminders of standard equations that serve to integrate circulatory, respiratory and skeletal muscle functions. More specifically, the article provides a conceptual and quantitative framework linking the exercise-induced increase in whole body oxygen uptake to central circulatory and peripheral circulatory factors with the view to establish the normalcy of response. Thus, the article reviews the exercise response to cardiac output determinants and provides qualitative and quantitative perspective bases for making assumptions on the peripheral circulatory factors and oxygen use. Finally, the article demonstrates the usefulness of exercise testing as an effective integrative physiological approach to develop clinical reasoning or verify pathophysiological outcomes. Copyright © 2012 SPLF. Published by Elsevier Masson SAS. All rights reserved.

Gaussian-windowed frame based method of moments formulation of surface-integral-equation for extended apertures

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shlivinski, A., E-mail: amirshli@ee.bgu.ac.il; Lomakin, V., E-mail: vlomakin@eng.ucsd.edu

2016-03-01

Scattering or coupling of electromagnetic beam-field at a surface discontinuity separating two homogeneous or inhomogeneous media with different propagation characteristics is formulated using surface integral equation, which are solved by the Method of Moments with the aid of the Gabor-based Gaussian window frame set of basis and testing functions. The application of the Gaussian window frame provides (i) a mathematically exact and robust tool for spatial-spectral phase-space formulation and analysis of the problem; (ii) a system of linear equations in a transmission-line like form relating mode-like wave objects of one medium with mode-like wave objects of the second medium; (iii)more » furthermore, an appropriate setting of the frame parameters yields mode-like wave objects that blend plane wave properties (as if solving in the spectral domain) with Green's function properties (as if solving in the spatial domain); and (iv) a representation of the scattered field with Gaussian-beam propagators that may be used in many large (in terms of wavelengths) systems.« less

Numerical integration of ordinary differential equations of various orders

NASA Technical Reports Server (NTRS)

Gear, C. W.

1969-01-01

Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.

Laser dynamics: The system dynamics and network theory of optoelectronic integrated circuit design

NASA Astrophysics Data System (ADS)

Tarng, Tom Shinming-T. K.

Laser dynamics is the system dynamics, communication and network theory for the design of opto-electronic integrated circuit (OEIC). Combining the optical network theory and optical communication theory, the system analysis and design for the OEIC fundamental building blocks is considered. These building blocks include the direct current modulation, inject light modulation, wideband filter, super-gain optical amplifier, E/O and O/O optical bistability and current-controlled optical oscillator. Based on the rate equations, the phase diagram and phase portrait analysis is applied to the theoretical studies and numerical simulation. The OEIC system design methodologies are developed for the OEIC design. Stimulating-field-dependent rate equations are used to model the line-width narrowing/broadening mechanism for the CW mode and frequency chirp of semiconductor lasers. The momentary spectra are carrier-density-dependent. Furthermore, the phase portrait analysis and the nonlinear refractive index is used to simulate the single mode frequency chirp. The average spectra of chaos, period doubling, period pulsing, multi-loops and analog modulation are generated and analyzed. The bifurcation-chirp design chart with modulation depth and modulation frequency as parameters is provided for design purpose.

[Research on the method of interference correction for nondispersive infrared multi-component gas analysis].

PubMed

Sun, You-Wen; Liu, Wen-Qing; Wang, Shi-Mei; Huang, Shu-Hua; Yu, Xiao-Man

2011-10-01

A method of interference correction for nondispersive infrared multi-component gas analysis was described. According to the successive integral gas absorption models and methods, the influence of temperature and air pressure on the integral line strengths and linetype was considered, and based on Lorentz detuning linetypes, the absorption cross sections and response coefficients of H2O, CO2, CO, and NO on each filter channel were obtained. The four dimension linear regression equations for interference correction were established by response coefficients, the absorption cross interference was corrected by solving the multi-dimensional linear regression equations, and after interference correction, the pure absorbance signal on each filter channel was only controlled by the corresponding target gas concentration. When the sample cell was filled with gas mixture with a certain concentration proportion of CO, NO and CO2, the pure absorbance after interference correction was used for concentration inversion, the inversion concentration error for CO2 is 2.0%, the inversion concentration error for CO is 1.6%, and the inversion concentration error for NO is 1.7%. Both the theory and experiment prove that the interference correction method proposed for NDIR multi-component gas analysis is feasible.

Determination of constant-volume balloon capabilities for aeronautical research. [specifically measurement of atmospheric phenomena

NASA Technical Reports Server (NTRS)

Tatom, F. B.; King, R. L.

1977-01-01

The proper application of constant-volume balloons (CVB) for measurement of atmospheric phenomena was determined. And with the proper interpretation of the resulting data. A literature survey covering 176 references is included. the governing equations describing the three-dimensional motion of a CVB immersed in a flow field are developed. The flowfield model is periodic, three-dimensional, and nonhomogeneous, with mean translational motion. The balloon motion and flow field equations are cast into dimensionless form for greater generality, and certain significant dimensionless groups are identified. An alternate treatment of the balloon motion, based on first-order perturbation analysis, is also presented. A description of the digital computer program, BALLOON, used for numerically integrating the governing equations is provided.

The unified acoustic and aerodynamic prediction theory of advanced propellers in the time domain

NASA Technical Reports Server (NTRS)

Farassat, F.

1984-01-01

This paper presents some numerical results for the noise of an advanced supersonic propeller based on a formulation published last year. This formulation was derived to overcome some of the practical numerical difficulties associated with other acoustic formulations. The approach is based on the Ffowcs Williams-Hawkings equation and time domain analysis is used. To illustrate the method of solution, a model problem in three dimensions and based on the Laplace equation is solved. A brief sketch of derivation of the acoustic formula is then given. Another model problem is used to verify validity of the acoustic formulation. A recent singular integral equation for aerodynamic applications derived from the acoustic formula is also presented here.

Mathematical analysis of thermal diffusion shock waves

NASA Astrophysics Data System (ADS)

Gusev, Vitalyi; Craig, Walter; Livoti, Roberto; Danworaphong, Sorasak; Diebold, Gerald J.

2005-10-01

Thermal diffusion, also known as the Ludwig-Soret effect, refers to the separation of mixtures in a temperature gradient. For a binary mixture the time dependence of the change in concentration of each species is governed by a nonlinear partial differential equation in space and time. Here, an exact solution of the Ludwig-Soret equation without mass diffusion for a sinusoidal temperature field is given. The solution shows that counterpropagating shock waves are produced which slow and eventually come to a halt. Expressions are found for the shock time for two limiting values of the starting density fraction. The effects of diffusion on the development of the concentration profile in time and space are found by numerical integration of the nonlinear differential equation.

Recent Advances in the Method of Forces: Integrated Force Method of Structural Analysis

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Coroneos, Rula M.; Hopkins, Dale A.

1998-01-01

Stress that can be induced in an elastic continuum can be determined directly through the simultaneous application of the equilibrium equations and the compatibility conditions. In the literature, this direct stress formulation is referred to as the integrated force method. This method, which uses forces as the primary unknowns, complements the popular equilibrium-based stiffness method, which considers displacements as the unknowns. The integrated force method produces accurate stress, displacement, and frequency results even for modest finite element models. This version of the force method should be developed as an alternative to the stiffness method because the latter method, which has been researched for the past several decades, may have entered its developmental plateau. Stress plays a primary role in the development of aerospace and other products, and its analysis is difficult. Therefore, it is advisable to use both methods to calculate stress and eliminate errors through comparison. This paper examines the role of the integrated force method in analysis, animation and design.

Control volume based hydrocephalus research; analysis of human data

NASA Astrophysics Data System (ADS)

Cohen, Benjamin; Wei, Timothy; Voorhees, Abram; Madsen, Joseph; Anor, Tomer

2010-11-01

Hydrocephalus is a neuropathophysiological disorder primarily diagnosed by increased cerebrospinal fluid volume and pressure within the brain. To date, utilization of clinical measurements have been limited to understanding of the relative amplitude and timing of flow, volume and pressure waveforms; qualitative approaches without a clear framework for meaningful quantitative comparison. Pressure volume models and electric circuit analogs enforce volume conservation principles in terms of pressure. Control volume analysis, through the integral mass and momentum conservation equations, ensures that pressure and volume are accounted for using first principles fluid physics. This approach is able to directly incorporate the diverse measurements obtained by clinicians into a simple, direct and robust mechanics based framework. Clinical data obtained for analysis are discussed along with data processing techniques used to extract terms in the conservation equation. Control volume analysis provides a non-invasive, physics-based approach to extracting pressure information from magnetic resonance velocity data that cannot be measured directly by pressure instrumentation.

A Tensor-Train accelerated solver for integral equations in complex geometries

NASA Astrophysics Data System (ADS)

Corona, Eduardo; Rahimian, Abtin; Zorin, Denis

2017-04-01

We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O (log ⁡ N) and once the inverse is computed, it can be applied in O (Nlog ⁡ N) . We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.

Feynman path integral application on deriving black-scholes diffusion equation for european option pricing

NASA Astrophysics Data System (ADS)

Utama, Briandhika; Purqon, Acep

2016-08-01

Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.

An integral equation-based numerical solver for Taylor states in toroidal geometries

NASA Astrophysics Data System (ADS)

O'Neil, Michael; Cerfon, Antoine J.

2018-04-01

We present an algorithm for the numerical calculation of Taylor states in toroidal and toroidal-shell geometries using an analytical framework developed for the solution to the time-harmonic Maxwell equations. Taylor states are a special case of what are known as Beltrami fields, or linear force-free fields. The scheme of this work relies on the generalized Debye source representation of Maxwell fields and an integral representation of Beltrami fields which immediately yields a well-conditioned second-kind integral equation. This integral equation has a unique solution whenever the Beltrami parameter λ is not a member of a discrete, countable set of resonances which physically correspond to spontaneous symmetry breaking. Several numerical examples relevant to magnetohydrodynamic equilibria calculations are provided. Lastly, our approach easily generalizes to arbitrary geometries, both bounded and unbounded, and of varying genus.

Numerical simulation for flow and heat transfer to Carreau fluid with magnetic field effect: Dual nature study

NASA Astrophysics Data System (ADS)

Hashim; Khan, Masood; Alshomrani, Ali Saleh

2017-12-01

This article considers a realistic approach to examine the magnetohydrodynamics (MHD) flow of Carreau fluid induced by the shrinking sheet subject to the stagnation-point. This study also explores the impacts of non-linear thermal radiation on the heat transfer process. The governing equations of physical model are expressed as a system of partial differential equations and are transformed into non-linear ordinary differential equations by introducing local similarity variables. The economized equations of the problem are numerically integrated using the Runge-Kutta Fehlberg integration scheme. In this study, we explore the condition of existence, non-existence, uniqueness and dual nature for obtaining numerical solutions. It is found that the solutions may possess multiple natures, upper and lower branch, for a specific range of shrinking parameter. Results indicate that due to an increment in the magnetic parameter, range of shrinking parameter where a dual solution exists, increases. Further, strong magnetic field enhances the thickness of the momentum boundary layer in case of the second solution while for first solution it reduces. We further note that the fluid suction diminishes the fluid velocity and therefore the thickness of the hydrodynamic boundary layer decreases as well. A critical analysis with existing works is performed which shows that outcome are benchmarks with these works.

TOPICA/TORIC integration for self-consistent antenna and plasma analysis

NASA Astrophysics Data System (ADS)

Maggiora, Riccardo; Lancellotti, Vito; Milanesio, Daniele; Kyrytsya, Volodymyr; Vecchi, Giuseppe; Bonoli, Paul T.; Wright, John C.

2006-10-01

TOPICA [1] is a numerical suite conceived for prediction and analysis of plasma-facing antennas. It can handle real-life 3D antenna geometries (with housing, Faraday screen, etc.) as well as a realistic plasma model, including measured density and temperature profiles. TORIC [2] solves the finite Larmor radius wave equations in the ICRF regime in arbitrary axisymmetric toroidal plasmas. Due to the approach followed in developing TOPICA (i.e. the formal splitting of the problem in the vacuum region around the antenna and the plasma region inside the toroidal chamber), the code lends itself to handle toroidal plasmas, provided TORIC is run independently to yield the plasma surface admittance tensorsY (m,m',n). The latter enter directly into the integral equations solved by TOPICA, thus allowing a far more accurate plasma description that accounts for curvature effects. TOPICA outputs comprise, among others, the EM fields in front of the plasma: these can in turn be input to TORIC, in order to self-consistently determine the EM field propagation in the plasma. In this work, we report on the theory underlying the TOPICA/TORIC integration and the ongoing evolution of the two codes. [1] V. Lancellotti et al., Nucl. Fusion, 46 (2006) S476 [2] M. Brambilla, Plasma Phys. Contr. Fusion (1999) 41 1

User's Guide for ECAP2D: an Euler Unsteady Aerodynamic and Aeroelastic Analysis Program for Two Dimensional Oscillating Cascades, Version 1.0

NASA Technical Reports Server (NTRS)

Reddy, T. S. R.

1995-01-01

This guide describes the input data required for using ECAP2D (Euler Cascade Aeroelastic Program-Two Dimensional). ECAP2D can be used for steady or unsteady aerodynamic and aeroelastic analysis of two dimensional cascades. Euler equations are used to obtain aerodynamic forces. The structural dynamic equations are written for a rigid typical section undergoing pitching (torsion) and plunging (bending) motion. The solution methods include harmonic oscillation method, influence coefficient method, pulse response method, and time integration method. For harmonic oscillation method, example inputs and outputs are provided for pitching motion and plunging motion. For the rest of the methods, input and output for pitching motion only are given.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Astrophysics Data System (ADS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30 degree inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Technical Reports Server (NTRS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-01-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30 degree inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Astrophysics Data System (ADS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-03-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center-line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30-deg inclination angle.

Analysis for predicting adiabatic wall temperatures with single hole coolant injection into a low speed crossflow

NASA Technical Reports Server (NTRS)

Wang, C. R.; Papell, S. S.; Graham, R. W.

1981-01-01

Assuming the local adiabatic wall temperature equals the local total temperature in a low speed coolant mixing layer, integral conservation equations with and without the boundary layer effects are formulated for the mixing layer downstream of a single coolant injection hole oriented at a 30 degree angle to the crossflow. These equations are solved numerically to determine the center-line local adiabatic wall temperature and the effective coolant coverage area. Comparison of the numerical results with an existing film cooling experiment indicates that the present analysis permits a simplified but reasonably accurate prediction of the centerline effectiveness and coolant coverage area downstream of a single hole crossflow streamwise injection at 30-deg inclination angle.

Cage Regional Energy Budgets from the GLAS 4TH Order Model

NASA Technical Reports Server (NTRS)

Herman, G. F.; Alexder, M. A.; Shubert, S. D.

1984-01-01

The status and future plans of a study to (1) assess the accuracy of regional energy balance calculations obtained from the 4th-order model, (2) determine the impact of satellite data on the calculations, and (3) determine their utility for ocean energy transport studies are discussed. An equation is presented which models the vertically-integrated, time and areally-averaged total energy content of a region of the atmosphere extending from the surface to the top of the atmosphere. All of the terms of the equation were evaluated using early versions of the GLAS FGGE IIIb analysis, and analysis with satellite data deleted. Results show that the budget is dominated by the surface fluxes, net radiation, and horizontal atmospoheric divergence.

Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem

NASA Astrophysics Data System (ADS)

Li, Lei; Liu, Jian-Guo; Lu, Jianfeng

2017-10-01

We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the `fluctuation-dissipation theorem', the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the `fluctuation-dissipation theorem' is satisfied, and this verifies that satisfying `fluctuation-dissipation theorem' indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.

Generalized quantum Fokker-Planck, diffusion, and Smoluchowski equations with true probability distribution functions.

PubMed

Banik, Suman Kumar; Bag, Bidhan Chandra; Ray, Deb Shankar

2002-05-01

Traditionally, quantum Brownian motion is described by Fokker-Planck or diffusion equations in terms of quasiprobability distribution functions, e.g., Wigner functions. These often become singular or negative in the full quantum regime. In this paper a simple approach to non-Markovian theory of quantum Brownian motion using true probability distribution functions is presented. Based on an initial coherent state representation of the bath oscillators and an equilibrium canonical distribution of the quantum mechanical mean values of their coordinates and momenta, we derive a generalized quantum Langevin equation in c numbers and show that the latter is amenable to a theoretical analysis in terms of the classical theory of non-Markovian dynamics. The corresponding Fokker-Planck, diffusion, and Smoluchowski equations are the exact quantum analogs of their classical counterparts. The present work is independent of path integral techniques. The theory as developed here is a natural extension of its classical version and is valid for arbitrary temperature and friction (the Smoluchowski equation being considered in the overdamped limit).

On Reductions of the Hirota-Miwa Equation

NASA Astrophysics Data System (ADS)

Hone, Andrew N. W.; Kouloukas, Theodoros E.; Ward, Chloe

2017-07-01

The Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence) is a bilinear partial difference equation in three independent variables. It is integrable in the sense that it arises as the compatibility condition of a linear system (Lax pair). The Hirota-Miwa equation has infinitely many reductions of plane wave type (including a quadratic exponential gauge transformation), defined by a triple of integers or half-integers, which produce bilinear ordinary difference equations of Somos/Gale-Robinson type. Here it is explained how to obtain Lax pairs and presymplectic structures for these reductions, in order to demonstrate Liouville integrability of some associated maps, certain of which are related to reductions of discrete Toda and discrete KdV equations.

Symmetric and arbitrarily high-order Birkhoff-Hermite time integrators and their long-time behaviour for solving nonlinear Klein-Gordon equations

NASA Astrophysics Data System (ADS)

Liu, Changying; Iserles, Arieh; Wu, Xinyuan

2018-03-01

The Klein-Gordon equation with nonlinear potential occurs in a wide range of application areas in science and engineering. Its computation represents a major challenge. The main theme of this paper is the construction of symmetric and arbitrarily high-order time integrators for the nonlinear Klein-Gordon equation by integrating Birkhoff-Hermite interpolation polynomials. To this end, under the assumption of periodic boundary conditions, we begin with the formulation of the nonlinear Klein-Gordon equation as an abstract second-order ordinary differential equation (ODE) and its operator-variation-of-constants formula. We then derive a symmetric and arbitrarily high-order Birkhoff-Hermite time integration formula for the nonlinear abstract ODE. Accordingly, the stability, convergence and long-time behaviour are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix, subject to suitable temporal and spatial smoothness. A remarkable characteristic of this new approach is that the requirement of temporal smoothness is reduced compared with the traditional numerical methods for PDEs in the literature. Numerical results demonstrate the advantage and efficiency of our time integrators in comparison with the existing numerical approaches.

Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation

NASA Astrophysics Data System (ADS)

Ma, Li-Yuan; Shen, Shou-Feng; Zhu, Zuo-Nong

2017-10-01

In this paper, we prove that an integrable nonlocal complex modified Korteweg-de Vries (mKdV) equation introduced by Ablowitz and Musslimani [Nonlinearity 29, 915-946 (2016)] is gauge equivalent to a spin-like model. From the gauge equivalence, one can see that there exists significant difference between the nonlocal complex mKdV equation and the classical complex mKdV equation. Through constructing the Darboux transformation for nonlocal complex mKdV equation, a variety of exact solutions including dark soliton, W-type soliton, M-type soliton, and periodic solutions are derived.

Simplified combustion noise theory yielding a prediction of fluctuating pressure level

NASA Technical Reports Server (NTRS)

Huff, R. G.

1984-01-01

The first order equations for the conservation of mass and momentum in differential form are combined for an ideal gas to yield a single second order partial differential equation in one dimension and time. Small perturbation analysis is applied. A Fourier transformation is performed that results in a second order, constant coefficient, nonhomogeneous equation. The driving function is taken to be the source of combustion noise. A simplified model describing the energy addition via the combustion process gives the required source information for substitution in the driving function. This enables the particular integral solution of the nonhomogeneous equation to be found. This solution multiplied by the acoustic pressure efficiency predicts the acoustic pressure spectrum measured in turbine engine combustors. The prediction was compared with the overall sound pressure levels measured in a CF6-50 turbofan engine combustor and found to be in excellent agreement.

Solvability of a Nonlinear Integral Equation in Dynamical String Theory

NASA Astrophysics Data System (ADS)

Khachatryan, A. Kh.; Khachatryan, Kh. A.

2018-04-01

We investigate an integral equation of the convolution type with a cubic nonlinearity on the entire real line. This equation has a direct application in open-string field theory and in p-adic string theory and describes nonlocal interactions. We prove that there exists a one-parameter family of bounded monotonic solutions and calculate the limits of solutions constructed at infinity.

On the Analysis Methods for the Time Domain and Frequency Domain Response of a Buried Objects*

NASA Astrophysics Data System (ADS)

Poljak, Dragan; Šesnić, Silvestar; Cvetković, Mario

2014-05-01

There has been a continuous interest in the analysis of ground-penetrating radar systems and related applications in civil engineering [1]. Consequently, a deeper insight of scattering phenomena occurring in a lossy half-space, as well as the development of sophisticated numerical methods based on Finite Difference Time Domain (FDTD) method, Finite Element Method (FEM), Boundary Element Method (BEM), Method of Moments (MoM) and various hybrid methods, is required, e.g. [2], [3]. The present paper deals with certain techniques for time and frequency domain analysis, respectively, of buried conducting and dielectric objects. Time domain analysis is related to the assessment of a transient response of a horizontal straight thin wire buried in a lossy half-space using a rigorous antenna theory (AT) approach. The AT approach is based on the space-time integral equation of the Pocklington type (time domain electric field integral equation for thin wires). The influence of the earth-air interface is taken into account via the simplified reflection coefficient arising from the Modified Image Theory (MIT). The obtained results for the transient current induced along the electrode due to the transmitted plane wave excitation are compared to the numerical results calculated via an approximate transmission line (TL) approach and the AT approach based on the space-frequency variant of the Pocklington integro-differential approach, respectively. It is worth noting that the space-frequency Pocklington equation is numerically solved via the Galerkin-Bubnov variant of the Indirect Boundary Element Method (GB-IBEM) and the corresponding transient response is obtained by the aid of inverse fast Fourier transform (IFFT). The results calculated by means of different approaches agree satisfactorily. Frequency domain analysis is related to the assessment of frequency domain response of dielectric sphere using the full wave model based on the set of coupled electric field integral equations for surfaces. The numerical solution is carried out by means of the improved variant of the Method of Moments (MoM) providing numerically stable and an efficient procedure for the extraction of singularities arising in integral expressions. The proposed analysis method is compared to the results obtained by using some commercial software packages. A satisfactory agreement has been achieved. Both approaches discussed throughout this work and demonstrated on canonical geometries could be also useful for benchmark purpose. References [1] L. Pajewski et al., Applications of Ground Penetrating Radar in Civil Engineering - COST Action TU1208, 2013. [2] U. Oguz, L. Gurel, Frequency Responses of Ground-Penetrating Radars Operating Over Highly Lossy Grounds, IEEE Trans. Geosci. and Remote sensing, Vol. 40, No 6, 2002. [3] D.Poljak, Advanced Modeling in Computational electromagnetic Compatibility, John Wiley and Sons, New York 2007. *This work benefited from networking activities carried out within the EU funded COST Action TU1208 "Civil Engineering Applications of Ground Penetrating Radar."

Modelling gas dynamics in 1D ducts with abrupt area change

NASA Astrophysics Data System (ADS)

Menina, R.; Saurel, R.; Zereg, M.; Houas, L.

2011-09-01

Most gas dynamic computations in industrial ducts are done in one dimension with cross-section-averaged Euler equations. This poses a fundamental difficulty as soon as geometrical discontinuities are present. The momentum equation contains a non-conservative term involving a surface pressure integral, responsible for momentum loss. Definition of this integral is very difficult from a mathematical standpoint as the flow may contain other discontinuities (shocks, contact discontinuities). From a physical standpoint, geometrical discontinuities induce multidimensional vortices that modify the surface pressure integral. In the present paper, an improved 1D flow model is proposed. An extra energy (or entropy) equation is added to the Euler equations expressing the energy and turbulent pressure stored in the vortices generated by the abrupt area variation. The turbulent energy created by the flow-area change interaction is determined by a specific estimate of the surface pressure integral. Model's predictions are compared with 2D-averaged results from numerical solution of the Euler equations. Comparison with shock tube experiments is also presented. The new 1D-averaged model improves the conventional cross-section-averaged Euler equations and is able to reproduce the main flow features.

Some integrable maps and their Hirota bilinear forms

NASA Astrophysics Data System (ADS)

Hone, A. N. W.; Kouloukas, T. E.; Quispel, G. R. W.

2018-01-01

We introduce a two-parameter family of birational maps, which reduces to a family previously found by Demskoi, Tran, van der Kamp and Quispel (DTKQ) when one of the parameters is set to zero. The study of the singularity confinement pattern for these maps leads to the introduction of a tau function satisfying a homogeneous recurrence which has the Laurent property, and the tropical (or ultradiscrete) analogue of this homogeneous recurrence confirms the quadratic degree growth found empirically by Demskoi et al. We prove that the tau function also satisfies two different bilinear equations, each of which is a reduction of the Hirota-Miwa equation (also known as the discrete KP equation, or the octahedron recurrence). Furthermore, these bilinear equations are related to reductions of particular two-dimensional integrable lattice equations, of discrete KdV or discrete Toda type. These connections, as well as the cluster algebra structure of the bilinear equations, allow a direct construction of Poisson brackets, Lax pairs and first integrals for the birational maps. As a consequence of the latter results, we show how each member of the family can be lifted to a system that is integrable in the Liouville sense, clarifying observations made previously in the original DTKQ case.

Sixth-Order Lie Group Integrators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Forest, E.

1990-03-01

In this paper we present the coefficients of several 6th order symplectic integrator of the type developed by R. Ruth. To get these results we fully exploit the connection with Lie groups. This integrator, as well as all the explicit integrators of Ruth, may be used in any equation where some sort of Lie bracket is preserved. In fact, if the Lie operator governing the equation of motion is separable into two solvable parts, the Ruth integrators can be used.

Monotonic Derivative Correction for Calculation of Supersonic Flows

ERIC Educational Resources Information Center

Bulat, Pavel V.; Volkov, Konstantin N.

2016-01-01

Aim of the study: This study examines numerical methods for solving the problems in gas dynamics, which are based on an exact or approximate solution to the problem of breakdown of an arbitrary discontinuity (the Riemann problem). Results: Comparative analysis of finite difference schemes for the Euler equations integration is conducted on the…

Aeroelastic analysis of a troposkien-type wind turbine blade

NASA Technical Reports Server (NTRS)

Nitzsche, F.

1981-01-01

The linear aeroelastic equations for one curved blade of a vertical axis wind turbine in state vector form are presented. The method is based on a simple integrating matrix scheme together with the transfer matrix idea. The method is proposed as a convenient way of solving the associated eigenvalue problem for general support conditions.

Fast Time Domain Integral Equation Solvers for Large-Scale Electromagnetic Analysis

DTIC Science & Technology

2004-10-01

this topic coauthored by Mingyu Lu and Eric Michielssen received the Best Student Paper award at the 2001 IEEE Antennas and Propagation International...Yu Zhong, current Ph.D. student at UIUC. 18. Yujia Li, current Ph.D. student at UIUC. 19. Mingyu Lu, current Postdoctoral Fellow at UIUC. 20

Extension of the KLI approximation toward the exact optimized effective potential.

PubMed

Iafrate, G J; Krieger, J B

2013-03-07

The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

Extension of the KLI approximation toward the exact optimized effective potential

NASA Astrophysics Data System (ADS)

Iafrate, G. J.; Krieger, J. B.

2013-03-01

The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

Efficient Solution of Three-Dimensional Problems of Acoustic and Electromagnetic Scattering by Open Surfaces

NASA Technical Reports Server (NTRS)

Turc, Catalin; Anand, Akash; Bruno, Oscar; Chaubell, Julian

2011-01-01

We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.

Explicit integration of Friedmann's equation with nonlinear equations of state

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong, E-mail: chensx@henu.edu.cn, E-mail: gwg1@damtp.cam.ac.uk, E-mail: yisongyang@nyu.edu

2015-05-01

In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in generalmore » settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.« less

Interaction of Kelvin waves and nonlocality of energy transfer in superfluids

NASA Astrophysics Data System (ADS)

Laurie, Jason; L'Vov, Victor S.; Nazarenko, Sergey; Rudenko, Oleksii

2010-03-01

We argue that the physics of interacting Kelvin Waves (KWs) is highly nontrivial and cannot be understood on the basis of pure dimensional reasoning. A consistent theory of KW turbulence in superfluids should be based upon explicit knowledge of their interactions. To achieve this, we present a detailed calculation and comprehensive analysis of the interaction coefficients for KW turbuelence, thereby, resolving previous mistakes stemming from unaccounted contributions. As a first application of this analysis, we derive a local nonlinear (partial differential) equation. This equation is much simpler for analysis and numerical simulations of KWs than the Biot-Savart equation, and in contrast to the completely integrable local induction approximation (in which the energy exchange between KWs is absent), describes the nonlinear dynamics of KWs. Second, we show that the previously suggested Kozik-Svistunov energy spectrum for KWs, which has often been used in the analysis of experimental and numerical data in superfluid turbulence, is irrelevant, because it is based upon an erroneous assumption of the locality of the energy transfer through scales. Moreover, we demonstrate the weak nonlocality of the inverse cascade spectrum with a constant particle-number flux and find resulting logarithmic corrections to this spectrum.

Particle paths and phase plane for time-dependent similarity solutions of the one-dimensional Vlasov-Maxwell equations

NASA Technical Reports Server (NTRS)

Roberts, Dana Aaron; Abraham-Shrauner, Barbara

1987-01-01

The phase trajectories of particles in a plasma described by the one-dimensional Vlasov-Maxwell equations are determined qualitatively, analyzing exact general similarity solutions for the cases of temporally damped and growing (sinusoidal or localized) electric fields. The results of numerical integration in both untransformed and Lie-group point-transformed coordinates are presented in extensive graphs and characterized in detail. The implications of the present analysis for the stability of BGK equilibria are explored, and the existence of nonlinear solutions arbitrarily close to and significantly different from the BGK solutions is demonstrated.

Low-thrust trajectory analysis for the geosynchronous mission

NASA Technical Reports Server (NTRS)

Jasper, T. P.

1973-01-01

Methodology employed in development of a computer program designed to analyze optimal low-thrust trajectories is described, and application of the program to a Solar Electric Propulsion Stage (SEPS) geosynchronous mission is discussed. To avoid the zero inclination and eccentricity singularities which plague many small-force perturbation techniques, a special set of state variables (equinoctial) is used. Adjoint equations are derived for the minimum time problem and are also free from the singularities. Solutions to the state and adjoint equations are obtained by both orbit averaging and precision numerical integration; an evaluation of these approaches is made.

Elementary functions in thermodynamic Bethe ansatz

NASA Astrophysics Data System (ADS)

Suzuki, J.

2015-05-01

Some years ago, Fendley found an explicit solution to the thermodynamic Bethe ansatz (TBA) equation for an N=2 supersymmetric theory in 2D with a specific F-term. Motivated by this, we seek explicit solutions for other super-potential cases utilizing the idea from the ODE/IM correspondence. We find that the TBA equations, corresponding to a wider class of super-potentials, admit solutions in terms of elementary functions such as modified Bessel functions and confluent hyper-geometric series. Based on talks given at ‘Infinite Analysis 2014’ (Tokyo, 2014) and at ‘Integrable lattice models and quantum field theories’ (Bad Honnef, 2014).

A linear quadratic tracker for Control Moment Gyro based attitude control of the Space Station

NASA Technical Reports Server (NTRS)

Kaidy, J. T.

1986-01-01

The paper discusses a design for an attitude control system for the Space Station which produces fast response, with minimal overshoot and cross-coupling with the use of Control Moment Gyros (CMG). The rigid body equations of motion are linearized and discretized and a Linear Quadratic Regulator (LQR) design and analysis study is performed. The resulting design is then modified such that integral and differential terms are added to the state equations to enhance response characteristics. Methods for reduction of computation time through channelization are discussed as well as the reduction of initial torque requirements.

Combustion-acoustic stability analysis for premixed gas turbine combustors

NASA Technical Reports Server (NTRS)

Darling, Douglas; Radhakrishnan, Krishnan; Oyediran, Ayo; Cowan, Lizabeth

1995-01-01

Lean, prevaporized, premixed combustors are susceptible to combustion-acoustic instabilities. A model was developed to predict eigenvalues of axial modes for combustion-acoustic interactions in a premixed combustor. This work extends previous work by including variable area and detailed chemical kinetics mechanisms, using the code LSENS. Thus the acoustic equations could be integrated through the flame zone. Linear perturbations were made of the continuity, momentum, energy, chemical species, and state equations. The qualitative accuracy of our approach was checked by examining its predictions for various unsteady heat release rate models. Perturbations in fuel flow rate are currently being added to the model.

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras

PubMed Central

Gazizov, R. K.

2017-01-01

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures. PMID:28265184

Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras.

PubMed

Gainetdinova, A A; Gazizov, R K

2017-01-01

We suggest an algorithm for integrating systems of two second-order ordinary differential equations with four symmetries. In particular, if the admitted transformation group has two second-order differential invariants, the corresponding system can be integrated by quadratures using invariant representation and the operator of invariant differentiation. Otherwise, the systems reduce to partially uncoupled forms and can also be integrated by quadratures.

A multi-domain spectral method for time-fractional differential equations

NASA Astrophysics Data System (ADS)

Chen, Feng; Xu, Qinwu; Hesthaven, Jan S.

2015-07-01

This paper proposes an approach for high-order time integration within a multi-domain setting for time-fractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.

Differential Galois theory and non-integrability of planar polynomial vector fields

NASA Astrophysics Data System (ADS)

Acosta-Humánez, Primitivo B.; Lázaro, J. Tomás; Morales-Ruiz, Juan J.; Pantazi, Chara

2018-06-01

We study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential Galois Theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check whether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrability with some higher transcendent functions, like the error function.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Yanai, Takeshi; Fann, George I.; Beylkin, Gregory

Using the fully numerical method for time-dependent Hartree–Fock and density functional theory (TD-HF/DFT) with the Tamm–Dancoff (TD) approximation we use a multiresolution analysis (MRA) approach to present our findings. From a reformulation with effective use of the density matrix operator, we obtain a general form of the HF/DFT linear response equation in the first quantization formalism. It can be readily rewritten as an integral equation with the bound-state Helmholtz (BSH) kernel for the Green's function. The MRA implementation of the resultant equation permits excited state calculations without virtual orbitals. Moreover, the integral equation is efficiently and adaptively solved using amore » numerical multiresolution solver with multiwavelet bases. Our implementation of the TD-HF/DFT methods is applied for calculating the excitation energies of H 2, Be, N 2, H 2O, and C 2H 4 molecules. The numerical errors of the calculated excitation energies converge in proportion to the residuals of the equation in the molecular orbitals and response functions. The energies of the excited states at a variety of length scales ranging from short-range valence excitations to long-range Rydberg-type ones are consistently accurate. It is shown that the multiresolution calculations yield the correct exponential asymptotic tails for the response functions, whereas those computed with Gaussian basis functions are too diffuse or decay too rapidly. Finally, we introduce a simple asymptotic correction to the local spin-density approximation (LSDA) so that in the TDDFT calculations, the excited states are correctly bound.« less

Complex quantum Hamilton-Jacobi equation with Bohmian trajectories: Application to the photodissociation dynamics of NOCl

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chou, Chia-Chun, E-mail: ccchou@mx.nthu.edu.tw

2014-03-14

The complex quantum Hamilton-Jacobi equation-Bohmian trajectories (CQHJE-BT) method is introduced as a synthetic trajectory method for integrating the complex quantum Hamilton-Jacobi equation for the complex action function by propagating an ensemble of real-valued correlated Bohmian trajectories. Substituting the wave function expressed in exponential form in terms of the complex action into the time-dependent Schrödinger equation yields the complex quantum Hamilton-Jacobi equation. We transform this equation into the arbitrary Lagrangian-Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation describing the rate of change in the complex action transported along Bohmian trajectories is simultaneouslymore » integrated with the guidance equation for Bohmian trajectories, and the time-dependent wave function is readily synthesized. The spatial derivatives of the complex action required for the integration scheme are obtained by solving one moving least squares matrix equation. In addition, the method is applied to the photodissociation of NOCl. The photodissociation dynamics of NOCl can be accurately described by propagating a small ensemble of trajectories. This study demonstrates that the CQHJE-BT method combines the considerable advantages of both the real and the complex quantum trajectory methods previously developed for wave packet dynamics.« less

SALLY LEVEL II- COMPUTE AND INTEGRATE DISTURBANCE AMPLIFICATION RATES ON SWEPT AND TAPERED LAMINAR FLOW CONTROL WINGS WITH SUCTION

NASA Technical Reports Server (NTRS)

Srokowski, A. J.

1994-01-01

The computer program SALLY was developed to compute the incompressible linear stability characteristics and integrate the amplification rates of boundary layer disturbances on swept and tapered wings. For some wing designs, boundary layer disturbance can significantly alter the wing performance characteristics. This is particularly true for swept and tapered laminar flow control wings which incorporate suction to prevent boundary layer separation. SALLY should prove to be a useful tool in the analysis of these wing performance characteristics. The first step in calculating the disturbance amplification rates is to numerically solve the compressible laminar boundary-layer equation with suction for the swept and tapered wing. A two-point finite-difference method is used to solve the governing continuity, momentum, and energy equations. A similarity transformation is used to remove the wall normal velocity as a boundary condition and place it into the governing equations as a parameter. Thus the awkward nonlinear boundary condition is avoided. The resulting compressible boundary layer data is used by SALLY to compute the incompressible linear stability characteristics. The local disturbance growth is obtained from temporal stability theory and converted into a local growth rate for integration. The direction of the local group velocity is taken as the direction of integration. The amplification rate, or logarithmic disturbance amplitude ratio, is obtained by integration of the local disturbance growth over distance. The amplification rate serves as a measure of the growth of linear disturbances within the boundary layer and can serve as a guide in transition prediction. This program is written in FORTRAN IV and ASSEMBLER for batch execution and has been implemented on a CDC CYBER 70 series computer with a central memory requirement of approximately 67K (octal) of 60 bit words. SALLY was developed in 1979.

Manipulator interactive design with interconnected flexible elements

NASA Technical Reports Server (NTRS)

Singh, R. P.; Likins, P. W.

1983-01-01

This paper describes the development of an analysis tool for the interactive design of control systems for manipulators and similar electro-mechanical systems amenable to representation as structures in a topological chain. The chain consists of a series of elastic bodies subject to small deformations and arbitrary displacements. The bodies are connected by hinges which permit kinematic constraints, control, or relative motion with six degrees of freedom. The equations of motion for the chain configuration are derived via Kane's method, extended for application to interconnected flexible bodies with time-varying boundary conditions. A corresponding set of modal coordinates has been selected. The motion equations are imbedded within a simulation that transforms the vector-dyadic equations into scalar form for numerical integration. The simulation also includes a linear, time-invariant controler specified in transfer function format and a set of sensors and actuators that interface between the structure and controller. The simulation is driven by an interactive set-up program resulting in an easy-to-use analysis tool.

A multivariate variational objective analysis-assimilation method. Part 2: Case study results with and without satellite data

NASA Technical Reports Server (NTRS)

Achtemeier, Gary L.; Kidder, Stanley Q.; Scott, Robert W.

1988-01-01

The variational multivariate assimilation method described in a companion paper by Achtemeier and Ochs is applied to conventional and conventional plus satellite data. Ground-based and space-based meteorological data are weighted according to the respective measurement errors and blended into a data set that is a solution of numerical forms of the two nonlinear horizontal momentum equations, the hydrostatic equation, and an integrated continuity equation for a dry atmosphere. The analyses serve first, to evaluate the accuracy of the model, and second to contrast the analyses with and without satellite data. Evaluation criteria measure the extent to which: (1) the assimilated fields satisfy the dynamical constraints, (2) the assimilated fields depart from the observations, and (3) the assimilated fields are judged to be realistic through pattern analysis. The last criterion requires that the signs, magnitudes, and patterns of the hypersensitive vertical velocity and local tendencies of the horizontal velocity components be physically consistent with respect to the larger scale weather systems.

Analysis of the long and short-period terms due the nonsphericity of the central body: applications for Mercury

NASA Astrophysics Data System (ADS)

Carvalho, J. P. S.

2017-10-01

In this work, we present an approach taking into account the single-averaged equations and unaveraged equations to investigate the dynamics of artificial satellites on the effect due to the non-spherical shape of the planet Mercury. An analysis considering the long-period terms and another taking into account the short-period terms is presented. The numerical integrations of the equations developed are performed using the Maple software. We consider the numerical values of the most updated spherical harmonic coefficients in the literature. Emphasis is given to analyze the effect of the C22 term in the dynamics of the spacecraft. We show that the two techniques are in agreement (average or not average). We found orbits that librates around an equilibrium point with small variation of the orbital elements, in particular the eccentricity and argument of the pericenter. We also note that the C22 term contributes to reduce the growth of the orbital eccentricity.

Dynamic Analysis and Control of Lightweight Manipulators with Flexible Parallel Link Mechanisms. Ph.D. Thesis

NASA Technical Reports Server (NTRS)

Lee, Jeh Won

1990-01-01

The objective is the theoretical analysis and the experimental verification of dynamics and control of a two link flexible manipulator with a flexible parallel link mechanism. Nonlinear equations of motion of the lightweight manipulator are derived by the Lagrangian method in symbolic form to better understand the structure of the dynamic model. The resulting equation of motion have a structure which is useful to reduce the number of terms calculated, to check correctness, or to extend the model to higher order. A manipulator with a flexible parallel link mechanism is a constrained dynamic system whose equations are sensitive to numerical integration error. This constrained system is solved using singular value decomposition of the constraint Jacobian matrix. Elastic motion is expressed by the assumed mode method. Mode shape functions of each link are chosen using the load interfaced component mode synthesis. The discrepancies between the analytical model and the experiment are explained using a simplified and a detailed finite element model.

A Note on ;New Hierarchies of Integrable Lattice Equations and Associated Properties: Darboux Transformation Conservation Laws and Integrable Coupling; [Rep. Math. Phys. 67 (2011), 259

NASA Astrophysics Data System (ADS)

Xu, Xi-Xiang

2016-12-01

We prove that two new hierarchies of integrable lattice equations in [Rep. Math. Phys.67 (2011), 259] can be respectively changed into the famous relativistic Toda lattice hierarchies in the polynomial and the rational forms by means of a simple transformation.

Proceedings of the 1993 Scientific Conference on Obscuration and Aerosol Research Held in Aberdeen Proving Ground on 22-24 Jun 1993

DTIC Science & Technology

1994-03-01

equation (4.1.27) is not a finite rank integral equation, it suggests that an approxi- mate finite rank integral equation Pf" = Pg + APL(K~p) Pfa 4 \\PNPfa...Harry Salem Richard Farrell Mark Seaver Dennis F Flanigan George Sehmel David Freund Jungshik Shin Robert H Frickel Orazio I Sindoni Edward Fry Michael

Variational iteration method — a promising technique for constructing equivalent integral equations of fractional order

NASA Astrophysics Data System (ADS)

Wang, Yi-Hong; Wu, Guo-Cheng; Baleanu, Dumitru

2013-10-01

The variational iteration method is newly used to construct various integral equations of fractional order. Some iterative schemes are proposed which fully use the method and the predictor-corrector approach. The fractional Bagley-Torvik equation is then illustrated as an example of multi-order and the results show the efficiency of the variational iteration method's new role.

Boundary regularized integral equation formulation of the Helmholtz equation in acoustics.

PubMed

Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y C

2015-01-01

A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals.

Boundary regularized integral equation formulation of the Helmholtz equation in acoustics

PubMed Central

Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y. C.

2015-01-01

A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals. PMID:26064591

Integrability of the one dimensional Schrödinger equation

NASA Astrophysics Data System (ADS)

Combot, Thierry

2018-02-01

We present a definition of integrability for the one-dimensional Schrödinger equation, which encompasses all known integrable systems, i.e., systems for which the spectrum can be explicitly computed. For this, we introduce the class of rigid functions, built as Liouvillian functions, but containing all solutions of rigid differential operators in the sense of Katz, and a notion of natural of boundary conditions. We then make a complete classification of rational integrable potentials. Many new integrable cases are found, some of them physically interesting.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

Traveling wave solutions to a reaction-diffusion equation

NASA Astrophysics Data System (ADS)

Feng, Zhaosheng; Zheng, Shenzhou; Gao, David Y.

2009-07-01

In this paper, we restrict our attention to traveling wave solutions of a reaction-diffusion equation. Firstly we apply the Divisor Theorem for two variables in the complex domain, which is based on the ring theory of commutative algebra, to find a quasi-polynomial first integral of an explicit form to an equivalent autonomous system. Then through this first integral, we reduce the reaction-diffusion equation to a first-order integrable ordinary differential equation, and a class of traveling wave solutions is obtained accordingly. Comparisons with the existing results in the literature are also provided, which indicates that some analytical results in the literature contain errors. We clarify the errors and instead give a refined result in a simple and straightforward manner.

Module-based multiscale simulation of angiogenesis in skeletal muscle

PubMed Central

2011-01-01

Background Mathematical modeling of angiogenesis has been gaining momentum as a means to shed new light on the biological complexity underlying blood vessel growth. A variety of computational models have been developed, each focusing on different aspects of the angiogenesis process and occurring at different biological scales, ranging from the molecular to the tissue levels. Integration of models at different scales is a challenging and currently unsolved problem. Results We present an object-oriented module-based computational integration strategy to build a multiscale model of angiogenesis that links currently available models. As an example case, we use this approach to integrate modules representing microvascular blood flow, oxygen transport, vascular endothelial growth factor transport and endothelial cell behavior (sensing, migration and proliferation). Modeling methodologies in these modules include algebraic equations, partial differential equations and agent-based models with complex logical rules. We apply this integrated model to simulate exercise-induced angiogenesis in skeletal muscle. The simulation results compare capillary growth patterns between different exercise conditions for a single bout of exercise. Results demonstrate how the computational infrastructure can effectively integrate multiple modules by coordinating their connectivity and data exchange. Model parameterization offers simulation flexibility and a platform for performing sensitivity analysis. Conclusions This systems biology strategy can be applied to larger scale integration of computational models of angiogenesis in skeletal muscle, or other complex processes in other tissues under physiological and pathological conditions. PMID:21463529

Soliton solutions for ABS lattice equations: I. Cauchy matrix approach

NASA Astrophysics Data System (ADS)

Nijhoff, Frank; Atkinson, James; Hietarinta, Jarmo

2009-10-01

In recent years there have been new insights into the integrability of quadrilateral lattice equations, i.e. partial difference equations which are the natural discrete analogues of integrable partial differential equations in 1+1 dimensions. In the scalar (i.e. single-field) case, there now exist classification results by Adler, Bobenko and Suris (ABS) leading to some new examples in addition to the lattice equations 'of KdV type' that were known since the late 1970s and early 1980s. In this paper, we review the construction of soliton solutions for the KdV-type lattice equations and use those results to construct N-soliton solutions for all lattice equations in the ABS list except for the elliptic case of Q4, which is left to a separate treatment.

Reaction formulation for radiation and scattering from plates, corner reflectors and dielectric-coated cylinders

NASA Technical Reports Server (NTRS)

Wang, N. N.

1974-01-01

The reaction concept is employed to formulate an integral equation for radiation and scattering from plates, corner reflectors, and dielectric-coated conducting cylinders. The surface-current density on the conducting surface is expanded with subsectional bases. The dielectric layer is modeled with polarization currents radiating in free space. Maxwell's equation and the boundary conditions are employed to express the polarization-current distribution in terms of the surface-current density on the conducting surface. By enforcing reaction tests with an array of electric test sources, the moment method is employed to reduce the integral equation to a matrix equation. Inversion of the matrix equation yields the current distribution, and the scattered field is then obtained by integrating the current distribution. The theory, computer program and numerical results are presented for radiation and scattering from plates, corner reflectors, and dielectric-coated conducting cylinders.

An arbitrary-order staggered time integrator for the linear acoustic wave equation

NASA Astrophysics Data System (ADS)

Lee, Jaejoon; Park, Hyunseo; Park, Yoonseo; Shin, Changsoo

2018-02-01

We suggest a staggered time integrator whose order of accuracy can arbitrarily be extended to solve the linear acoustic wave equation. A strategy to select the appropriate order of accuracy is also proposed based on the error analysis that quantitatively predicts the truncation error of the numerical solution. This strategy not only reduces the computational cost several times, but also allows us to flexibly set the modelling parameters such as the time step length, grid interval and P-wave speed. It is demonstrated that the proposed method can almost eliminate temporal dispersive errors during long term simulations regardless of the heterogeneity of the media and time step lengths. The method can also be successfully applied to the source problem with an absorbing boundary condition, which is frequently encountered in the practical usage for the imaging algorithms or the inverse problems.

Lattice properties of the Phase I BNL x-ray lithography source obtained from fits to magnetic measurement data

DOE Office of Scientific and Technical Information (OSTI.GOV)

Blumberg, L.N.; Murphy, J.B.; Reusch, M.F.

1991-01-01

The orbit, tune, chromaticity and {beta} values for the Phase 1 XLS ring were computed by numerical integration of equations of motion using fields obtained from the coefficients of the 3-dimensional solution of Laplace's Equation evaluated by fits to magnetic measurements. The results are in good agreement with available data. The method has been extended to higher order fits of TOSCA generated fields in planes normal to the reference axis using the coil configuration proposed for the Superconducting X-Ray Lithography Source. Agreement with results from numerical integration through fields given directly by TOSCA is excellent. The formulation of the normalmore » multipole expansion presented by Brown and Servranckx has been extended to include skew multipole terms. The method appears appropriate for analysis of magnetic measurements of the SXLS. 8 refs. , 2 figs., 2 tabs.« less

One-dimensional analysis of plane and radial thin film flows including solid-body rotation

NASA Technical Reports Server (NTRS)

Thomas, S.; Hankey, W.; Faghri, A.; Swanson, T.

1989-01-01

The flow of a thin liquid film with a free surface along a horizontal plate which emanates from a pressurized vessel is examined by integrating the equations of motion across the thin liquid layer and discretizing the integrated equations using finite difference techniques. The effects of 0-g and solid-body rotation will be discussed. The two cases of interest are plane flow and radial flow. In plane flow, the liquid is considered to be flowing along a channel with no change in the width of the channel, whereas in radial flow the liquid spreads out radially over a disk, so that the area changes along the radius. It is desired to determine the height of the liquid film at any location along the plate of disk, so that the heat transfer from the plate or disk can be found. The possibility that the flow could encounter a hydraulic jump is accounted for.

Full non-linear treatment of the global thermospheric wind system. I - Mathematical method and analysis of forces. II - Results and comparison with observations

NASA Technical Reports Server (NTRS)

Blum, P. W.; Harris, I.

1975-01-01

The equations of horizontal motion of the neutral atmosphere between 120 and 500 km are integrated with the inclusion of all nonlinear terms of the convective derivative and the viscous forces due to vertical and horizontal velocity gradients. Empirical models of the distribution of neutral and charged particles are assumed to be known. The model of velocities developed is a steady state model. In Part I the mathematical method used in the integration of the Navier-Stokes equations is described and the various forces are analyzed. Results of the method given in Part I are presented with comparison with previous calculations and observations of upper atmospheric winds. Conclusions are that nonlinear effects are only significant in the equatorial region, especially at solstice conditions and that nonlinear effects do not produce any superrotation.

An approximate analysis of the diffusing flow in a self-controlled heat pipe.

NASA Technical Reports Server (NTRS)

Somogyi, D.; Yen, H. H.

1973-01-01

Constant-density two-dimensional axisymmetric equations are presented for the diffusing flow of a class of self-controlled heat pipes. The analysis is restricted to the vapor space. Condensation of the vapor is related to its mass fraction at the wall by the gas kinetic formula. The Karman-Pohlhausen integral method is applied to obtain approximate solutions. Solutions are presented for a water heat pipe with neon control gas.

Integrated tools for control-system analysis

NASA Technical Reports Server (NTRS)

Ostroff, Aaron J.; Proffitt, Melissa S.; Clark, David R.

1989-01-01

The basic functions embedded within a user friendly software package (MATRIXx) are used to provide a high level systems approach to the analysis of linear control systems. Various control system analysis configurations are assembled automatically to minimize the amount of work by the user. Interactive decision making is incorporated via menu options and at selected points, such as in the plotting section, by inputting data. There are five evaluations such as the singular value robustness test, singular value loop transfer frequency response, Bode frequency response, steady-state covariance analysis, and closed-loop eigenvalues. Another section describes time response simulations. A time response for random white noise disturbance is available. The configurations and key equations used for each type of analysis, the restrictions that apply, the type of data required, and an example problem are described. One approach for integrating the design and analysis tools is also presented.

Variation objective analyses for cyclone studies

NASA Technical Reports Server (NTRS)

Achtemeier, G. L.; Kidder, S. Q.; Ochs, H. T.

1985-01-01

The objectives were to: (1) develop an objective analysis technique that will maximize the information content of data available from diverse sources, with particular emphasis on the incorporation of observations from satellites with those from more traditional immersion techniques; and (2) to develop a diagnosis of the state of the synoptic scale atmosphere on a much finer scale over a much broader region than is presently possible to permit studies of the interactions and energy transfers between global, synoptic and regional scale atmospheric processes. The variational objective analysis model consists of the two horizontal momentum equations, the hydrostatic equation, and the integrated continuity equation for a dry hydrostatic atmosphere. Preliminary tests of the model with the SESMAE I data set are underway for 12 GMT 10 April 1979. At this stage of purpose of the analysis is not the diagnosis of atmospheric structures but rather the validation of the model. Model runs for rawinsonde data and with the precision modulus weights set to force most of the adjustment of the wind field to the mass field have produced 90 to 95 percent reductions in the imbalance of the initial data after only 4-cycles through the Euler-Lagrange equations. Sensitivity tests for linear stability of the 11 Euler-Lagrange equations that make up the VASP Model 1 indicate that there will be a lower limit to the scales of motion that can be resolved by this method. Linear stability criteria are violated where there is large horizontal wind shear near the upper tropospheric jet.

Substructure method in high-speed monorail dynamic problems

NASA Astrophysics Data System (ADS)

Ivanchenko, I. I.

2008-12-01

The study of actions of high-speed moving loads on bridges and elevated tracks remains a topical problem for transport. In the present study, we propose a new method for moving load analysis of elevated tracks (monorail structures or bridges), which permits studying the interaction between two strained objects consisting of rod systems and rigid bodies with viscoelastic links; one of these objects is the moving load (monorail rolling stock), and the other is the carrying structure (monorail elevated track or bridge). The methods for moving load analysis of structures were developed in numerous papers [1-15]. At the first stage, when solving the problem about a beam under the action of the simplest moving load such as a moving weight, two fundamental methods can be used; the same methods are realized for other structures and loads. The first method is based on the use of a generalized coordinate in the expansion of the deflection in the natural shapes of the beam, and the problem is reduced to solving a system of ordinary differential equations with variable coefficients [1-3]. In the second method, after the "beam-weight" system is decomposed, just as in the problem with the weight impact on the beam [4], solving the problem is reduced to solving an integral equation for the dynamic weight reaction [6, 7]. In [1-3], an increase in the number of retained forms leads to an increase in the order of the system of equations; in [6, 7], difficulties arise when solving the integral equations related to the conditional stability of the step procedures. The method proposed in [9, 14] for beams and rod systems combines the above approaches and eliminates their drawbacks, because it permits retaining any necessary number of shapes in the deflection expansion and has a resolving system of equations with an unconditionally stable integration scheme and with a minimum number of unknowns, just as in the method of integral equations [6, 7]. This method is further developed for combined schemes modeling a strained elastic compound moving structure and a monorail elevated track. The problems of development of methods for dynamic analysis of monorails are very topical, especially because of increasing speeds of the rolling stock motion. These structures are studied in [16-18]. In the present paper, the above problem is solved by using the method for the moving load analysis and a step procedure of integration with respect to time, which were proposed in [9, 19], respectively. Further, these components are used to enlarge the possibilities of the substructure method in problems of dynamics. In the approach proposed for moving load analysis of structures, for a substructure (having the shape of a boundary element or a superelement) we choose an object moving at a constant speed (a monorail rolling stock); in this case, we use rod boundary elements of large length, which are gathered in a system modeling these objects. In particular, sets of such elements form a model of a monorail rolling stock, namely, carriage hulls, wheeled carts, elements of the wheel spring suspension, models of continuous beams of monorail ways and piers with foundations admitting emergency subsidence and unilateral links. These specialized rigid finite elements with linear and nonlinear links, included into the set of earlier proposed finite elements [14, 19], permit studying unsteady vibrations in the "monorail train-elevated track" (MTET) system taking into account various irregularities on the beam-rail, the pier emergency subsidence, and their elastic support by the basement. In this case, a high degree of the structure spatial digitization is obtained by using rods with distributed parameters in the analysis. The displacements are approximated by linear functions and trigonometric Fourier series, which, as was already noted, permits increasing the number of degrees of freedom of the system under study simultaneously preserving the order of the resolving system of equations. This approach permits studying the stress-strain state in the MTET system and determining accelerations at the desired points of the rolling stock. The proposed numerical procedure permits uniquely solving linear and nonlinear differential equations describing the operation of the model, which replaces the system by a monorail rolling stock consisting of several specialized mutually connected cars and a system of continuous beams on elastic inertial supports. This approach (based on the use of a moving substructure, which is also modeled by a system of boundary rod elements) permits maximally reducing the number of unknowns in the resolving system of equations at each step of its solution [11]. The authors of the preceding investigations of this problem, when studying the simultaneous vibrations of bridges and moving loads, considered only the case in which the rolling stock was represented by sufficiently complicated systems of rigid bodies connected by viscoelastic links [3-18] and the rolling stock motion was described by systems of ordinary differential equations. A specific characteristic of the proposed method is that it is convenient to derive the equations of motion of both the rolling stock and the bridge structure. The method [9, 14] permits obtaining the equations of interaction between the structures as two separate finite-element structures. Hence the researcher need not traditionally write out the system of equations of motion, for example, for the rolling stock (of cars) with finitely many degrees of freedom [3-18].We note several papers where simultaneous vibrations of an elastic moving load and an elastic carrying structure are considered in a rather narrow region and have a specific character. For example, the motion of an elastic rod along an elastic infinite rod on an elastic foundation is studied in [20], and the body of a car moving along a beam is considered as a rod with ten concentrated masses in [21].

PREFACE Integrability and nonlinear phenomena Integrability and nonlinear phenomena

NASA Astrophysics Data System (ADS)

Gómez-Ullate, David; Lombardo, Sara; Mañas, Manuel; Mazzocco, Marta; Nijhoff, Frank; Sommacal, Matteo

2010-10-01

Back in 1967, Clifford Gardner, John Greene, Martin Kruskal and Robert Miura published a seminal paper in Physical Review Letters which was to become a cornerstone in the theory of integrable systems. In 2006, the authors of this paper received the AMS Steele Prize. In this award the AMS pointed out that `In applications of mathematics, solitons and their descendants (kinks, anti-kinks, instantons, and breathers) have entered and changed such diverse fields as nonlinear optics, plasma physics, and ocean, atmospheric, and planetary sciences. Nonlinearity has undergone a revolution: from a nuisance to be eliminated, to a new tool to be exploited.' From this discovery the modern theory of integrability bloomed, leading scientists to a deep understanding of many nonlinear phenomena which is by no means reachable by perturbation methods or other previous tools from linear theories. Nonlinear phenomena appear everywhere in nature, their description and understanding is therefore of great interest both from the theoretical and applicative point of view. If a nonlinear phenomenon can be represented by an integrable system then we have at our disposal a variety of tools to achieve a better mathematical description of the phenomenon. This special issue is largely dedicated to investigations of nonlinear phenomena which are related to the concept of integrability, either involving integrable systems themselves or because they use techniques from the theory of integrability. The idea of this special issue originated during the 18th edition of the Nonlinear Evolution Equations and Dynamical Systems (NEEDS) workshop, held at Isola Rossa, Sardinia, Italy, 16-23 May 2009 (http://needs-conferences.net/2009/). The issue benefits from the occasion offered by the meeting, in particular by its mini-workshops programme, and contains invited review papers and contributed papers. It is worth pointing out that there was an open call for papers and all contributions were peer reviewed according to the standards of the journal. The selection of papers in this issue aims to bring together recent developments and findings, even though it consists of only a fraction of the impressive developments in recent years which have affected a broad range of fields, including the theory of special functions, quantum integrable systems, numerical analysis, cellular automata, representations of quantum groups, symmetries of difference equations, discrete geometry, among others. The special issue begins with four review papers: Integrable models in nonlinear optics and soliton solutions Degasperis [1] reviews integrable models in nonlinear optics. He presents a number of approximate models which are integrable and illustrates the links between the mathematical and applicative aspects of the theory of integrable dynamical systems. In particular he discusses the recent impact of boomeronic-type wave equations on applications arising in the context of the resonant interaction of three waves. Hamiltonian PDEs: deformations, integrability, solutions Dubrovin [2] presents classification results for systems of nonlinear Hamiltonian partial differential equations (PDEs) in one spatial dimension. In particular he uses a perturbative approach to the theory of integrability of these systems and discusses their solutions. He conjectures universality of the critical behaviour for the solutions, where the notion of universality refers to asymptotic independence of the structure of solutions (at the point of gradient catastrophe) from the choice of generic initial data as well as from the choice of a generic PDE. KP solitons in shallow water Kodama [3] presents a survey of recent studies on soliton solutions of the Kadomtsev-Petviashvili (KP) equation. A large variety of exact soliton solutions of the KP equation are presented and classified. The study includes numerical analysis of the stability of the found solution as well as numerical simulations of the initial value problems which indicate that a certain class of initial waves approach asymptotically these exact solutions of the KP equation. The author discusses an application of the theory to the problem of the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, known as the Mach reflection problem in shallow water. A beautiful explanation of the problem was presented in a swimming pool experiment during NEEDS 2009. Smooth and peaked solitons of the CH equation Holm and Ivanov [4] discuss the relations between smooth and peaked soliton solutions for the Camassa-Holm (CH) shallow water wave equation in one spatial dimension. They first present the derivation of the soliton solution for the CH equation by means of inverse scattering transform (IST); the solution is obtained in a form that admits the peakon limit. The canonical Hamiltonian formulation of the CH equation in action-angle variables is recovered using the scattering data. The authors review some of the geometric properties of the CH equation and conclude their review with the higher dimensional generalization of the dispersionless CH equation, known as EPDiff. They also consider the possible extensions of their approach in three open problems. Regular contributions to this issue cover a wide range of topics related to integrable systems. Let us briefly illustrate some of the topics covered by this issue. One of the main topics is the study of hierarchies of integrable equations. The multifaceted idea of integrability of a particular PDE includes an approach whose aim is to find an infinite set of independent conserved quantities, much in the spirit of Liouville integrability in classical mechanics. The existence of these conserved quantities in involution, or of the corresponding infinite set of commuting symmetries, leads to an infinite set of commuting flows; i.e., to the construction of a hierarchy of compatible PDEs with respect to an infinite set of times. Obviously one can generalize or adapt this construction to different settings like the integro-differential, discrete or super-symmetric ones. The emphasis is usually to find auxiliary linear systems defining an infinite set of linear commuting flows whose solutions, if some asymptotic conditions are imposed, are named wave or Baker-Akhiezer functions. These linear flows determine the so called Lax equations, another infinite set of commuting equations whose compatibility leads to the so called Zakharov-Shabat system. An alternative description of the hierarchies is achieved with the use of the bilinear equations directly linked with the tau-function description of the hierarchy. There are two paradigmatic integrable hierarchies, namely the KP and 2-dimensional Toda lattice (2DTL). These hierarchies are treated within this volume in three contributions. In particular, Takasaki [5] reconsiders the extended Toda hierarchy of Carlet, Dubrovin and Zhang in the light of Ogawa's 2 + 1D extension of the 1D Toda hierarchy. It turns out that the former may be thought of as some sort of dimensional reduction of the latter. This explains the structure of the bilinear formalism proposed by Milanov. Carlet and Manas [6] study the 2-component KP and 2D Toda hierarchies and solve explicitly several implicit constraints present in the usual Lax formulation of the hierarchy, thus identifying a set of free dependent variables for such hierarchies. Finally, the KP hierarchy is considered in the paper by Lin et al [7], which explores the extended flows of a q-deformed modified KP hierarchy leading to the introduction of self-consistent sources. By a combination of the dressing method and the method of variation of constants, the authors are able through a dressing approach to find a scheme for the construction of solutions of the corresponding integrable equations with self-consistent sources. The study of dispersionless integrable hierarchies is an active field of research, and this special issue includes two papers devoted to the subject. Konopelchenko et al [8] describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation in terms of the hodograph solutions of the dispersionless coupled Korteweg-de Vries hierarchies. Finally, Bogdanov [9] considers 2-component integrable generalizations of the dispersionless 2D Toda lattice hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov-Santini hierarchy generalizing the dKP hierarchy. He presents the simplest 2-component generalization of the dispersionless 2DTL equation, being its differential reduction analogous to the Dunajski interpolating system. Some papers in the issue are concerned with methods to construct solutions of integrable systems, while others place more emphasis on studying properties of specific solutions of applicative interest. Among the first approach, the paper by Kaup and van Gorder [10] describes perturbation theory applied to the Inverse Scattering Transform in 3x eigenvalue problems of Zakharov-Shabat's type. Schiebold [11] studies a projection method to construct solutions of the Ablowitz-Kaup-Newell-Segur (AKNS) system, which enables her to write explicit N-soliton solutions in closed form. An example of the second kind is the paper by Biondini and Wang [12], who study in detail the behaviour of line soliton solutions of the 2DTL, describing their directions and amplitudes and also the richness of their interactions, which include resonant soliton interactions and web structure. An important field of study in integrable systems relates to the singularity structure of the solutions to nonlinear equations. When all movable singularities are poles, the system is said to have the Painleve property. The solutions may be multivalued but they can be analytically continued to meromorphic functions on the universal cover of the punctured Riemann sphere (the punctures being the fixed singularities) and the spectral curve is an affine algebraic curve. Benes and Previato [13] study the connection between the Painleve property and algebras of differential operators, extending an approach initiated by Flaschka. Solutions to some integrable systems can be constructed in terms of analytic objects associated to a spectral algebraic curve. It is therefore of interest to study the Riemann surfaces of algebraic functions, a program illustrated in the paper by Braden and Northover [14], who have implemented some algorithms for this purpose in a popular symbolic computation software. In the paper by Zhilinski [15], the critical points of the energy momentum map in classical Hamiltonian problems with nontrivial monodromy are shown to form regular lattices. The quantum mechanical counterpart has similar lattices for the joint spectrum of the commuting observables. Some examples are given in which these points form special geometric patterns. Claeys [16] uses analytic techniques and Riemann-Hilbert problems to study the asymptotic behaviour when x and t tend to infinity of a solution to the second member of the Painleve I hierarchy, which arises in multicritical string model theory and random matrix theory. This solution is conjectured to describe the universal asymptotics for Hamiltonian perturbations of hyperbolic equations near the point of gradient catastrophe for the unperturbed equation. Darboux and Backlund transformations were born more than a century ago in the context of the geometric theory of surfaces. In the past few decades they have become a useful element in the theory of integrability, with applications in different guises. Typically, they appear in dressing methods that show how to construct new interesting solutions from known simple ones. A few of the contributed papers to the issue make use of these transformations as one of their fundamental objects. Liu et al [17] use iterated Darboux transformations to construct compact representations of the multi-soliton solutions to the derivative nonlinear Schroedinger (DNLS) equation. Ragnisco and Zullo [18] construct Backlund transformations for the trigonometric classical Gaudin magnet in the partially anisotropic (xxz) case, identifying the subcase of transformations that preserve the real character of the variables. The recently discovered exceptional polynomials are complete polynomial systems that satisfy Sturm-Liouville problems but differ from the classical families of Hermite, Laguerre and Jacobi. Gomez-Ullate et al [19] prove that the families of exceptional orthogonal polynomials known to date can be obtained from the classical ones via a Darboux transformation, which becomes a useful tool to derive some of their properties. Integrability in the context of classical mechanics is associated to the existence of a sufficient number of conserved quantities, which allows sometimes an explicit integration of the equations of motion. This is the case for the motion of the Chaplygin sleigh, a rigid body motion on a fluid with nonholonomic constraints studied in the paper by Fedorov and Garcia-Naranjo [20], who derive explicit solutions and study their asymptotic behaviour. In connection with classical mechanics, some techniques of KAM theory have been used by Procesi [21] to derive normal forms for the NLS equation in its Hamiltonian formulation and prove existence and stability of quasi-periodic solutions in the case of periodic boundary conditions. Algebraic and group theoretic aspects of integrability are covered in a number of papers in the issue. The quest for symmetries of a system of differential equations usually allows us to reduce the order or the number of equations or to find special solutions possesing that symmetry, but algebraic aspects of integrable systems encompass a wide and rich spectrum of techniques, as evidenced by the following contributions. Muriel and Romero [22] perform a systematic study of all second order nonlinear ODEs that are linearizable by generalized Sundman and point transformations, showing that the two classes are inequivalent and providing an explicit characterization thereof. Lie algebras are also prominent in the work of Gerdjikov et al [23], where a class of integrable PDEs associated to symmetric spaces is studied in detail. In their approach, systems of nonlinear integrable PDEs are obtained as reductions of generic integrable systems corresponding to Lax operators with matrix coefficients. The reduction here is carried out using a reduction group which reflects symmetries of the Lax operator. These symmetries allow also a characterization of the corresponding Riemann-Hilbert data. Habibullin [24] employs algebraic techniques to study discrete chains of differential-difference equations that are Darboux integrable, i.e. that admit a certain number of nontrivial first integrals. Musso [25] provides a unified algebraic framework for the rational, trigonometric and elliptic Gaudin models. The results are achieved using a generalization of the Gaudin algebras and of the so-called coproduct method. Odesskii and Sokolov [26] present a classification of all infinite (1+1)-dimensional hydrodynamic-type chains of shift one. They establish a one-to-one correspondence between integrable chains and infinite triangular Gibbons-Tsarev (GT) systems and thus reduce the classification problem to a description of all GT-systems. In Korff's paper [27] we find a study of various algebraic and combinatorial structures that emerge in the statistical vertex model with infinite spin, an integrable model associated to a certain quantum affine algebra. In the crystal limit, this model is connected with the WZNW model in conformal field theory. The motivation for some of the submitted contributions arises also from field theories in theoretical physics. Ferreira et al [28] construct soliton solutions with non-zero topological charges to the Skyrme-Faddeev model in Yang-Mills theory. Using techniques of differential geometry and complex analysis, Manton and Rink [29] explore vortex solutions on hyperbolic surfaces extending an approach by Witten. These solutions can be interpreted as self-dual SU(2) Yang-Mills fields on R4. Shah and Woodhouse [30] use the Penrose-Ward correspondence from twistor theory to relate generalized anti self-duality equations to certain isomonodromic problems whose solutions are expressed in terms of generalized hypergeometric functions. Applications of integrable systems and nonlinear phenomena in other fields are also present in some of the papers. Kanna et al [31] study the collision of soliton solutions to coherently coupled NLS equations using a variant of the Hirota bilinearization method. Their results have applications in pulse shaping in nonlinear optics. Calogero et al [32] present examples of systems of ODEs with quadratic nonlinearities that could describe rate equations in chemical dynamics. They derive explicit conditions on the parameters of the problem for which the solutions are periodic and isochronous. Ablowitz and Haut [33] study the motion of large amplitude water waves with surface tension using asymptotic expansions and providing a comparison with experimental results. This issue is the result of the collaboration of many individuals. We would like to thank the editors and staff of the Journal of Physics A: Mathematical and Theoretical for their enthusiastic support and efficient help during the preparation of this issue. A key factor has been the work of many anonymous referees who performed careful analysis and scrutiny of the research papers submitted to this issue, often making remarks which helped to improve their quality and readability. They carried out dedicated, altruistic work with a very high standard and this issue would not exist without their contribution. Finally, we would like to thank the authors who responded to our open call, sending us their most recent results and sharing with us the enthusiasm and interest for this fascinating field of research. We hope that this collection of papers will provide a good overview for anyone interested in recent developments in the field of integrability and nonlinear phenomena. [1] Integrable models in nonlinear optics and soliton solutions Degasperis A [2] Hamiltonian PDEs: deformations, integrability, solutions Dubrovin B [3] Smooth and peaked solitons of the CH equation Holm D D and Ivanov R I [4] KP solitons in shallow water Kodama Y [5] Two extensions of 1D Toda hierarchy Takasaki K [6] On the Lax representation of the 2-component KP and 2D Toda hierarchies Guido Carlet and Manuel Manas [7] The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons Lin R L, Peng H and Manas M [8] Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation Konopelchenko B, Martinez Alonso L and E Medina [9] Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy Bogdanov L V [10] Squared eigenfunctions and the perturbation theory for the nondegenerate N x N operator: a general outline Kaup D J and Van Gorder R A [11] The noncommutative AKNS system: projection to matrix systems, countable superposition and soliton-like solutions Schiebold C [12] On the soliton solutions of the two-dimensional Toda lattice Biondini G and Wang D [13] Differential algebra of the Painleve property Benes G N and Previato E [14] Klein's curve Braden H W and Northover T P [15] Quantum monodromy and pattern formation Zhilinskii B [16] A symptotics for a special solution to the second member of the Painleve I hierarchy Claeys T [17] Darboux transformation for a two-component derivative nonlinear Schroedinger equation Ling L and Liu Q P [18] Backlund transformations as exact integrable time discretizations for the trigonometric Gaudin model Ragnisco O and Zullo F [19] Exceptional orthogonal polynomials and the Darboux transformation Gomez-Ullate D, Kamran N and Milson R [20] The hydrodynamic Chaplygin sleigh Fedorov Y N and Garcia-Naranjo L C [21] A normal form for beam and non-local nonlinear Schroedinger equations Procesi M [22] Nonlocal transformations and linearization of second-order ordinary differential equations Muriel and Romero J L [23] Reductions of integrable equations on A.III-type symmetric spaces Gerdjikov V S, Mikhailov A V and Valchev T I [24] On Darboux-integrable semi-discrete chains Habibullin I, Zheltukhina N and Sakieva A [25] Loop coproducts, Gaudin models and Poisson coalgebras Musso F [26] Classification of integrable hydrodynamic chains Odesskii A V and Sokolov V V [27] Noncommutative Schur polynomials and the crystal limit of the Uq sl(2)-vertex model Korff C [28] Axially symmetric soliton solutions in a Skyrme-Faddeev-type model with Gies's extension Ferreira L A, Sawado N and Toda K [29] Vortices on hyperbolic surfaces Manton N S and Rink N A [30] Multivariate hypergeometric cascades, isomonodromy problems and Ward ansatze Shah M R and Woodhouse N J M [31] Coherently coupled bright optical solitons and their collisions Kanna T, Vijayajayanthi M and Lakshmanan M [32] Isochronous rate equations describing chemical reactions Calogero F, Leyvraz F and Sommacal M [33] Asymptotic expansions for solitary gravity-capillary waves in two and three dimensions Ablowitz M J and Haut T S

Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems

NASA Astrophysics Data System (ADS)

Konopelchenko, B. G.; Ortenzi, G.

2013-12-01

The structure and properties of families of critical points for classes of functions W(z,{\\overline{z}}) obeying the elliptic Euler-Poisson-Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the ‘inverse’ hierarchy and equations associated with the real-analytic Eisenstein series E(\\beta ,{\\overline{\\beta }};1/2) among them. The specific bi-Hamiltonian structure of these equations is also discussed.

Series: Utilization of Differential Equations and Methods for Solving Them in Medical Physics (4).

PubMed

Murase, Kenya

2016-01-01

Partial differential equations are often used in the field of medical physics. In this (final) issue, the methods for solving the partial differential equations were introduced, which include separation of variables, integral transform (Fourier and Fourier-sine transforms), Green's function, and series expansion methods. Some examples were also introduced, in which the integral transform and Green's function methods were applied to solving Pennes' bioheat transfer equation and the Fourier series expansion method was applied to Navier-Stokes equation for analyzing the wall shear stress in blood vessels.Finally, the author hopes that this series will be helpful for people who engage in medical physics.

Simulating nonlinear steady-state traveling waves on the falling liquid film entrained by a gas flow

NASA Astrophysics Data System (ADS)

Tsvelodub, O. Yu; Bocharov, A. A.

2017-09-01

The article is devoted to the simulation of nonlinear waves on a liquid film flowing under gravity in the known stress field at the interface. The paper studies nonlinear waves on a liquid film, flowing under the action of gravity in a known stress field at the interface. In the case of small Reynolds numbers the problem is reduced to the consideration of solutions of the nonlinear integral-differential equation for film thickness deviation from the undisturbed level. The periodic and soliton steady-state traveling solutions of this equation have been numerically found. The analysis of branching of new families of steady-state traveling solutions has been performed. In particular, it is shown that this model equation has solutions in the form of solitons-humps.

Near-wall turbulence model and its application to fully developed turbulent channel and pipe flows

NASA Technical Reports Server (NTRS)

Kim, S.-W.

1990-01-01

A near-wall turbulence model and its incorporation into a multiple-timescale turbulence model are presented. The near-wall turbulence model is obtained from a k-equation turbulence model and a near-wall analysis. In the method, the equations for the conservation of mass, momentum, and turbulent kinetic energy are integrated up to the wall, and the energy transfer and the dissipation rates inside the near-wall layer are obtained from algebraic equations. Fully developed turbulent channel and pipe flows are solved using a finite element method. The computational results compare favorably with experimental data. It is also shown that the turbulence model can resolve the overshoot phenomena of the turbulent kinetic energy and the dissipation rate in the region very close to the wall.

A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type,

DTIC Science & Technology

NONLINEAR DIFFERENTIAL EQUATIONS, INTEGRATION), (*PARTIAL DIFFERENTIAL EQUATIONS, BOUNDARY VALUE PROBLEMS), BANACH SPACE , MAPPING (TRANSFORMATIONS), SET THEORY, TOPOLOGY, ITERATIONS, STABILITY, THEOREMS

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration.

PubMed

Straube, Ronny

2017-12-01

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks. Copyright © 2017 Elsevier B.V. All rights reserved.

Integral Equations and Scattering Solutions for a Square-Well Potential.

ERIC Educational Resources Information Center

Bagchi, B.; Seyler, R. G.

1979-01-01

Derives Green's functions and integral equations for scattering solutions subject to a variety of boundary conditions. Exact solutions are obtained for the case of a finite spherical square-well potential, and properties of these solutions are discussed. (Author/HM)

The statistical theory of the fracture of fragile bodies. Part 2: The integral equation method

NASA Technical Reports Server (NTRS)

Kittl, P.

1984-01-01

It is demonstrated how with the aid of a bending test, the Weibull fracture risk function can be determined - without postulating its analytical form - by resolving an integral equation. The respective solutions for rectangular and circular section beams are given. In the first case the function is expressed as an algorithm and in the second, in the form of series. Taking into account that the cumulative fracture probability appearing in the solution to the integral equation must be continuous and monotonically increasing, any case of fabrication or selection of samples can be treated.

Inclusion of trial functions in the Langevin equation path integral ground state method: Application to parahydrogen clusters and their isotopologues

DOE Office of Scientific and Technical Information (OSTI.GOV)

Schmidt, Matthew; Constable, Steve; Ing, Christopher

2014-06-21

We developed and studied the implementation of trial wavefunctions in the newly proposed Langevin equation Path Integral Ground State (LePIGS) method [S. Constable, M. Schmidt, C. Ing, T. Zeng, and P.-N. Roy, J. Phys. Chem. A 117, 7461 (2013)]. The LePIGS method is based on the Path Integral Ground State (PIGS) formalism combined with Path Integral Molecular Dynamics sampling using a Langevin equation based sampling of the canonical distribution. This LePIGS method originally incorporated a trivial trial wavefunction, ψ{sub T}, equal to unity. The present paper assesses the effectiveness of three different trial wavefunctions on three isotopes of hydrogen formore » cluster sizes N = 4, 8, and 13. The trial wavefunctions of interest are the unity trial wavefunction used in the original LePIGS work, a Jastrow trial wavefunction that includes correlations due to hard-core repulsions, and a normal mode trial wavefunction that includes information on the equilibrium geometry. Based on this analysis, we opt for the Jastrow wavefunction to calculate energetic and structural properties for parahydrogen, orthodeuterium, and paratritium clusters of size N = 4 − 19, 33. Energetic and structural properties are obtained and compared to earlier work based on Monte Carlo PIGS simulations to study the accuracy of the proposed approach. The new results for paratritium clusters will serve as benchmark for future studies. This paper provides a detailed, yet general method for optimizing the necessary parameters required for the study of the ground state of a large variety of systems.« less

Dispersion analysis of leaky guided waves in fluid-loaded waveguides of generic shape.

PubMed

Mazzotti, M; Marzani, A; Bartoli, I

2014-01-01

A fully coupled 2.5D formulation is proposed to compute the dispersive parameters of waveguides with arbitrary cross-section immersed in infinite inviscid fluids. The discretization of the waveguide is performed by means of a Semi-Analytical Finite Element (SAFE) approach, whereas a 2.5D BEM formulation is used to model the impedance of the surrounding infinite fluid. The kernels of the boundary integrals contain the fundamental solutions of the space Fourier-transformed Helmholtz equation, which governs the wave propagation process in the fluid domain. Numerical difficulties related to the evaluation of singular integrals are avoided by using a regularization procedure. To improve the numerical stability of the discretized boundary integral equations for the external Helmholtz problem, the so called CHIEF method is used. The discrete wave equation results in a nonlinear eigenvalue problem in the complex axial wavenumbers that is solved at the frequencies of interest by means of a contour integral algorithm. In order to separate physical from non-physical solutions and to fulfill the requirement of holomorphicity of the dynamic stiffness matrix inside the complex wavenumber contour, the phase of the radial bulk wavenumber is uniquely defined by enforcing the Snell-Descartes law at the fluid-waveguide interface. Three numerical applications are presented. The computed dispersion curves for a circular bar immersed in oil are in agreement with those extracted using the Global Matrix Method. Novel results are presented for viscoelastic steel bars of square and L-shaped cross-section immersed in water. Copyright © 2013 Elsevier B.V. All rights reserved.

Asian International Students at an Australian University: Mapping the Paths between Integrative Motivation, Competence in L2 Communication, Cross-Cultural Adaptation and Persistence with Structural Equation Modelling

ERIC Educational Resources Information Center

Yu, Baohua

2013-01-01

This study examined the interrelationships of integrative motivation, competence in second language (L2) communication, sociocultural adaptation, academic adaptation and persistence of international students at an Australian university. Structural equation modelling demonstrated that the integrative motivation of international students has a…

Development of volume equations using data obtained by upper stem dendrometry with Monte Carlo integration: preliminary results for eastern redcedar

Treesearch

Thomas B. Lynch; Rodney E. Will; Rider Reynolds

2013-01-01

Preliminary results are given for development of an eastern redcedar (Juniperus virginiana) cubic-volume equation based on measurements of redcedar sample tree stem volume using dendrometry with Monte Carlo integration. Monte Carlo integration techniques can be used to provide unbiased estimates of stem cubic-foot volume based on upper stem diameter...

PREFACE: Symmetries and integrability of difference equations Symmetries and integrability of difference equations

NASA Astrophysics Data System (ADS)

Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel

2009-11-01

The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first meeting with the name `Symmetries and Integrability of Discrete Equations (SIDE)' was held in Estérel, Québec, Canada. This was organized by D Levi, P Winternitz and L Vinet. After the success of the first meeting the scientific community decided to hold bi-annual SIDE meetings. They were held in 1996 at the University of Kent (UK), 1998 in Sabaudia (Italy), 2000 at the University of Tokyo (Japan), 2002 in Giens (France), 2004 in Helsinki (Finland) and in 2006 at the University of Melbourne (Australia). In 2008 the SIDE 8 meeting was again organized near Montreal, in Ste-Adèle, Québec, Canada. The SIDE 8 International Advisory Committee (also the SIDE steering committee) consisted of Frank Nijhoff, Alexander Bobenko, Basil Grammaticos, Jarmo Hietarinta, Nalini Joshi, Decio Levi, Vassilis Papageorgiou, Junkichi Satsuma, Yuri Suris, Claude Vialet and Pavel Winternitz. The local organizing committee consisted of Pavel Winternitz, John Harnad, Véronique Hussin, Decio Levi, Peter Olver and Luc Vinet. Financial support came from the Centre de Recherches Mathématiques in Montreal and the National Science Foundation (through the University of Minnesota). Proceedings of the first three SIDE meetings were published in the LMS Lecture Note series. Since 2000 the emphasis has been on publishing selected refereed articles in response to a general call for papers issued after the conference. This allows for a wider author base, since the call for papers is not restricted to conference participants. The SIDE topics thus are represented in special issues of Journal of Physics A: Mathematical and General 34 (48) and Journal of Physics A: Mathematical and Theoretical, 40 (42) (SIDE 4 and SIDE 7, respectively), Journal of Nonlinear Mathematical Physics 10 (Suppl. 2) and 12 (Suppl. 2) (SIDE 5 and SIDE 6 respectively). The SIDE 8 meeting was organized around several topics and the contributions to this special issue reflect the diversity presented during the meeting. The papers presented at the SIDE 8 meeting were organized into the following special sessions: geometry of discrete and continuous Painlevé equations; continuous symmetries of discrete equations—theory and computational applications; algebraic aspects of discrete equations; singularity confinement, algebraic entropy and Nevanlinna theory; discrete differential geometry; discrete integrable systems and isomonodromy transformations; special functions as solutions of difference and q-difference equations. This special issue of the journal is organized along similar lines. The first three articles are topical review articles appearing in alphabetical order (by first author). The article by Doliwa and Nieszporski describes the Darboux transformations in a discrete setting, namely for the discrete second order linear problem. The article by Grammaticos, Halburd, Ramani and Viallet concentrates on the integrability of the discrete systems, in particular they describe integrability tests for difference equations such as singularity confinement, algebraic entropy (growth and complexity), and analytic and arithmetic approaches. The topical review by Konopelchenko explores the relationship between the discrete integrable systems and deformations of associative algebras. All other articles are presented in alphabetical order (by first author). The contributions were solicited from all participants as well as from the general scientific community. The contributions published in this special issue can be loosely grouped into several overlapping topics, namely: •Geometry of discrete and continuous Painlevé equations (articles by Spicer and Nijhoff and by Lobb and Nijhoff). •Continuous symmetries of discrete equations—theory and applications (articles by Dorodnitsyn and Kozlov; Levi, Petrera and Scimiterna; Scimiterna; Ste-Marie and Tremblay; Levi and Yamilov; Rebelo and Winternitz). •Yang--Baxter maps (article by Xenitidis and Papageorgiou). •Algebraic aspects of discrete equations (articles by Doliwa and Nieszporski; Konopelchenko; Tsarev and Wolf). •Singularity confinement, algebraic entropy and Nevanlinna theory (articles by Grammaticos, Halburd, Ramani and Viallet; Grammaticos, Ramani and Tamizhmani). •Discrete integrable systems and isomonodromy transformations (article by Dzhamay). •Special functions as solutions of difference and q-difference equations (articles by Atakishiyeva, Atakishiyev and Koornwinder; Bertola, Gekhtman and Szmigielski; Vinet and Zhedanov). •Other topics (articles by Atkinson; Grünbaum Nagai, Kametaka and Watanabe; Nagiyev, Guliyeva and Jafarov; Sahadevan and Uma Maheswari; Svinin; Tian and Hu; Yao, Liu and Zeng). This issue is the result of the collaboration of many individuals. We would like to thank the authors who contributed and everyone else involved in the preparation of this special issue.

Solving modal equations of motion with initial conditions using MSC/NASTRAN DMAP. Part 1: Implementing exact mode superposition

NASA Technical Reports Server (NTRS)

Abdallah, Ayman A.; Barnett, Alan R.; Ibrahim, Omar M.; Manella, Richard T.

1993-01-01

Within the MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) module TRD1, solving physical (coupled) or modal (uncoupled) transient equations of motion is performed using the Newmark-Beta or mode superposition algorithms, respectively. For equations of motion with initial conditions, only the Newmark-Beta integration routine has been available in MSC/NASTRAN solution sequences for solving physical systems and in custom DMAP sequences or alters for solving modal systems. In some cases, one difficulty with using the Newmark-Beta method is that the process of selecting suitable integration time steps for obtaining acceptable results is lengthy. In addition, when very small step sizes are required, a large amount of time can be spent integrating the equations of motion. For certain aerospace applications, a significant time savings can be realized when the equations of motion are solved using an exact integration routine instead of the Newmark-Beta numerical algorithm. In order to solve modal equations of motion with initial conditions and take advantage of efficiencies gained when using uncoupled solution algorithms (like that within TRD1), an exact mode superposition method using MSC/NASTRAN DMAP has been developed and successfully implemented as an enhancement to an existing coupled loads methodology at the NASA Lewis Research Center.

Hypersonic three-dimensional nonequilibrium boundary-layer equations in generalized curvilinear coordinates

NASA Technical Reports Server (NTRS)

Lee, Jong-Hun

1993-01-01

The basic governing equations for the second-order three-dimensional hypersonic thermal and chemical nonequilibrium boundary layer are derived by means of an order-of-magnitude analysis. A two-temperature concept is implemented into the system of boundary-layer equations by simplifying the rather complicated general three-temperature thermal gas model. The equations are written in a surface-oriented non-orthogonal curvilinear coordinate system, where two curvilinear coordinates are non-orthogonial and a third coordinate is normal to the surface. The equations are described with minimum use of tensor expressions arising from the coordinate transformation, to avoid unnecessary confusion for readers. The set of equations obtained will be suitable for the development of a three-dimensional nonequilibrium boundary-layer code. Such a code could be used to determine economically the aerodynamic/aerothermodynamic loads to the surfaces of hypersonic vehicles with general configurations. In addition, the basic equations for three-dimensional stagnation flow, of which solution is required as an initial value for space-marching integration of the boundary-layer equations, are given along with the boundary conditions, the boundary-layer parameters, and the inner-outer layer matching procedure. Expressions for the chemical reaction rates and the thermodynamic and transport properties in the thermal nonequilibrium environment are explicitly given.

Computation of steady and unsteady quasi-one-dimensional viscous/inviscid interacting internal flows at subsonic, transonic, and supersonic Mach numbers

NASA Technical Reports Server (NTRS)

Swafford, Timothy W.; Huddleston, David H.; Busby, Judy A.; Chesser, B. Lawrence

1992-01-01

Computations of viscous-inviscid interacting internal flowfields are presented for steady and unsteady quasi-one-dimensional (Q1D) test cases. The unsteady Q1D Euler equations are coupled with integral boundary-layer equations for unsteady, two-dimensional (planar or axisymmetric), turbulent flow over impermeable, adiabatic walls. The coupling methodology differs from that used in most techniques reported previously in that the above mentioned equation sets are written as a complete system and solved simultaneously; that is, the coupling is carried out directly through the equations as opposed to coupling the solutions of the different equation sets. Solutions to the coupled system of equations are obtained using both explicit and implicit numerical schemes for steady subsonic, steady transonic, and both steady and unsteady supersonic internal flowfields. Computed solutions are compared with measurements as well as Navier-Stokes and inverse boundary-layer methods. An analysis of the eigenvalues of the coefficient matrix associated with the quasi-linear form of the coupled system of equations indicates the presence of complex eigenvalues for certain flow conditions. It is concluded that although reasonable solutions can be obtained numerically, these complex eigenvalues contribute to the overall difficulty in obtaining numerical solutions to the coupled system of equations.

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Matrix Methods for Solving Hartree-Fock Equations in Atomic Structure Calculations and Line Broadening

DOE PAGES

Gomez, Thomas; Nagayama, Taisuke; Fontes, Chris; ...

2018-04-23

Atomic structure of N-electron atoms is often determined by solving the Hartree-Fock equations, which are a set of integro-differential equations. The integral part of the Hartree-Fock equations treats electron exchange, but the Hartree-Fock equations are not often treated as an integro-differential equation. The exchange term is often approximated as an inhomogeneous or an effective potential so that the Hartree-Fock equations become a set of ordinary differential equations (which can be solved using the usual shooting methods). Because the Hartree-Fock equations are an iterative-refinement method, the inhomogeneous term relies on the previous guess of the wavefunction. In addition, there are numericalmore » complications associated with solving inhomogeneous differential equations. This work uses matrix methods to solve the Hartree-Fock equations as an integro-differential equation. It is well known that a derivative operator can be expressed as a matrix made of finite-difference coefficients; energy eigenvalues and eigenvectors can be obtained by using linear-algebra packages. The integral (exchange) part of the Hartree-Fock equation can be approximated as a sum and written as a matrix. The Hartree-Fock equations can be solved as a matrix that is the sum of the differential and integral matrices. We compare calculations using this method against experiment and standard atomic structure calculations. This matrix method can also be used to solve for free-electron wavefunctions, thus improving how the atoms and free electrons interact. Here, this technique is important for spectral line broadening in two ways: it improves the atomic structure calculations, and it improves the motion of the plasma electrons that collide with the atom.« less

Effect of turbulent eddy viscosity on the unstable surface mode above an acoustic liner

NASA Astrophysics Data System (ADS)

Marx, David; Aurégan, Yves

2013-07-01

Lined ducts are used to reduce noise radiation from ducts in turbofan engines. In certain conditions they may sustain hydrodynamic instabilities. A local linear stability analysis of the flow in a 2D lined channel is performed using a numerical integration of the governing equations. Several model equations are used, one of them taking into account turbulent eddy viscosity, and a realistic turbulent mean flow profile is used that vanishes at the wall. The stability analysis results are compared to published experimental results. Both the model and the experiments show the existence of an unstable mode, and the importance of taking into account eddy viscosity in the model is shown. When this is done, quantities such as the growth rate and the velocity eigenfunctions are shown to agree correctly.

Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis.

PubMed

Gutiérrez-Vega, Julio C; Rodríguez-Masegosa, Rodolfo; Chávez-Cerda, Sabino

2003-11-01

A detailed study of the axicon-based Bessel-Gauss resonator with concave output coupler is presented. We employ a technique to convert the Huygens-Fresnel integral self-consistency equation into a matrix equation and then find the eigenvalues and the eigenfields of the resonator at one time. A paraxial ray analysis is performed to find the self-consistency condition to have stable periodic ray trajectories after one or two round trips. The fast-Fourier-transform-based Fox and Li algorithm is applied to describe the three-dimensional intracavity field distribution. Special attention was directed to the dependence of the output transverse profiles, the losses, and the modal-frequency changes on the curvature of the output coupler and the cavity length. The propagation of the output beam is discussed.

Efficient three-dimensional resist profile-driven source mask optimization optical proximity correction based on Abbe-principal component analysis and Sylvester equation

NASA Astrophysics Data System (ADS)

Lin, Pei-Chun; Yu, Chun-Chang; Chen, Charlie Chung-Ping

2015-01-01

As one of the critical stages of a very large scale integration fabrication process, postexposure bake (PEB) plays a crucial role in determining the final three-dimensional (3-D) profiles and lessening the standing wave effects. However, the full 3-D chemically amplified resist simulation is not widely adopted during the postlayout optimization due to the long run-time and huge memory usage. An efficient simulation method is proposed to simulate the PEB while considering standing wave effects and resolution enhancement techniques, such as source mask optimization and subresolution assist features based on the Sylvester equation and Abbe-principal component analysis method. Simulation results show that our algorithm is 20× faster than the conventional Gaussian convolution method.

Fully-coupled analysis of jet mixing problems. Part 1. Shock-capturing model, SCIPVIS

NASA Technical Reports Server (NTRS)

Dash, S. M.; Wolf, D. E.

1984-01-01

A computational model, SCIPVIS, is described which predicts the multiple cell shock structure in imperfectly expanded, turbulent, axisymmetric jets. The model spatially integrates the parabolized Navier-Stokes jet mixing equations using a shock-capturing approach in supersonic flow regions and a pressure-split approximation in subsonic flow regions. The regions are coupled using a viscous-characteristic procedure. Turbulence processes are represented via the solution of compressibility-corrected two-equation turbulence models. The formation of Mach discs in the jet and the interactive analysis of the wake-like mixing process occurring behind Mach discs is handled in a rigorous manner. Calculations are presented exhibiting the fundamental interactive processes occurring in supersonic jets and the model is assessed via comparisons with detailed laboratory data for a variety of under- and overexpanded jets.

A High Precision Prediction Model Using Hybrid Grey Dynamic Model

ERIC Educational Resources Information Center

Li, Guo-Dong; Yamaguchi, Daisuke; Nagai, Masatake; Masuda, Shiro

2008-01-01

In this paper, we propose a new prediction analysis model which combines the first order one variable Grey differential equation Model (abbreviated as GM(1,1) model) from grey system theory and time series Autoregressive Integrated Moving Average (ARIMA) model from statistics theory. We abbreviate the combined GM(1,1) ARIMA model as ARGM(1,1)…

SPREADSHEET METHOD FOR EVALUATION OF BIOCHEMICAL REACTION RATE COEFFICIENTS AND THEIR UNCERTAINTIES BY WEIGHTED NONLINEAR LEAST-SQUARES ANALYSIS OF THE INTEGRATED MONOD EQUATION. (R825689C036)

EPA Science Inventory

The perspectives, information and conclusions conveyed in research project abstracts, progress reports, final reports, journal abstracts and journal publications convey the viewpoints of the principal investigator and may not represent the views and policies of ORD and EPA. Concl...

An interval precise integration method for transient unbalance response analysis of rotor system with uncertainty

NASA Astrophysics Data System (ADS)

Fu, Chao; Ren, Xingmin; Yang, Yongfeng; Xia, Yebao; Deng, Wangqun

2018-07-01

A non-intrusive interval precise integration method (IPIM) is proposed in this paper to analyze the transient unbalance response of uncertain rotor systems. The transfer matrix method (TMM) is used to derive the deterministic equations of motion of a hollow-shaft overhung rotor. The uncertain transient dynamic problem is solved by combing the Chebyshev approximation theory with the modified precise integration method (PIM). Transient response bounds are calculated by interval arithmetic of the expansion coefficients. Theoretical error analysis of the proposed method is provided briefly, and its accuracy is further validated by comparing with the scanning method in simulations. Numerical results show that the IPIM can keep good accuracy in vibration prediction of the start-up transient process. Furthermore, the proposed method can also provide theoretical guidance to other transient dynamic mechanical systems with uncertainties.

Evaluating Feynman integrals by the hypergeometry

NASA Astrophysics Data System (ADS)

Feng, Tai-Fu; Chang, Chao-Hsi; Chen, Jian-Bin; Gu, Zhi-Hua; Zhang, Hai-Bin

2018-02-01

The hypergeometric function method naturally provides the analytic expressions of scalar integrals from concerned Feynman diagrams in some connected regions of independent kinematic variables, also presents the systems of homogeneous linear partial differential equations satisfied by the corresponding scalar integrals. Taking examples of the one-loop B0 and massless C0 functions, as well as the scalar integrals of two-loop vacuum and sunset diagrams, we verify our expressions coinciding with the well-known results of literatures. Based on the multiple hypergeometric functions of independent kinematic variables, the systems of homogeneous linear partial differential equations satisfied by the mentioned scalar integrals are established. Using the calculus of variations, one recognizes the system of linear partial differential equations as stationary conditions of a functional under some given restrictions, which is the cornerstone to perform the continuation of the scalar integrals to whole kinematic domains numerically with the finite element methods. In principle this method can be used to evaluate the scalar integrals of any Feynman diagrams.

Orbit determination based on meteor observations using numerical integration of equations of motion

NASA Astrophysics Data System (ADS)

Dmitriev, V.; Lupovka, V.; Gritsevich, M.

2014-07-01

We review the definitions and approaches to orbital-characteristics analysis applied to photographic or video ground-based observations of meteors. A number of camera networks dedicated to meteors registration were established all over the word, including USA, Canada, Central Europe, Australia, Spain, Finland and Poland. Many of these networks are currently operational. The meteor observations are conducted from different locations hosting the network stations. Each station is equipped with at least one camera for continuous monitoring of the firmament (except possible weather restrictions). For registered multi-station meteors, it is possible to accurately determine the direction and absolute value for the meteor velocity and thus obtain the topocentric radiant. Based on topocentric radiant one further determines the heliocentric meteor orbit. We aim to reduce total uncertainty in our orbit-determination technique, keeping it even less than the accuracy of observations. The additional corrections for the zenith attraction are widely in use and are implemented, for example, here [1]. We propose a technique for meteor-orbit determination with higher accuracy. We transform the topocentric radiant in inertial (J2000) coordinate system using the model recommended by IAU [2]. The main difference if compared to the existing orbit-determination techniques is integration of ordinary differential equations of motion instead of addition correction in visible velocity for zenith attraction. The attraction of the central body (the Sun), the perturbations by Earth, Moon and other planets of the Solar System, the Earth's flattening (important in the initial moment of integration, i.e. at the moment when a meteoroid enters the atmosphere), atmospheric drag may be optionally included in the equations. In addition, reverse integration of the same equations can be performed to analyze orbital evolution preceding to meteoroid's collision with Earth. To demonstrate the developed technique, we provide calculated orbits for several cases, including well-known meteorite-producing fireballs. A comparison of our estimates with previously published ones is also provided.

On the Formulation of Weakly Singular Displacement/Traction Integral Equations; and Their Solution by the MLPG Method

NASA Technical Reports Server (NTRS)

Atluri, Satya N.; Shen, Shengping

2002-01-01

In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.

The geometric approach to sets of ordinary differential equations and Hamiltonian dynamics

NASA Technical Reports Server (NTRS)

Estabrook, F. B.; Wahlquist, H. D.

1975-01-01

The calculus of differential forms is used to discuss the local integration theory of a general set of autonomous first order ordinary differential equations. Geometrically, such a set is a vector field V in the space of dependent variables. Integration consists of seeking associated geometric structures invariant along V: scalar fields, forms, vectors, and integrals over subspaces. It is shown that to any field V can be associated a Hamiltonian structure of forms if, when dealing with an odd number of dependent variables, an arbitrary equation of constraint is also added. Families of integral invariants are an immediate consequence. Poisson brackets are isomorphic to Lie products of associated CT-generating vector fields. Hamilton's variational principle follows from the fact that the maximal regular integral manifolds of a closed set of forms must include the characteristics of the set.

High-order solution methods for grey discrete ordinates thermal radiative transfer

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maginot, Peter G., E-mail: maginot1@llnl.gov; Ragusa, Jean C., E-mail: jean.ragusa@tamu.edu; Morel, Jim E., E-mail: morel@tamu.edu

This work presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

High-order solution methods for grey discrete ordinates thermal radiative transfer

DOE Office of Scientific and Technical Information (OSTI.GOV)

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

This paper presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

High-order solution methods for grey discrete ordinates thermal radiative transfer

DOE PAGES

Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E.

2016-09-29

This paper presents a solution methodology for solving the grey radiative transfer equations that is both spatially and temporally more accurate than the canonical radiative transfer solution technique of linear discontinuous finite element discretization in space with implicit Euler integration in time. We solve the grey radiative transfer equations by fully converging the nonlinear temperature dependence of the material specific heat, material opacities, and Planck function. The grey radiative transfer equations are discretized in space using arbitrary-order self-lumping discontinuous finite elements and integrated in time with arbitrary-order diagonally implicit Runge–Kutta time integration techniques. Iterative convergence of the radiation equation ismore » accelerated using a modified interior penalty diffusion operator to precondition the full discrete ordinates transport operator.« less

AKNS hierarchy, Darboux transformation and conservation laws of the 1D nonautonomous nonlinear Schroedinger equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhao Dun; Center for Interdisciplinary Studies, Lanzhou University, Lanzhou 730000; Zhang Yujuan

2011-04-15

By constructing nonisospectral Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, we investigate the nonautonomous nonlinear Schroedinger (NLS) equations which have been used to describe the Feshbach resonance management in matter-wave solitons in Bose-Einstein condensate and the dispersion and nonlinearity managements for optical solitons. It is found that these equations are some special cases of a new integrable model of nonlocal nonautonomous NLS equations. Based on the Lax pairs, the Darboux transformation and conservation laws are explored. It is shown that the local external potentials would break down the classical infinite number of conservation laws. The result indicates that the integrability of the nonautonomous NLSmore » systems may be nontrivial in comparison to the conventional concept of integrability in the canonical case.« less