The Study of Gay-Berne Fluid:. Integral Equations Method
NASA Astrophysics Data System (ADS)
Khordad, Reza; Mohebbi, Mehran; Keshavarzi, Abolla; Poostforush, Ahmad; Ghajari Haghighi, Farnaz
We study a classical fluid of nonspherical molecules. The components of the fluid are the ellipsoidal molecules interacting through the Gay-Berne potential model. A method is described, which allows the Percus-Yevick (PY) and hypernetted-chain (HNC) integral equation theories to be solved numerically for this fluid. Explicit results are given and comparisons are made with recent Monte Carlo (MC) simulations. It is found that, at lower cutoff lmax, the HNC and the PY closures give significantly different results. The HNC and PY (approximately) theories, at higher cutoff lmax, are superior in predicting the existence of the phase transition in a qualitative agreement with computer simulation.
Integral Equations in the Study of Polar and Ionic Interaction Site Fluids
NASA Astrophysics Data System (ADS)
Howard, Jesse J.; Pettitt, B. Montgomery
2011-10-01
We consider some of the current integral equation approaches and application to model polar liquid mixtures. We show 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.
Integral equations in the study of polar and ionic interaction site fluids
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
Properties of the Lennard-Jones dimeric fluid in two dimensions: An integral equation study
Urbic, Tomaz; Dias, Cristiano L.
2014-03-07
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 can be used to predict the phase diagram of dimers whose molecular parameters are known.
Properties of the Lennard-Jones dimeric fluid in two dimensions: An integral equation study
Urbic, Tomaz; Dias, Cristiano L.
2014-01-01
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 can be used to predict the phase diagram of dimers whose molecular parameters are known. PMID:24606372
The application of the integral equation theory to study the hydrophobic interaction
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
Integral equation study of particle confinement effects in a polymer/particle mixture
Henderson, D; Trokhymchuk, A; Kalyuzhnyi, Y; Gee, R; Lacevic, N
2007-05-09
Integral equation theory techniques are applied to evaluate the structuring of the polymer when large solid particles are embedded into a bulk polymer melt. The formalism presented here is applied to obtain an insight into the filler particle aggregation tendency. We find that with the employed polymer-particle interaction model it is very unlikely that the particles will aggregate. We believe that in such a system aggregation and clustering can occur when the filler particles are dressed by tightly bound polymer layers.
Fresnel Integral Equations: Numerical Properties
Adams, R J; Champagne, N J II; Davis, B A
2003-07-22
A spatial-domain solution to the problem of electromagnetic scattering from a dielectric half-space is outlined. The resulting half-space operators are referred to as Fresnel surface integral operators. When used as preconditioners for nonplanar geometries, the Fresnel operators yield surface Fresnel integral equations (FIEs) which are stable with respect to dielectric constant, discretization, and frequency. Numerical properties of the formulations are discussed.
Structure and discrimination in chiral fluids: A molecular dynamics and integral equation study
NASA Astrophysics Data System (ADS)
Cann, N. M.; Das, B.
2000-08-01
An analysis of structure and discrimination in simple chiral fluids is presented. The chiral molecules consist of a central carbon bonded to four distinct groups. Molecular-dynamics simulations have been performed on a one-component chiral fluid and on two racemic mixtures. For the racemates, discrimination, as measured by differences in pair distribution functions, is present but found to be small. Intermolecular pair interaction energies are found to be good predictors of the magnitude and the sign (mirror-image pairs favored) of the differences observed in site-site distribution functions. For the one-component fluid, the quality of structural predictions from the reference-interaction-site method and Chandler-Silbey-Ladanyi (CSL) integral equation theories, with the hypernetted chain (HNC) and Percus-Yevick closures, has been examined. These theories generally provide a qualitatively correct description of the site-site distributions. Extensions beyond the HNC level have been explored: Two-field-point bridge diagrams have been explicitly evaluated and included in the CSL theory. The inclusion of these diagrams significantly improves the quality of the integral equation theories. Since the CSL theory has not been used extensively, and bridge diagrams have been evaluated in only a few instances, a detailed analysis of their impact is presented. For racemic mixtures, diagram evaluation is shown to be crucial. Specifically, the differences in site-site distributions for sites on identical and mirror-image molecules are found to originate from bridge diagrams which involve interactions between four-site, or larger, clusters. Discrimination cannot be predicted from an integral equation theory which neglects these diagrams.
A Collocation Method for Volterra Integral Equations
NASA Astrophysics Data System (ADS)
Kolk, Marek
2010-09-01
We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with logarithmic kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration. An attainable order of the convergence of the method is studied. We illustrate our results with a numerical example.
Explicit integration of Friedmann's equation with nonlinear equations of state
NASA Astrophysics Data System (ADS)
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong
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 general 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.
Integrable (2 k)-Dimensional Hitchin Equations
NASA Astrophysics Data System (ADS)
Ward, R. S.
2016-07-01
This letter describes a completely integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4 k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k = 2 case are described.
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.
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
NASA Astrophysics Data System (ADS)
Klinger, B.; Baur, O.; Mayer-Gürr, T.
2014-02-01
The NASA mission GRAIL (Gravity Recovery And Interior Laboratory) makes use of low-low satellite-to-satellite tracking between the spacecraft GRAIL-A (Ebb) and GRAIL-B (Flow) to determine high-resolution lunar gravity field features. The inter-satellite measurements are independent of the visibility of the spacecraft from Earth, and hence provide data for both the nearside and the farside of the Moon. We propose to exploit this ranging data by an integral equation approach using short orbital arcs; it is based on the reformulation of Newton's equation of motion as a boundary value problem. This technique has been successfully applied for the recovery of the gravity field of the Earth from the Gravity Recovery And Climate Experiment (GRACE) project-the terrestrial sibling of GRAIL. By means of a series of simulation studies we demonstrate the potential of the approach. We pay particular attention on a priori gravity field information, orbital arc length, observation noise and the impact of spectral aliasing (omission error). Finally, we compute a first lunar gravity model (GrazLGM200a) from real data of the primary mission phase (March 1, 2012 to May 29, 2012). The unconstrained model is expanded up to spherical harmonic degree and order 200. From our simulations and real data results we conclude that the integral equation approach is well suited for GRAIL gravity field recovery.
Integral equations for resonance and virtual states
Orlov, Y.V.; Turovtsev, V.V.
1984-05-01
Integral equations are derived for the resonance and virtual (antibound) states consisting of two or three bodies. The derivation is based on the analytic continuation of the integral equations of scattering theory to nonphysical energy sheets. The resulting equations can be used to exhibit the analytic properties of amplitudes that are necessary for practical calculations using the equations for the quasistationary levels and Gamov wave functions derived in this paper. The Fourier transformation and the normalization rule for the wave function are generalized to the case of nonstationary states. The energy of the antibound state of the tritium nucleus is calculated for a ''realistic'' local potential.
Engineering integrable nonautonomous nonlinear Schroedinger equations
He Xugang; Zhao Dun; Li Lin; Luo Honggang
2009-05-15
We investigate Painleve integrability of a generalized nonautonomous one-dimensional nonlinear Schroedinger (NLS) equation with time- and space-dependent dispersion, nonlinearity, and external potentials. Through the Painleve analysis some explicit requirements on the dispersion, nonlinearity, dissipation/gain, and the external potential as well as the constraint conditions are identified. It provides an explicit way to engineer integrable nonautonomous NLS equations at least in the sense of Painleve integrability. Furthermore analytical solutions of this class of integrable nonautonomous NLS equations can be obtained explicitly from the solutions of the standard NLS equation by a general transformation. The result provides a significant way to control coherently the soliton dynamics in the corresponding nonlinear systems, as that in Bose-Einstein condensate experiments. We analyze explicitly the soliton dynamics under the nonlinearity management and the external potentials and discuss its application in the matter-wave dynamics. Some comparisons with the previous works have also been discussed.
Computation of virial coefficients from integral equations.
Zhang, Cheng; Lai, Chun-Liang; Pettitt, B Montgomery
2015-06-01
A polynomial-time method of computing the virial coefficients from an integral equation framework is presented. The method computes the truncated density expansions of the correlation functions by series transformations, and then extracts the virial coefficients from the density components. As an application, the method was used in a hybrid-closure integral equation with a set of self-consistent conditions, which produced reasonably accurate virial coefficients for the hard-sphere fluid and Gaussian model in high dimensions. PMID:26049482
Computation of virial coefficients from integral equations
NASA Astrophysics Data System (ADS)
Zhang, Cheng; Lai, Chun-Liang; Pettitt, B. Montgomery
2015-06-01
A polynomial-time method of computing the virial coefficients from an integral equation framework is presented. The method computes the truncated density expansions of the correlation functions by series transformations, and then extracts the virial coefficients from the density components. As an application, the method was used in a hybrid-closure integral equation with a set of self-consistent conditions, which produced reasonably accurate virial coefficients for the hard-sphere fluid and Gaussian model in high dimensions.
Application of wavelets to singular integral scattering equations
Kessler, B.M.; Payne, G.L.; Polyzou, W.N.
2004-09-01
The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms are demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems.
A SYMPLECTIC INTEGRATOR FOR HILL'S EQUATIONS
Quinn, Thomas; Barnes, Rory; Perrine, Randall P.; Richardson, Derek C.
2010-02-15
Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV, and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.
Integral equations for flows in wind tunnels
NASA Technical Reports Server (NTRS)
Fromme, J. A.; Golberg, M. A.
1979-01-01
This paper surveys recent work on the use of integral equations for the calculation of wind tunnel interference. Due to the large number of possible physical situations, the discussion is limited to two-dimensional subsonic and transonic flows. In the subsonic case, the governing boundary value problems are shown to reduce to a class of Cauchy singular equations generalizing the classical airfoil equation. The theory and numerical solution are developed in some detail. For transonic flows nonlinear singular equations result, and a brief discussion of the work of Kraft and Kraft and Lo on their numerical solution is given. Some typical numerical results are presented and directions for future research are indicated.
Effective density terms in proper integral equations
NASA Astrophysics Data System (ADS)
Dyer, Kippi M.; Perkyns, John S.; Pettitt, B. Montgomery
2005-11-01
Two complementary routes to a new integral equation theory for site-site molecular fluids are presented. First, a simple approximation to a subset of the atomic site bridge functions in the diagrammatically proper integral equation theory is presented. This in turn leads to a form analogous to the reactive fluid theory, in which the normalization of the intramolecular distribution function and the value of the off-diagonal elements in the density matrix of the proper integral equations are the means of propagating the bridge function approximation. Second, a derivation from a topological expansion of a model for the single-site activity followed by a topological reduction and low-order truncation is given. This leads to an approximate numerical value for the new density coefficient. The resulting equations give a substantial improvement over the standard construction as shown with a series of simple diatomic model calculations.
Lomba, Enrique; Bores, Cecilia; Notario, Rafael; Sánchez-Gil, V
2016-09-01
In this work we have assessed the ability of a recently proposed three-dimensional integral equation approach to describe the explicit spatial distribution of molecular hydrogen confined in a crystal formed by short-capped nanotubes of C50 H10. To that aim we have resorted to extensive molecular simulation calculations whose results have been compared with our three-dimensional integral equation approximation. We have first tested the ability of a single C50 H10 nanocage for the encapsulation of H2 by means of molecular dynamics simulations, in particular using targeted molecular dynamics to estimate the binding Gibbs energy of a host hydrogen molecule inside the nanocage. Then, we have investigated the adsorption isotherm of the nanocage crystal using grand canonical Monte Carlo simulations in order to evaluate the maximum load of molecular hydrogen. For a packing close to the maximum load explicit hydrogen density maps and density profiles have been determined using molecular dynamics simulations and the three-dimensional Ornstein-Zernike equation with a hypernetted chain closure. In these conditions of extremely tight confinement the theoretical approach has shown to be able to reproduce the three-dimensional structure of the adsorbed fluid with accuracy down to the finest details. PMID:27367179
NASA Astrophysics Data System (ADS)
Lomba, Enrique; Bores, Cecilia; Notario, Rafael; Sánchez-Gil, V.
2016-09-01
In this work we have assessed the ability of a recently proposed three-dimensional integral equation approach to describe the explicit spatial distribution of molecular hydrogen confined in a crystal formed by short-capped nanotubes of C50 H10. To that aim we have resorted to extensive molecular simulation calculations whose results have been compared with our three-dimensional integral equation approximation. We have first tested the ability of a single C50 H10 nanocage for the encapsulation of H2 by means of molecular dynamics simulations, in particular using targeted molecular dynamics to estimate the binding Gibbs energy of a host hydrogen molecule inside the nanocage. Then, we have investigated the adsorption isotherm of the nanocage crystal using grand canonical Monte Carlo simulations in order to evaluate the maximum load of molecular hydrogen. For a packing close to the maximum load explicit hydrogen density maps and density profiles have been determined using molecular dynamics simulations and the three-dimensional Ornstein–Zernike equation with a hypernetted chain closure. In these conditions of extremely tight confinement the theoretical approach has shown to be able to reproduce the three-dimensional structure of the adsorbed fluid with accuracy down to the finest details.
Classification of integrable B-equations
NASA Astrophysics Data System (ADS)
van der Kamp, Peter H.
We classify integrable equations which have the form u t=a 1u n+K(v 0,v 1,…), v t=a 2v n, where a 1,a 2∈ C, n∈ N and K a quadratic polynomial in derivatives of v. This is done using the symbolic calculus, biunit coordinates and the Lech-Mahler theorem. Furthermore we present a method, based on resultants, to determine whether an equation is in a hierarchy of lower order.
Solution of a system of dual integral equations.
NASA Technical Reports Server (NTRS)
Buell, J.; Kagiwada, H.; Kalaba, R.; Ruspini, E.; Zagustin, E.
1972-01-01
The solution of a presented system of differential equations with initial values is shown to satisfy a system of dual integral equations of a type appearing in the study of axisymmetric problems of potential theory. Of practical interest are possible applications in biomechanics, particularly, for the case of trauma due to impact.
Radial symmetry and monotonicity for an integral equation
NASA Astrophysics Data System (ADS)
Ma, Li; Chen, Dezhong
2008-06-01
In this paper we study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space . Instead of the usual method of moving planes, we use a new Hardy-Littlewood-Sobolev (HLS) type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.
Exact Solutions and Conservation Laws for a New Integrable Equation
Gandarias, M. L.; Bruzon, M. S.
2010-09-30
In this work we study a generalization of an integrable equation proposed by Qiao and Liu from the point of view of the theory of symmetry reductions in partial differential equations. Among the solutions we obtain a travelling wave with decaying velocity and a smooth soliton solution. We determine the subclass of these equations which are quasi-self-adjoint and we get a nontrivial conservation law.
Huš, Matej; Urbic, Tomaz; Munaò, Gianmarco
2014-10-28
Thermodynamic and structural properties of a coarse-grained model of methanol are examined by Monte Carlo simulations and reference interaction site model (RISM) integral equation theory. Methanol particles are described as dimers formed from an apolar Lennard-Jones sphere, mimicking the methyl group, and a sphere with a core-softened potential as the hydroxyl group. Different closure approximations of the RISM theory are compared and discussed. The liquid structure of methanol is investigated by calculating site-site radial distribution functions and static structure factors for a wide range of temperatures and densities. Results obtained show a good agreement between RISM and Monte Carlo simulations. The phase behavior of methanol is investigated by employing different thermodynamic routes for the calculation of the RISM free energy, drawing gas-liquid coexistence curves that match the simulation data. Preliminary indications for a putative second critical point between two different liquid phases of methanol are also discussed.
Huš, Matej; Munaò, Gianmarco; Urbic, Tomaz
2014-01-01
Thermodynamic and structural properties of a coarse-grained model of methanol are examined by Monte Carlo simulations and reference interaction site model (RISM) integral equation theory. Methanol particles are described as dimers formed from an apolar Lennard-Jones sphere, mimicking the methyl group, and a sphere with a core-softened potential as the hydroxyl group. Different closure approximations of the RISM theory are compared and discussed. The liquid structure of methanol is investigated by calculating site-site radial distribution functions and static structure factors for a wide range of temperatures and densities. Results obtained show a good agreement between RISM and Monte Carlo simulations. The phase behavior of methanol is investigated by employing different thermodynamic routes for the calculation of the RISM free energy, drawing gas-liquid coexistence curves that match the simulation data. Preliminary indications for a putative second critical point between two different liquid phases of methanol are also discussed. PMID:25362323
NASA Astrophysics Data System (ADS)
Bracken, Paul
2010-04-01
A system of evolution equations can be developed from the structure equations for a submanifold embedded in a three-dimensional space. It is seen how these same equations can be obtained from a generalized matrix Lax pair provided a single constraint equation is imposed. This can be done in Euclidean space as well as in Minkowski space. The integrable systems which result from this process can be thought of as generalizing the SO(3) and SO(2,1) Lax pairs which have been studied previously.
Integral equations for the electromagnetic field in dielectrics
NASA Astrophysics Data System (ADS)
Mostowski, Jan; Załuska-Kotur, Magdalena A.
2016-09-01
We study static the electric field and electromagnetic waves in dielectric media. In contrast to the standard approach, we use, formulate and solve integral equations for the field. We discuss the case of an electrostatic field of a point charge placed inside a dielectric; the integral equation approach allows us to find and interpret the dielectric constant in terms of molecular polarizability. Next we discuss propagation of electromagnetic waves using the same integral equation approach. We derive the dispersion relation and find the reflection and transmission coefficients at the boundary between the vacuum and the dielectric. The present approach supplements the standard approach based on macroscopic Maxwell equations and contributes to better a understanding of some electromagnetic effects.
Recursion operators, conservation laws, and integrability conditions for difference equations
NASA Astrophysics Data System (ADS)
Mikhailov, A. V.; Wang, Jing Ping; Xenitidis, P.
2011-04-01
We attempt to propose an algebraic approach to the theory of integrable difference equations. We define the concept of a recursion operator for difference equations and show that it generates an infinite sequence of symmetries and canonical conservation laws for a difference equation. As in the case of partial differential equations, these canonical densities can serve as integrability conditions for difference equations. We obtain the recursion operators for the Viallet equation and all the Adler-Bobenko-Suris equations.
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.
To the theory of volterra integral equations of the first kind with discontinuous kernels
NASA Astrophysics Data System (ADS)
Apartsin, A. S.
2016-05-01
A nonclassical Volterra linear integral equation of the first kind describing the dynamics of an developing system with allowance for its age structure is considered. The connection of this equation with the classical Volterra linear integral equation of the first kind with a piecewise-smooth kernel is studied. For solving such equations, the quadrature method is applied.
Alternative field representations and integral equations for modeling inhomogeneous dielectrics
NASA Technical Reports Server (NTRS)
Volakis, John L.
1992-01-01
New volume and volume-surface integral equations are presented for modeling inhomogeneous dielectric regions. The presented integral equations result in more efficient numerical implementations and should, therefore, be useful in a variety of electromagnetic applications.
A path integral approach to the Langevin equation
NASA Astrophysics Data System (ADS)
Das, Ashok K.; Panda, Sudhakar; Santos, J. R. L.
2015-02-01
We study the Langevin equation with both a white noise and a colored noise. We construct the Lagrangian as well as the Hamiltonian for the generalized Langevin equation which leads naturally to a path integral description from first principles. This derivation clarifies the meaning of the additional fields introduced by Martin, Siggia and Rose in their functional formalism. We show that the transition amplitude, in this case, is the generating functional for correlation functions. We work out explicitly the correlation functions for the Markovian process of the Brownian motion of a free particle as well as for that of the non-Markovian process of the Brownian motion of a harmonic oscillator (Uhlenbeck-Ornstein model). The path integral description also leads to a simple derivation of the Fokker-Planck equation for the generalized Langevin equation.
Evolution equations: Frobenius integrability, conservation laws and travelling waves
NASA Astrophysics Data System (ADS)
Prince, Geoff; Tehseen, Naghmana
2015-10-01
We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss ‘local’ conservations laws for evolution equations in general and demonstrate all the results for the Korteweg-de Vries equation.
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
NASA Astrophysics Data System (ADS)
Pankratov, Oleg; Kuvshinov, Alexey
2016-01-01
second part, we summarize modern trends in the development of efficient 3-D EM forward modelling schemes with special emphasis on recent advances in the integral equation approach.
Distribution theory for Schrödinger’s integral equation
Lange, Rutger-Jan
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 both 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.
Distribution theory for Schrödinger's integral equation
NASA Astrophysics Data System (ADS)
Lange, Rutger-Jan
2015-12-01
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 both 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.
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
Bezerra, Rui M F; Pinto, Paula A; Fraga, Irene; Dias, Albino A
2016-03-01
To determine initial velocities of enzyme catalyzed reactions without theoretical errors it is necessary to consider the use of the integrated Michaelis-Menten equation. When the reaction product is an inhibitor, this approach is particularly important. Nevertheless, kinetic studies usually involved the evaluation of other inhibitors beyond the reaction product. The occurrence of these situations emphasizes the importance of extending the integrated Michaelis-Menten equation, assuming the simultaneous presence of more than one inhibitor because reaction product is always present. This methodology is illustrated with the reaction catalyzed by alkaline phosphatase inhibited by phosphate (reaction product, inhibitor 1) and urea (inhibitor 2). The approach is explained in a step by step manner using an Excel spreadsheet (available as a template in Appendix). Curve fitting by nonlinear regression was performed with the Solver add-in (Microsoft Office Excel). Discrimination of the kinetic models was carried out based on Akaike information criterion. This work presents a methodology that can be used to develop an automated process, to discriminate in real time the inhibition type and kinetic constants as data (product vs. time) are achieved by the spectrophotometer. PMID:26777432
Penke, Lars; Deary, Ian J
2010-09-01
Charlton et al. (2008) (Charlton, R.A., Landua, S., Schiavone, F., Barrick, T.R., Clark, C.A., Markus, H.S., Morris, R.G.A., 2008. Structural equation modelling investigation of age-related variance in executive function and DTI-measured white matter change. Neurobiol. Aging 29, 1547-1555) presented a model that suggests a specific age-related effect of white matter integrity on working memory. We illustrate potential pitfalls of structural equation modelling by criticizing their model for (a) its neglect of latent variables, (b) its complexity, (c) its questionable causal assumptions, (d) the use of empirical model reduction, (e) the mix-up of theoretical perspectives, and (f) the failure to compare alternative models. We show that a more parsimonious model, based solely on the well-established general factor of cognitive ability, fits their data at least as well. Importantly, when modelled this way there is no support for a role of white matter integrity in cognitive aging in this sample, indicating that their conclusion is strongly dependent on how the data are analysed. We suggest that evidence from more conclusive study designs is needed. PMID:20079555
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.
Darboux integrability of determinant and equations for principal minors
NASA Astrophysics Data System (ADS)
Demskoi, D. K.; Tran, D. T.
2016-07-01
We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian–Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order—this implies Darboux integrability. On the other hand, the recurrent formulae for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulae and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while, in the non-vanishing case, they are given in terms of solutions of ordinary linear equations.
Applying Quadrature Rules with Multiple Nodes to Solving Integral Equations
Hashemiparast, S. M.; Avazpour, L.
2008-09-01
There are many procedures for the numerical solution of Fredholm integral equations. The main idea in these procedures is accuracy of the solution. In this paper, we use Gaussian quadrature with multiple nodes to improve the solution of these integral equations. The application of this method is illustrated via some examples, the related tables are given at the end.
Poisson structures for lifts and periodic reductions of integrable lattice equations
NASA Astrophysics Data System (ADS)
Kouloukas, Theodoros E.; Tran, Dinh T.
2015-02-01
We introduce and study suitable Poisson structures for four-dimensional maps derived as lifts and specific periodic reductions of integrable lattice equations. These maps are Poisson with respect to these structures and the corresponding integrals are in involution.
An integrable shallow water equation with peaked solitons
Camassa, R.; Holm, D.D. )
1993-09-13
We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.
Calculation of unsteady transonic flows using the integral equation method
NASA Technical Reports Server (NTRS)
Nixon, D.
1978-01-01
The basic integral equations for a harmonically oscillating airfoil in a transonic flow with shock waves are derived; the reduced frequency is assumed to be small. The problems associated with shock wave motion are treated using a strained coordinate system. The integral equation is linear and consists of both line integrals and surface integrals over the flow field which are evaluated by quadrature. This leads to a set of linear algebraic equations that can be solved directly. The shock motion is obtained explicitly by enforcing the condition that the flow is continuous except at a shock wave. Results obtained for both lifting and nonlifting oscillatory flows agree satisfactorily with other accurate results.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Numerical integration of asymptotic solutions of ordinary differential equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Phase-integral solution of the radial Dirac equation
Linnaeus, Staffan
2010-03-15
A phase-integral (WKB) solution of the radial Dirac equation is constructed, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the classical transition points. The potential is allowed to be the time component of a four-vector, a Lorentz scalar, a pseudoscalar, or any combination of these. The key point in the construction is the transformation from two coupled first-order equations constituting the radial Dirac equation to a single second-order Schroedinger-type equation. This transformation can be carried out in infinitely many ways, giving rise to different second-order equations but with the same spectrum. A unique transformation is found that produces a particularly simple second-order equation and correspondingly simple and well-behaved phase-integral solutions. The resulting phase-integral formulas are applied to unbound and bound states of the Coulomb potential. For bound states, the exact energy levels are reproduced.
Exponential Methods for the Time Integration of Schroedinger Equation
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.
NASA Astrophysics Data System (ADS)
Bagderina, Yulia Yu
2016-04-01
Scalar second-order ordinary differential equations with cubic nonlinearity in the first-order derivative are considered. Lie symmetries admitted by an arbitrary equation are described in terms of the invariants of this family of equations. Constructing the first integrals is discussed. We study also the equations which have the first integral rational in the first-order derivative.
Stability of negative solitary waves for an integrable modified Camassa-Holm equation
Yin Jiuli; Tian Lixin; Fan Xinghua
2010-05-15
In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.
Integrable cosmological models from higher dimensional Einstein equations
Sano, Masakazu; Suzuki, Hisao
2007-09-15
We consider the cosmological models for the higher dimensional space-time which includes the curvatures of our space as well as the curvatures of the internal space. We find that the condition for the integrability of the cosmological equations is that the total space-time dimensions are D=10 or D=11 which is exactly the conditions for superstrings or M theory. We obtain analytic solutions with generic initial conditions in the four-dimensional Einstein frame and study the accelerating universe when both our space and the internal space have negative curvatures.
Integrability of the Wong Equations in the Class of Linear Integrals of Motion
NASA Astrophysics Data System (ADS)
Magazev, A. A.
2016-04-01
The Wong equations, which describe the motion of a classical charged particle with isospin in an external gauge field, are considered. The structure of the Lie algebra of the linear integrals of motion of these equations is investigated. An algebraic condition for integrability of the Wong equations is formulated. Some examples are considered.
LETTER TO THE EDITOR: Deconstructing an integrable lattice equation
NASA Astrophysics Data System (ADS)
Ramani, A.; Joshi, N.; Grammaticos, B.; Tamizhmani, T.
2006-02-01
We show that an integrable lattice equation, obtained by J Hietarinta using the 'consistency around a cube' method without the tetrahedron assumption, is indeed solvable by linearization. We also present its nonautonomous extension.
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.
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 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.
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…
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.
Dirac equation for particles with arbitrary half-integral spin
NASA Astrophysics Data System (ADS)
Guseinov, I. I.
2011-11-01
The sets of ? -component irreducible and Clifford algebraic Hermitian and unitary matrices through the two-component Pauli matrices are suggested, where s = 1/2, 3/2, 5/2, … . Using these matrix sets, the eigenvalues of which are ? , the ? -component generalized Dirac equation for a description of arbitrary half-integral spin particles is constructed. In accordance with the correspondence principle, the generalized Dirac equation suggested arises from the condition of relativistic invariance. This equation is reduced to the sets of two-component matrix equations the number of which is equal to ? . The new relativistic invariant equation of motion leads to an equation of continuity with a positive-definite probability density and also to the Klein-Gordon equation. This relativistic equation is causal in the presence of an external electromagnetic field interaction. It is shown that, in the case of nonrelativistic limit, the relativistic equation presented is reduced to the Pauli equation describing the motion of half-integral spin particle in the electromagnetic field.
Non-isotropic solution of an OZ equation: matrix methods for integral equations
NASA Astrophysics Data System (ADS)
Chen, Zhuo-Min; Pettitt, B. Montgomery
1995-02-01
Integral equations of the Ornstein-Zernike (OZ) type have been useful constructs in the theory of liquids for nearly a century. Only a limited number of model systems yield an analytic solution; the rest must be solved numerically. For anisotropic systems the numerical problems are heightened by the coupling of more unknowns and equations. A matrix method for solving the full anisotropic OZ integral equation is presented. The method is compared in the isotropic limit with traditional approaches. Examples are given for a 1-D fluid with a corrugated (periodic) external potential. The full two point correlation functions for both isotropic and anisotropic systems are given and discussed.
An integrable shallow water equation with linear and nonlinear dispersion.
Dullin, H R; Gottwald, G A; Holm, D D
2001-11-01
We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases. PMID:11690414
On constructing accurate approximations of first integrals for difference equations
NASA Astrophysics Data System (ADS)
Rafei, M.; Van Horssen, W. T.
2013-04-01
In this paper, a perturbation method based on invariance factors and multiple scales will be presented for weakly nonlinear, regularly perturbed systems of ordinary difference equations. Asymptotic approximations of first integrals will be constructed on long iteration-scales, that is, on iteration-scales of order ɛ-1, where ɛ is a small parameter. It will be shown that all invariance factors have to satisfy a functional equation. To show how this perturbation method works, the method is applied to a Van der Pol equation, and a Rayleigh equation. It will be explicitly shown for the first time in the literature how these multiple scales should be introduced for systems of difference equations to obtain very accurate approximations of first integrals on long iteration-scales.
Numerical integration of ordinary differential equations on manifolds
NASA Astrophysics Data System (ADS)
Crouch, P. E.; Grossman, R.
1993-12-01
This paper is concerned with the problem of developing numerical integration algorithms for differential equations that, when viewed as equations in some Euclidean space, naturally evolve on some embedded submanifold. It is desired to construct algorithms whose iterates also evolve on the same manifold. These algorithms can therefore be viewed as integrating ordinary differential equations on manifolds. The basic method “decouples” the computation of flows on the submanifold from the numerical integration process. It is shown that two classes of single-step and multistep algorithms can be posed and analyzed theoretically, using the concept of “freezing” the coefficients of differential operators obtained from the defining vector field. Explicit third-order algorithms are derived, with additional equations augmenting those of their classical counterparts, obtained from “obstructions” defined by nonvanishing Lie brackets.
Phase space lattices and integrable nonlinear wave equations
NASA Astrophysics Data System (ADS)
Tracy, Eugene; Zobin, Nahum
2003-10-01
Nonlinear wave equations in fluids and plasmas that are integrable by Inverse Scattering Theory (IST), such as the Korteweg-deVries and nonlinear Schrodinger equations, are known to be infinite-dimensional Hamiltonian systems [1]. These are of interest physically because they predict new phenomena not present in linear wave theories, such as solitons and rogue waves. The IST method provides solutions of these equations in terms of a special class of functions called Riemann theta functions. The usual approach to the theory of theta functions tends to obscure the underlying phase space structure. A theory due to Mumford and Igusa [2], however shows that the theta functions arise naturally in the study of phase space lattices. We will describe this theory, as well as potential applications to nonlinear signal processing and the statistical theory of nonlinear waves. 1] , S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of solitons: the inverse scattering method (Consultants Bureau, New York, 1984). 2] D. Mumford, Tata lectures on theta, Vols. I-III (Birkhauser); J. Igusa, Theta functions (Springer-Verlag, New York, 1972).
Computing the Casimir force using regularized boundary integral equations
NASA Astrophysics Data System (ADS)
Kilen, Isak; Jakobsen, Per Kristen
2014-11-01
In this paper we use a novel regularization procedure to reduce the calculation of the Casimir force for 2D scalar fields between compact objects to the solution of a classical integral equation defined on the boundaries of the objects. The scalar fields are subject to Dirichlet boundary conditions on the object boundaries. We test the integral equation by comparing with what we get for parallel plates, concentric circles and adjacent circles using mode summation and the functional integral method. We show how symmetries in the shapes and configuration of boundaries can easily be incorporated into our method and that it leads to fast evaluation of the Casimir force for symmetric situations.
Numerical solution of boundary-integral equations for molecular electrostatics.
Bardhan, Jaydeep P
2009-03-01
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. PMID:19275391
Coupling finite element and integral equation solutions using decoupled boundary meshes
NASA Technical Reports Server (NTRS)
Cwik, Tom
1992-01-01
A method is outlined for calculating scattered fields from inhomogeneous penetrable objects using a coupled finite element-integral equation solution. The finite element equation can efficiently model fields in penetrable and inhomogeneous regions, while the integral equation exactly models fields on the finite element mesh boundary and in the exterior region. By decoupling the interior finite element and exterior integral equation meshes, considerable flexibility is found in both the number of field expansion points as well as their density. Only the nonmetal portions of the object need be modeled using a finite element expansion; exterior perfect conducting surfaces are modeled using an integral equation with a single unknown field since E(tan) is identically zero on these surfaces. Numerical convergence, accuracy, and stability at interior resonant frequencies are studied in detail.
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.
Integration of the equations of movement in dead reckoning navigation
NASA Astrophysics Data System (ADS)
Banachowicz, A.; Wolski, A.
2012-04-01
Calculations of position coordinates in dead reckoning navigation essentially comes down to the integration of ship movements assuming an initial condition (position) of the ship. This corresponds to Cauchy's problem. However, in this case the ship's velocity vector as a derivative of its track (trajectory) is not a given function, but comes from navigational measurements performed in discrete time instants. Due to the discrete character of velocity vector or acceleration measurements, ship's movement equations particularly qualify for numerical calculations. In this case the equation nodes are the time instants of measurements and navigational parameter values read out at those instants. This article presents the applications of numerical integration of differential equations (movement) for measurements of velocity vectors and acceleration vector (inertial navigation systems). The considerations are illustrated with navigational measurements recorded during sea trials of the rescue ship integrated system.
An integral equation solution for multistage turbomachinery design calculations
NASA Technical Reports Server (NTRS)
Mcfarland, Eric R.
1993-01-01
A method was developed to calculate flows in multistage turbomachinery. The method is an extension of quasi-three-dimensional blade-to-blade solution methods. Governing equations for steady compressible inviscid flow are linearized by introducing approximations. The linearized flow equations are solved using integral equation techniques. The flows through both stationary and rotating blade rows are determined in a single calculation. Multiple bodies can be modelled for each blade row, so that arbitrary blade counts can be analyzed. The method's benefits are its speed and versatility.
Local Integral Estimates for Quasilinear Equations with Measure Data
Tian, Qiaoyu; Zhang, Shengzhi; Xu, Yonglin; Mu, Jia
2016-01-01
Local integral estimates as well as local nonexistence results for a class of quasilinear equations −Δpu = σP(u) + ω for p > 1 and Hessian equations Fk[−u] = σP(u) + ω were established, where σ is a nonnegative locally integrable function or, more generally, a locally finite measure, ω is a positive Radon measure, and P(u) ~ expαuβ with α > 0 and β ≥ 1 or P(u) = up−1. PMID:27294190
Local Integral Estimates for Quasilinear Equations with Measure Data.
Tian, Qiaoyu; Zhang, Shengzhi; Xu, Yonglin; Mu, Jia
2016-01-01
Local integral estimates as well as local nonexistence results for a class of quasilinear equations -Δ p u = σP(u) + ω for p > 1 and Hessian equations F k [-u] = σP(u) + ω were established, where σ is a nonnegative locally integrable function or, more generally, a locally finite measure, ω is a positive Radon measure, and P(u) ~ expαu (β) with α > 0 and β ≥ 1 or P(u) = u (p-1). PMID:27294190
The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation
NASA Astrophysics Data System (ADS)
Dong, Huanhe; Zhang, Yong; Zhang, Xiaoen
2016-07-01
A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.
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.
Comparison of four stable numerical methods for Abel's integral equation
NASA Technical Reports Server (NTRS)
Murio, Diego A.; Mejia, Carlos E.
1991-01-01
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.
Time Integration Schemes for the Unsteady Navier-stokes Equations
NASA Technical Reports Server (NTRS)
Bijl, Hester; Carpenter, Mark H.; Vatsa, Veer N.
2001-01-01
The efficiency and accuracy of several time integration schemes are investigated for the unsteady Navier-Stokes equations. This study focuses on the efficiency of higher-order Runge-Kutta schemes in comparison with the popular Backward Differencing Formulations. For this comparison an unsteady two-dimensional laminar flow problem is chosen, i.e., flow around a circular cylinder at Re = 1200. It is concluded that for realistic error tolerances (smaller than 10(exp -1)) fourth-and fifth-order Runge-Kutta schemes are the most efficient. For reasons of robustness and computer storage, the fourth-order Runge-Kutta method is recommended. The efficiency of the fourth-order Runge-Kutta scheme exceeds that of second-order Backward Difference Formula by a factor of 2.5 at engineering error tolerance levels (10(exp -1) to 10(exp -2)). Efficiency gains are more dramatic at smaller tolerances.
NASA Astrophysics Data System (ADS)
Jiang, Shidong; Luo, Li-Shi
2016-07-01
The integral equation for the flow velocity u (x ; k) in the steady Couette flow derived from the linearized Bhatnagar-Gross-Krook-Welander kinetic equation is studied in detail both theoretically and numerically in a wide range of the Knudsen number k between 0.003 and 100.0. First, it is shown that the integral equation is a Fredholm equation of the second kind in which the norm of the compact integral operator is less than 1 on Lp for any 1 ≤ p ≤ ∞ and thus there exists a unique solution to the integral equation via the Neumann series. Second, it is shown that the solution is logarithmically singular at the endpoints. More precisely, if x = 0 is an endpoint, then the solution can be expanded as a double power series of the form ∑n=0∞∑m=0∞cn,mxn(xln x) m about x = 0 on a small interval x ∈ (0 , a) for some a > 0. And third, a high-order adaptive numerical algorithm is designed to compute the solution numerically to high precision. The solutions for the flow velocity u (x ; k), the stress Pxy (k), and the half-channel mass flow rate Q (k) are obtained in a wide range of the Knudsen number 0.003 ≤ k ≤ 100.0; and these solutions are accurate for at least twelve significant digits or better, thus they can be used as benchmark solutions.
Volume integrals of ellipsoids associated with the inhomogeneous Helmholtz equation
NASA Technical Reports Server (NTRS)
Fu, L. S.; Mura, T.
1982-01-01
Problems of wave phenomena in the fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the inhomogeneous Helmholtz equation, for an ellipsoidal region. By using appropriate Taylor series expansions and the multinomial theorem, these volume integrals are obtained in series form for regions r greater than r-prime and r less than r-prime, where r and r-prime are the distances from the origin to the point of observation and the source. Derivatives of these integrals are easily evaluated. When the wavenumber approaches zero the results reduce directly to the potentials of ellipsoids of variable densities.
The pentabox Master Integrals with the Simplified Differential Equations approach
NASA Astrophysics Data System (ADS)
Papadopoulos, Costas G.; Tommasini, Damiano; Wever, Christopher
2016-04-01
We present the calculation of massless two-loop Master Integrals relevant to five-point amplitudes with one off-shell external leg and derive the complete set of planar Master Integrals with five on-mass-shell legs, that contribute to many 2 → 3 amplitudes of interest at the LHC, as for instance three jet production, γ , V, H + 2 jets etc., based on the Simplified Differential Equations approach.
NASA Astrophysics Data System (ADS)
Khalilov, E. H.
2016-07-01
The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.
Integral Equations and the Bound-State Problem.
ERIC Educational Resources Information Center
Bagchi, B.; Seyler, R. G.
1980-01-01
An integral equation for the s-wave bound-state solution is derived and then solved for a square-well potential. It is shown that the scattering solutions continue to exist at negative energies, and when evaluated at the energy of a bound state these solutions do reduce to the bound-state solution.
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.
Efficient Integration of Quantum Mechanical Wave Equations by Unitary Transforms
Bauke, Heiko; Keitel, Christoph H.
2009-08-13
The integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The energy and momentum spectrum of a wave function imposes fundamental limits on the performance of numerical algorithms for this problem. We demonstrate how unitary transforms can help to surmount these limitations.
On the Implementation of 3D Galerkin Boundary Integral Equations
Nintcheu Fata, Sylvain; Gray, Leonard J
2010-01-01
In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of singular and hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.
Integral equations for the microstructures of supercritical fluids
Lee, L.L.; Cochran, H.D.
1993-11-01
Molecular interactions and molecular distributions are at the heart of the supercritical behavior of fluid mixtures. The distributions, i.e. structure, can be obtained through any of the three routes: (1) scattering experiments, (2) Monte Carlo or molecular dynamics simulation, and (3) integral equations that govern the relation between the molecular interactions u(r) and the probability distributions g{sub ij}(r). Most integral equations are based on the Ornstein-Zernike relation connecting the total correlation to the direct correlation. The OZ relation requires a {open_quotes}closure{close_quotes} equation to be solvable. Thus the Percus-Yevick, hypernetted chain, and mean spherical approximations have been proposed. The authors outline the numerical methods of solution for these integral equations, including the Picard, Labik-Gillan, and Baxter methods. Solution of these equations yields the solvent-solute, solvent-solvent, and solute-solute pair correlation functions (pcf`s). Interestingly, these pcf`s exhibit characteristical signatures for supercritical mixtures that are classified as {open_quotes}attractive{close_quotes} or {open_quotes}repulsive{close_quotes} in nature. Close to the critical locus, the pcf shows enhanced first neighbor peaks with concomitant long-range build-ups (sic attractive behavior) or reduced first peaks plus long-range depletion (sic repulsive behavior) of neighbors. For ternary mixtures with entrainers, there are synergistic effects between solvent and cosolvent, or solute and cosolute. These are also detectable on the distribution function level. The thermodynamic consequences are deciphered through the Kirkwood-Buff fluctuation integrals (G{sub ij}) and their matrix inverses: the direct correlation function integrals (DCFI`s). These quantities connect the correlation functions to the chemical potential derivatives (macroscopic variables) thus acting as {open_quotes}bridges{close_quotes} between the two Weltanschauungen.
Boundary regularized integral equation formulation of the Helmholtz equation in acoustics.
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
Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation
NASA Astrophysics Data System (ADS)
Kazakov, A. Ya.; Slavyanov, S. Yu.
2008-05-01
Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painlevé PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation.
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.
Singularity Preserving Numerical Methods for Boundary Integral Equations
NASA Technical Reports Server (NTRS)
Kaneko, Hideaki (Principal Investigator)
1996-01-01
In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.
Hamiltonian time integrators for Vlasov-Maxwell equations
He, Yang; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian; Qin, Hong; Sun, Yajuan
2015-12-15
Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.
Podesta, John J.
2012-08-15
The electric field generated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform (static) flow velocity is studied in the electrostatic approximation which is often valid in the near field. For plasmas characterized by Maxwell distribution functions with isotropic temperatures, the linearized Vlasov-Poisson equations may be formulated in terms of an equivalent integral equation in the time domain. The kernel of the integral equation has a relatively simple mathematical form consisting of elementary functions such as exponential and trigonometric functions (sines and cosines), and contains no infinite sums of Bessel functions. Consequently, the integral equation is amenable to numerical solutions and may be useful for the study of the impulse response of magnetized plasmas and, more generally, the response to arbitrary waveforms.
ISDEP: Integrator of stochastic differential equations for plasmas
NASA Astrophysics Data System (ADS)
Velasco, J. L.; Bustos, A.; Castejón, F.; Fernández, L. A.; Martin-Mayor, V.; Tarancón, A.
2012-09-01
In this paper we present a general description of the ISDEP code (Integrator of Stochastic Differential Equations for Plasmas) and a brief overview of its physical results and applications so far. ISDEP is a Monte Carlo code that calculates the distribution function of a minority population of ions in a magnetized plasma. It solves the ion equations of motion taking into account the complex 3D structure of fusion devices, the confining electromagnetic field and collisions with other plasma species. The Monte Carlo method used is based on the equivalence between the Fokker-Planck and Langevin equations. This allows ISDEP to run in distributed computing platforms without communication between nodes with almost linear scaling. This paper intends to be a general description and a reference paper in ISDEP.
Solving integral equations for binary and ternary systems
NASA Astrophysics Data System (ADS)
Nader Lotfollahi, Mohammad; Modarress, Hamid
2002-02-01
Solving integral equations is an effective approach to obtain the radial distribution function (RDF) of multicomponent mixtures. In this work, by extending Gillan's approach [M. J. Gillan, Mol. Phys. 38(6), 1781 (1979)], the integral equation was solved by numerical method and was applied to both binary and ternary mixtures. The Lennard-Jones (LJ) potential function was used to express the pair molecular interactions in calculating the RDF and chemical potential. This allowed a comparison with available simulation data, on the RDF and the chemical potential, since the simulation data have been reported for the LJ potential function. The RDF and the chemical potential results indicated good agreement with the simulation data. The calculations were extended to the ternary system and the RDFs for carbon dioxide-octane-naphthalene were obtained. The numerical method used in solving integral equation was rapidly convergent and not sensitive to the first estimation. The method proposed in this work can be easily extended to more than the three-component systems.
Poisson's equation solution of Coulomb integrals in atoms and molecules
NASA Astrophysics Data System (ADS)
Weatherford, Charles A.; Red, Eddie; Joseph, Dwayne; Hoggan, Philip
The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behaviour of atomic orbitals. In this work, it is shown that the two-centre Coulomb integrals involved can be expressed as one-electron kinetic-energy-like integrals. This is accomplished using the fact that the Coulomb operator is a Green's function of the Laplacian. The ensuing integrals may be further simplified by defining Coulomb forms for the one-electron potential satisfying Poisson's equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. The remaining questions of translating orbitals involved in three and four centre integrals and the evaluation of exchange energy are also briefly discussed. The summation coefficients in Coulomb forms are evaluated using the LU decomposition. This algorithm is highly parallel. The Poisson method may be used to calculate Coulomb energy integrals efficiently. For a single processor, gains of CPU time for a given chemical accuracy exceed a factor of 40. This method lends itself to evaluation on a parallel computer.
Initial states in integrable quantum field theory quenches from an integral equation hierarchy
NASA Astrophysics Data System (ADS)
Horváth, D. X.; Sotiriadis, S.; Takács, G.
2016-01-01
We consider the problem of determining the initial state of integrable quantum field theory quenches in terms of the post-quench eigenstates. The corresponding overlaps are a fundamental input to most exact methods to treat integrable quantum quenches. We construct and examine an infinite integral equation hierarchy based on the form factor bootstrap, proposed earlier as a set of conditions determining the overlaps. Using quenches of the mass and interaction in Sinh-Gordon theory as a concrete example, we present theoretical arguments that the state has the squeezed coherent form expected for integrable quenches, and supporting an Ansatz for the solution of the hierarchy. Moreover we also develop an iterative method to solve numerically the lowest equation of the hierarchy. The iterative solution along with extensive numerical checks performed using the next equation of the hierarchy provides a strong numerical evidence that the proposed Ansatz gives a very good approximation for the solution.
One-way spatial integration of hyperbolic equations
NASA Astrophysics Data System (ADS)
Towne, Aaron; Colonius, Tim
2015-11-01
In this paper, we develop and demonstrate a method for constructing well-posed one-way approximations of linear hyperbolic systems. We use a semi-discrete approach that allows the method to be applied to a wider class of problems than existing methods based on analytical factorization of idealized dispersion relations. After establishing the existence of an exact one-way equation for systems whose coefficients do not vary along the axis of integration, efficient approximations of the one-way operator are constructed by generalizing techniques previously used to create nonreflecting boundary conditions. When physically justified, the method can be applied to systems with slowly varying coefficients in the direction of integration. To demonstrate the accuracy and computational efficiency of the approach, the method is applied to model problems in acoustics and fluid dynamics via the linearized Euler equations; in particular we consider the scattering of sound waves from a vortex and the evolution of hydrodynamic wavepackets in a spatially evolving jet. The latter problem shows the potential of the method to offer a systematic, convergent alternative to ad hoc regularizations such as the parabolized stability equations.
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…
NASA Astrophysics Data System (ADS)
Imai, Kenji
2014-02-01
In this paper, a new n-dimensional homogeneous Lotka-Volterra (HLV) equation, which possesses a Lie symmetry, is derived by the extension from a three-dimensional HLV equation. Its integrability is shown from the viewpoint of Lie symmetries. Furthermore, we derive dynamical systems of higher order, which possess the Lie symmetry, using the algebraic structure of this HLV equation.
NASA Technical Reports Server (NTRS)
Baker, A. J.; Soliman, M. O.
1978-01-01
A study of accuracy and convergence of linear functional finite element solution to linear parabolic and hyperbolic partial differential equations is presented. A variable-implicit integration procedure is employed for the resultant system of ordinary differential equations. Accuracy and convergence is compared for the consistent and two lumped assembly procedures for the identified initial-value matrix structure. Truncation error estimation is accomplished using Richardson extrapolation.
Darboux Transformation for the Vector Sine-Gordon Equation and Integrable Equations on a Sphere
NASA Astrophysics Data System (ADS)
Mikhailov, Alexander V.; Papamikos, Georgios; Wang, Jing Ping
2016-07-01
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its Bäcklund transformations. We show that there is a new Lax operator canonically associated with our Darboux transformation resulting an evolutionary differential-difference system on a sphere. The latter is a generalised symmetry for the chain of Bäcklund transformations. Using the re-factorisation approach and the Bianchi permutability of the Darboux transformations, we derive new vector Yang-Baxter map and integrable discrete vector sine-Gordon equation on a sphere.
Exponential integrators for the incompressible Navier-Stokes equations.
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
Investigation of ODE integrators using interactive graphics. [Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Brown, R. L.
1978-01-01
Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.
Phase-integral method for the radial Dirac equation
Linnæus, Staffan
2014-09-15
A phase-integral (WKB) solution of the radial Dirac equation is calculated up to the third order of approximation, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the zeroth-order transition points. The potential is allowed to be of scalar, vector, or tensor type, or any combination of these. The connection problem is investigated in detail. Explicit formulas are given for single-turning-point phase shifts and single-well energy levels.
Integral equation for gauge invariant quark Green's function
Sazdjian, H.
2008-08-29
We consider gauge invariant quark two-point Green's functions in which the gluonic phase factor follows a skew-polygonal line. Using a particular representation for the quark propagator in the presence of an external gluon field, functional relations between Green's functions with different numbers of segments of the polygonal lines are established. An integral equation is obtained for the Green's function having a phase factor along a single straight line. The related kernels involve Wilson loops with skew-polygonal contours and with functional derivatives along the sides of the contours.
NASA Technical Reports Server (NTRS)
Hayes, E. F.; Kouri, D. J.
1971-01-01
Coupled integral equations are derived for the full scattering amplitudes for both reactive and nonreactive channels. The equations do not involve any partial wave expansion and are obtained using channel operators for reactive and nonreactive collisions. These coupled integral equations are similar in nature to equations derived for purely nonreactive collisions of structureless particles. Using numerical quadrature techniques, these equations may be reduced to simultaneous algebraic equations which may then be solved.
The Integration of Teacher's Pedagogical Content Knowledge Components in Teaching Linear Equation
ERIC Educational Resources Information Center
Yusof, Yusminah Mohd.; Effandi, Zakaria
2015-01-01
This qualitative research aimed to explore the integration of the components of pedagogical content knowledge (PCK) in teaching Linear Equation with one unknown. For the purpose of the study, a single local case study with multiple participants was used. The selection of the participants was made based on various criteria: having more than 5 years…
Exact solutions for the fractional differential equations by using the first integral method
NASA Astrophysics Data System (ADS)
Aminikhah, Hossein; Sheikhani, A. Refahi; Rezazadeh, Hadi
2015-03-01
In this paper, we apply the first integral method to study the solutions of the nonlinear fractional modified Benjamin-Bona-Mahony equation, the nonlinear fractional modified Zakharov-Kuznetsov equation and the nonlinear fractional Whitham-Broer-Kaup-Like systems. This method is based on the ring theory of commutative algebra. The results obtained by the proposed method show that the approach is effective and general. This approach can also be applied to other nonlinear fractional differential equations, which are arising in the theory of solitons and other areas.
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.
Numerical solution of a class of integral equations arising in two-dimensional aerodynamics
NASA Technical Reports Server (NTRS)
Fromme, J.; Golberg, M. A.
1978-01-01
We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.
ERIC Educational Resources Information Center
Alejandra, Almirón; Fernando, Bifano; Leonardo, Lupinacci
2015-01-01
Solving systems of equations at school, at least in Argentina, is usually a task that students are given as a series of techniques that "allow" them to find a solution. How to overcome educational obstacles that are generated from a fragmented approach of knowledge? What can DGS do, in particular the CAS environment? What epistemic and…
On Lie symmetries, exact solutions and integrability to the KdV-Sawada-Kotera-Ramani equation
NASA Astrophysics Data System (ADS)
Ma, Pan-Li; Tian, Shou-Fu; Zhang, Tian-Tian; Zhang, Xing-Yong
2016-04-01
In this paper, the KdV-Sawada-Kotera-Ramani equation is investigated, which is used to describe the resonances of solitons in one-dimensional space. By using the Lie symmetry analysis method, the vector field and optimal system of the equation are derived, respectively. The optimal system is further used to study the symmetry reductions and exact solutions. Furthermore, the exact analytic solutions of the equation can be obtained by considering the power series theory. Finally, the complete integrability of the equation is systematically presented by using binary Bell's polynomials, which includes the bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws. Based on its bilinear representation, the N-soliton solutions of the equation are also constructed with exact analytic expression.
An integrated development of the equations of motion for elastic hypersonic flight vehicles
NASA Technical Reports Server (NTRS)
Bilimoria, Karl D.; Schmidt, David K.
1992-01-01
An integrated, consistent analytical framework is developed for modeling the dynamics of elastic hypersonic flight vehicles. A Lagrangian approach is used in order to capture the dynamics of rigid-body motion, elastic deformation, fluid flow, rotating machinery, wind, and a spherical rotating earth model, and to account for their interactions with each other. A vector form of the force, moment and elastic-deformation equations is developed from Lagrange's equation; a useable scalar form of these equations is also presented. The appropriate kinematic equations are developed, and are presented in a useable form. A preliminary study of the significance of selected terms in the equations of motion is conducted. Using generic data for a single-stage-to-orbit vehicle, it was found that the Coriolis force can reach values of up to 6 percent of the vehicle weight, and that the forces and moments attributable to fluid-flow terms can be significant.
Discretization of the induced-charge boundary integral equation.
Bardhan, J. P.; Eisenberg, R. S.; Gillespie, D.; Rush Univ. Medical Center
2009-07-01
Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch et al. [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.
Discretization of the induced-charge boundary integral equation
NASA Astrophysics Data System (ADS)
Bardhan, Jaydeep P.; Eisenberg, Robert S.; Gillespie, Dirk
2009-07-01
Boundary-element methods (BEMs) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein-solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch [IEEE Trans Comput.-Comput.-Aided Des. 20, 1398 (2001)] to compute the BEM matrix elements is always more accurate than the traditional centroid-collocation method. Qualocation is not more expensive to implement than collocation and can save significant computational time by reducing the number of boundary elements needed to discretize the dielectric interfaces.
Discretization of the Induced-Charge Boundary Integral Equation
Bardhan, Jaydeep P.; Eisenberg, Robert S.; Gillespie, Dirk
2013-01-01
Boundary-element methods (BEM) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few Angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein/solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch, Wang, and White (IEEE. Trans. Comput.-Aided Des. 20:1398, 2001) to compute the BEM matrix elements is always more accurate than the traditional centroid collocation method. Qualocation is no more expensive to implement than collocation and can save significant computional time by reducing the number of boundary elements needed to discretize the dielectric interfaces. PMID:19658728
Integrated optics technology study
NASA Technical Reports Server (NTRS)
Chen, B.
1982-01-01
The materials and processes available for the fabrication of single mode integrated electrooptical components are described. Issues included in the study are: (1) host material and orientation, (2) waveguide formation, (3) optical loss mechanisms, (4) wavelength selection, (5) polarization effects and control, (6) laser to integrated optics coupling,(7) fiber optic waveguides to integrated optics coupling, (8) souces, (9) detectors. The best materials, technology and processes for fabrication of integrated optical components for communications and fiber gyro applications are recommended.
Integration of the Equations of Classical Electrode-Effect Theory with Aerosols
NASA Astrophysics Data System (ADS)
Kalinin, A. V.; Leont'ev, N. V.; Terent'ev, A. M.; Umnikov, E. D.
2016-04-01
This paper is devoted to an analytical study of the one-dimensional stationary system of equations for modeling of the electrode effect in the Earth's atmospheric layer with aerosols. New integrals of the system are derived. Using these integrals, the expressions for solutions of the system and estimates of the electrode layer's thickness as a function of the aerosol concentration are obtained for numerical parameters close to real.
Integration of the Equations of Classical Electrode-Effect Theory with Aerosols
NASA Astrophysics Data System (ADS)
Kalinin, A. V.; Leont'ev, N. V.; Terent'ev, A. M.; Umnikov, E. D.
2016-05-01
This paper is devoted to an analytical study of the one-dimensional stationary system of equations for modeling of the electrode effect in the Earth's atmospheric layer with aerosols. New integrals of the system are derived. Using these integrals, the expressions for solutions of the system and estimates of the electrode layer's thickness as a function of the aerosol concentration are obtained for numerical parameters close to real.
Chen, Ke
1996-12-31
We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.
ERIC Educational Resources Information Center
Marsh, Herbert W.; Muthen, Bengt; Asparouhov, Tihomir; Ludtke, Oliver; Robitzsch, Alexander; Morin, Alexandre J. S.; Trautwein, Ulrich
2009-01-01
This study is a methodological-substantive synergy, demonstrating the power and flexibility of exploratory structural equation modeling (ESEM) methods that integrate confirmatory and exploratory factor analyses (CFA and EFA), as applied to substantively important questions based on multidimentional students' evaluations of university teaching…
Integration-free interval doubling for Riccati equation solutions
NASA Technical Reports Server (NTRS)
Sidhu, G. S.; Bierman, G. J.
1977-01-01
Starting with certain identities obtained by Reid (1972) and Redheffer (1962) for general matrix Riccati equations (RE's), we give various algorithms for the case of constant coefficients. The algorithms are based on two ideas - first, relate the RE solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant, the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration-free, superlinearly convergent iterative solution to the algebraic RE. Preliminary numerical experiments show that our algorithms, arranged in square-root form, provide a method that is numerically stable and appears to be competitive with other methods of solving the algebraic RE.
The reduced basis method for the electric field integral equation
Fares, M.; Hesthaven, J.S.; Maday, Y.; Stamm, B.
2011-06-20
We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.
Integral equation model for warm and hot dense mixtures.
Starrett, C E; Saumon, D; Daligault, J; Hamel, S
2014-09-01
In a previous work [C. E. Starrett and D. Saumon, Phys. Rev. E 87, 013104 (2013)] a model for the calculation of electronic and ionic structures of warm and hot dense matter was described and validated. In that model the electronic structure of one atom in a plasma is determined using a density-functional-theory-based average-atom (AA) model and the ionic structure is determined by coupling the AA model to integral equations governing the fluid structure. That model was for plasmas with one nuclear species only. Here we extend it to treat plasmas with many nuclear species, i.e., mixtures, and apply it to a carbon-hydrogen mixture relevant to inertial confinement fusion experiments. Comparison of the predicted electronic and ionic structures with orbital-free and Kohn-Sham molecular dynamics simulations reveals excellent agreement wherever chemical bonding is not significant. PMID:25314550
Inversion of airborne tensor VLF data using integral equations
NASA Astrophysics Data System (ADS)
Kamm, Jochen; Pedersen, Laust B.
2014-08-01
The Geological Survey of Sweden has been collecting airborne tensor very low frequency data (VLF) over several decades, covering large parts of the country. The data has been an invaluable source of information for identifying conductive structures that can among other things be related to water-filled fault zones, wet sediments that fill valleys or ore mineralizations. Because the method only uses two differently polarized plane waves of very similar frequency, vertical resolution is low and interpretation is in most cases limited to maps that are directly derived from the data. Occasionally, 2-D inversion is carried out along selected profiles. In this paper, we present for the first time a 3-D inversion for tensor VLF data in order to further increase the usefulness of the data set. The inversion is performed using a non-linear conjugate gradient scheme (Polak-Ribière) with an inexact line-search. The gradient is obtained by an algebraic adjoint method that requires one additional forward calculation involving the adjoint system matrix. The forward modelling is based on integral equations with an analytic formulation of the half-space Green's tensor. It avoids typically required Hankel transforms and is particularly amenable to singularity removal prior to the numerical integration over the volume elements. The system is solved iteratively, thus avoiding construction and storage of the dense system matrix. By using fast 3-D Fourier transforms on nested grids, subsequently farther away interactions are represented with less detail and therefore with less computational effort, enabling us to bridge the gap between the relatively short wavelengths of the fields (tens of metres) and the large model dimensions (several square kilometres). We find that the approximation of the fields can be off by several per cent, yet the transfer functions in the air are practically unaffected. We verify our code using synthetic calculations from well-established 2-D methods, and
NASA Astrophysics Data System (ADS)
Min, Xiaoyi
This thesis first presents the study of the interaction of electromagnetic waves with three-dimensional heterogeneous, dielectric, magnetic, and lossy bodies by surface integral equation modeling. Based on the equivalence principle, a set of coupled surface integral equations is formulated and then solved numerically by the method of moments. Triangular elements are used to model the interfaces of the heterogeneous body, and vector basis functions are defined to expand the unknown current in the formulation. The validity of this formulation is verified by applying it to concentric spheres for which an exact solution exists. The potential applications of this formulation to a partially coated sphere and a homogeneous human body are discussed. Next, this thesis also introduces an efficient new set of integral equations for treating the scattering problem of a perfectly conducting body coated with a thin magnetically lossy layer. These electric field integral equations and magnetic field integral equations are numerically solved by the method of moments (MoM). To validate the derived integral equations, an alternative method to solve the scattering problem of an infinite circular cylinder coated with a thin magnetic lossy layer has also been developed, based on the eigenmode expansion. Results for the radar cross section and current densities via the MoM and the eigenmode expansion method are compared. The agreement is excellent. The finite difference time domain method is subsequently implemented to solve a metallic object coated with a magnetic thin layer and numerical results are compared with that by the MoM. Finally, this thesis presents an application of the finite-difference time-domain approach to the problem of electromagnetic receiving and scattering by a cavity -backed antenna situated on an infinite conducting plane. This application involves modifications of Yee's model, which applies the difference approximations of field derivatives to differential
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. PMID:24329213
Modern integral equation techniques for quantum reactive scattering theory
Auerbach, S.M.
1993-11-01
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H{sub 2} {yields} H{sub 2}/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H{sub 2} state resolved integral cross sections {sigma}{sub v{prime}j{prime},vj}(E) for the transitions (v = 0,j = 0) to (v{prime} = 1,j{prime} = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence.
Xiong, Z.; Tripp, A.C.
1994-12-31
This paper presents an integral equation algorithm for 3D EM modeling at high frequencies for applications in engineering an environmental studies. The integral equation method remains the same for low and high frequencies, but the dominant roles of the displacements currents complicate both numerical treatments and interpretations. With singularity extraction technique they successively extended the application of the Hankel filtering technique to the computation of Hankel integrals occurring in high frequency EM modeling. Time domain results are calculated from frequency domain results via Fourier transforms. While frequency domain data are not obvious for interpretations, time domain data show wave-like pictures that resemble seismograms. Both 1D and 3D numerical results show clearly the layer interfaces.
A Study of Equating in NAEP. NAEP Validity Studies.
ERIC Educational Resources Information Center
Hedges, Larry V.; Vevea, Jack L.
This study investigates the amount of uncertainty added to National Assessment of Educational Progress (NAEP) estimates by equating error under both ideal and less than ideal circumstances. Data from past administrations are used to guide simulations of various equating designs and error due to equating is estimated empirically. The design…
Improved Integral Equation Solution for the First Passage Time of Leaky Integrate-and-Fire Neurons
Dong, `Yi; Mihalas, Stefan; Niebur, Ernst
2011-01-01
An accurate calculation of the first passage time probability density (FPTPD) is essential for computing the likelihood of solutions of the stochastic leaky integrate-and-fire model. The previously proposed numerical calculation of the FPTPD based on the integral equation method discretizes the probability current of the voltage crossing the threshold. While the method is accurate for high noise levels, we show that it results in large numerical errors for small noise. The problem is solved by analytically computing, in each time bin, the mean probability current. Efficiency is further improved by identifying and ignoring time bins with negligible mean probability current. PMID:21105825
A class of nonlinear differential equations with fractional integrable impulses
NASA Astrophysics Data System (ADS)
Wang, JinRong; Zhang, Yuruo
2014-09-01
In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki's normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki-Ulam's type stability are introduced and generalized Ulam-Hyers-Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.
The Application of a Boundary Integral Equation Method to the Prediction of Ducted Fan Engine Noise
NASA Technical Reports Server (NTRS)
Dunn, M. H.; Tweed, J.; Farassat, F.
1999-01-01
The prediction of ducted fan engine noise using a boundary integral equation method (BIEM) is considered. Governing equations for the BIEM are based on linearized acoustics and describe the scattering of incident sound by a thin, finite-length cylindrical duct in the presence of a uniform axial inflow. A classical boundary value problem (BVP) is derived that includes an axisymmetric, locally reacting liner on the duct interior. Using potential theory, the BVP is recast as a system of hypersingular boundary integral equations with subsidiary conditions. We describe the integral equation derivation and solution procedure in detail. The development of the computationally efficient ducted fan noise prediction program TBIEM3D, which implements the BIEM, and its utility in conducting parametric noise reduction studies are discussed. Unlike prediction methods based on spinning mode eigenfunction expansions, the BIEM does not require the decomposition of the interior acoustic field into its radial and axial components which, for the liner case, avoids the solution of a difficult complex eigenvalue problem. Numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and BIEM results. We demonstrate BIEM liner capability by examining radiation patterns for several cases of practical interest.
ERIC Educational Resources Information Center
Collins, Myrtle T.; Greenberg, Marvin
1974-01-01
Article described a plan to develop integrated study through music activities. Students learned to become more independent learners while concentrating on more complex and creative activities. (Author/RK)
Numerical comparison of spectral properties of volume-integral-equation formulations
NASA Astrophysics Data System (ADS)
Markkanen, Johannes; Ylä-Oijala, Pasi
2016-07-01
We study and compare spectral properties of various volume-integral-equation formulations. The equations are written for the electric flux, current, field, and potentials, and discretized with basis functions spanning the appropriate function spaces. Each formulation leads to eigenvalue distributions of different kind due to the effects of discretization procedure, namely, the choice of basis and testing functions. The discrete spectrum of the potential formulation reproduces the theoretically predicted spectrum almost exactly while the spectra of other formulations deviate from the ideal one. It is shown that the potential formulation has the spectral properties desired from the preconditioning perspective.
Cauchy-Jost function and hierarchy of integrable equations
NASA Astrophysics Data System (ADS)
Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.
2015-11-01
We describe the properties of the Cauchy-Jost (also known as Cauchy-Baker-Akhiezer) function of the Kadomtsev-Petviashvili-II equation. Using the bar partial -method, we show that for this function, all equations of the Kadomtsev-Petviashvili-II hierarchy are given in a compact and explicit form, including equations for the Cauchy-Jost function itself, time evolutions of the Jost solutions, and evolutions of the potential of the heat equation.
Integral Equation Theory for the Conformation of Polyelectrolytes
NASA Astrophysics Data System (ADS)
Shew, C.-Y.; Yethiraj, A.
1996-03-01
The equilibrium conformation properties of polyelectrolyes are explored using the integral equation theory. The polymer molecules are modeled as freely-jointed beads that interact via a hard sphere plus screened Coulomb potential. To obtain the intramolecuar correlation function ( and hence the chain conformations) the many chain system is replaced by a single chain whose beads interact via the bare interaction plus a solvent-induced potential, which approximately accounts for the presence of the other molecules. Since this solvent induced potential is a functional of the intramolecular correlations it is obtained iteratively in a self-consistent fashion. The intramolecular correlation functions for a given solvation potential are obtained via Monte Carlo simulation of a single chain. A thread model of the polymer molecules is also investigated, in which case the single chain conformations are obtained using a variational method. The predictions of the theory for these two models are similar. For single chains
Integrable systems of partial differential equations determined by structure equations and Lax pair
NASA Astrophysics Data System (ADS)
Bracken, Paul
2010-01-01
It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.
NASA Astrophysics Data System (ADS)
Chang, A. H.; Yee, K. S.; Prodan, J.
1992-08-01
To obtain an accurate solution in the method of moments (MM), it is vital that an appropriate integral equation be used. In solving the problem of scattering from bodies of revolution (BOR) with anisotropic surface impedance boundary conditions (IBC), different answers may result from seemingly minor differences in the integral equation formulation adopted. In this communication different types of integral equations are compared with one another when they are applied to bodies of revolution.
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
CALL FOR PAPERS: Special issue on Symmetries and Integrability of Difference Equations
NASA Astrophysics Data System (ADS)
Doliwa, Adam; Korhonen, Risto; Lafortune, Stephane
2006-10-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Special issue on Symmetries and Integrability of Difference Equations' as featured at the SIDE VII meeting held during July 2006 in Melbourne (http://web.maths.unsw.edu.au/%7Eschief/side/side.html). Participants at that meeting, as well as other researchers working in the field of difference equations and discrete systems, are invited to submit a research paper to this issue. This meeting was the seventh of a series of biennial meetings devoted to the study of integrable difference equations and related topics. 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, just as 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, quantum field theory, etc. It is thus crucial to develop tools to study and solve difference equations. While the theory of symmetry and integrability for differential equations is now well-established, this is not yet the case for discrete equations. The situation has undergone impressive development in recent years and has affected a broad range of fields, including the theory of special functions, quantum integrable systems, numerical analysis, cellular
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.
The geometric property of soliton solutions for the integrable KdV6 equations
NASA Astrophysics Data System (ADS)
Li, Jibin; Zhang, Yi
2010-04-01
The geometric property of soliton solutions of the three completely integrable sixth-order nonlinear equations (KdV6) is studied by using the method of dynamical systems and the work of Wazwaz [Appl. Math. Comput. 204, 963 (2008)]. This paper proved that a solitary wave solution corresponds to a homoclinic orbit of a four-dimensional dynamical system to a equilibrium point. The orbit lies on the intersection curve of two level set passing through the same equilibrium point.
Fitting integrated enzyme rate equations to progress curves with the use of a weighting matrix.
Franco, R; Aran, J M; Canela, E I
1991-01-01
A method is presented for fitting the pairs of values product formed-time taken from progress curves to the integrated rate equation. The procedure is applied to the estimation of the kinetic parameters of the adenosine deaminase system. Simulation studies demonstrate the capabilities of this strategy. A copy of the FORTRAN77 program used can be obtained from the authors by request. PMID:2006914
Fitting integrated enzyme rate equations to progress curves with the use of a weighting matrix.
Franco, R; Aran, J M; Canela, E I
1991-03-01
A method is presented for fitting the pairs of values product formed-time taken from progress curves to the integrated rate equation. The procedure is applied to the estimation of the kinetic parameters of the adenosine deaminase system. Simulation studies demonstrate the capabilities of this strategy. A copy of the FORTRAN77 program used can be obtained from the authors by request. PMID:2006914
Phase diagram of the hard-core Yukawa fluid within the integral equation method.
El Mendoub, E B; Wax, J-F; Jakse, N
2006-11-01
In this study, the integral equation method proposed recently by Sarkisov [J. Chem. Phys. 114, 9496 (2001).], which has proved accurate for continuous potentials, is extended successfully to the hard sphere potential plus an attractive Yukawa tail. By comparing the results of thermodynamic properties, including the liquid-vapor phase diagram, with available simulation data, it is found that this method remains reliable for this class of models of interaction often used in colloid science. PMID:17279956
The Reduction of Ducted Fan Engine Noise Via a Boundary Integral Equation Method
NASA Technical Reports Server (NTRS)
Tweed, John
2000-01-01
Engineering studies for reducing ducted fan engine noise were conducted using the noise prediction code TBIEM3D. To conduct parametric noise reduction calculations, it was necessary to advance certain theoretical and computational aspects of the boundary integral equation method (BIEM) described in and implemented in TBIEM3D. Also, enhancements and upgrades to TBIEM3D were made for facilitating the code's use in this research and by the aeroacoustics engineering community.
Differential Forms Basis Functions for Better Conditioned Integral Equations
Fasenfest, B; White, D; Stowell, M; Rieben, R; Sharpe, R; Madsen, N; Rockway, J D; Champagne, N J; Jandhyala, V; Pingenot, J
2005-01-13
Differential forms offer a convenient way to classify physical quantities and set up computational problems. By observing the dimensionality and type of derivatives (divergence,curl,gradient) applied to a quantity, an appropriate differential form can be chosen for that quantity. To use these differential forms in a simulation, the forms must be discretized using basis functions. The 0-form through 2-form basis functions are formed for surfaces. Twisted 1-form and 2-form bases will be presented in this paper. Twisted 1-form (1-forms) basis functions ({Lambda}) are divergence-conforming edge basis functions with units m{sup -1}. They are appropriate for representing vector quantities with continuous normal components, and they belong to the same function space as the commonly used RWG bases [1]. They are used here to formulate the frequency-domain EFIE with Galerkin testing. The 2-form basis functions (f) are scalar basis functions with units m{sup -2} and with no enforced continuity between elements. At lowest order, the 2-form basis functions are similar to pulse basis functions. They are used here to formulate an electrostatic integral equation. It should be noted that the derivative of an n-form differential form basis function is an (n+1)-form, i.e. the derivative of a 1-form basis function is a 2-form. Because the basis functions are constructed such that they have spatial units, the spatial units are removed from the degrees of freedom, leading to a better-conditioned system matrix. In this conference paper, we look at the performance of these differential forms and bases by examining the conditioning of matrix systems for electrostatics and the EFIE. The meshes used were refined across the object to consider the behavior of these basis transforms for elements of different sizes.
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation.
Mentink, J H; Tretyakov, M V; Fasolino, A; Katsnelson, M I; Rasing, Th
2010-05-01
We propose new semi-implicit numerical methods for the integration of the stochastic Landau-Lifshitz equation with built-in angular momentum conservation. The performance of the proposed integrators is tested on the 1D Heisenberg chain. For this system, our schemes show better stability properties and allow us to use considerably larger time steps than standard explicit methods. At the same time, these semi-implicit schemes are also of comparable accuracy to and computationally much cheaper than the standard midpoint implicit method. The results are of key importance for atomistic spin dynamics simulations and the study of spin dynamics beyond the macro spin approximation. PMID:21393676
A procedure on the first integrals of second-order nonlinear ordinary differential equations
NASA Astrophysics Data System (ADS)
Yasar, Emrullah; Yıldırım, Yakup
2015-12-01
In this article, we demonstrate the applicability of the integrating factor method to path equation describing minimum drag work, and a special Hamiltonian equation corresponding Riemann zeros for obtaining the first integrals. The effectiveness and powerfullness of this method is verified by applying it for two selected second-order nonlinear ordinary differential equations (NLODEs). As a result integrating factors and first integrals for them are succesfully established. The obtained results show that the integrating factor approach can also be applied to other NLODEs.
NASA Astrophysics Data System (ADS)
Gonczarek, Ryszard; Krzyzosiak, Mateusz; Gonczarek, Adam; Jacak, Lucjan
2015-06-01
In this paper, we discuss the mathematical structure of the s-wave superconducting gap and other quantitative characteristics of superconducting systems. In particular, we evaluate and discuss integrals inherent in fundamental equations describing superconducting systems. The results presented here extend the approach formulated by Abrikosov and Maki, which was restricted to the first-order expansion. A few infinite families of integrals are derived and allow us to express the fundamental equations by means of analytic formulas. They can be then exploited in order to find some quantitative characteristics of superconducting systems by the method of successive approximations. We show that the results can be applied to some modern formalisms in order to study high-Tc superconductors and other superconducting materials of the new generation.
Integration of a largest set of coupled differential equations on the CYBER 205 vector processor
NASA Astrophysics Data System (ADS)
Halcomb, Lawrence L.; Diestler, Dennis J.
1986-01-01
We utilize the Control Data CYBER 205 vector processor to solve s large setof coupled first-order ordinary differential equations arising in our theoretical studies of vibrational relaxation of molecules in solids. To describe the relaxation process, we employ a "hemiquantal" methodology that effectively mixes classical and quantum mechanics. The model used is that of a single diatomic molecule embedded in an otherwise pure one-dimensional lattice. The resulting hemiquantal equations (HQE) are integrated with a fully vectorized fourth-order Runge-Kutta routine. The details of the vectorization process are presented and representative results are included. The extension of the algorithm to allow simultaneous integration of sets of HQE is also discussed.
Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind
Eggermont, P. P. B.
1999-01-15
We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form Nf-exp(S*log f) , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (a la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm.
ERIC Educational Resources Information Center
Huang, Heng-Tsung Danny
2010-01-01
Thus far, few research studies have examined the practice of integrated speaking test tasks in the field of second/foreign language oral assessment. This dissertation utilized structural equation modeling (SEM) and qualitative techniques to explore the relationships among topical knowledge, anxiety, and integrated speaking test performance and to…
(2+1)-dimensional non-isospectral multi-component AKNS equations and its integrable couplings
Sun Yepeng
2010-03-08
(2+1)-dimensional non-isospectral multi-component AKNS equations are derived from an arbitrary order matrix spectral problem. As a reduction, (2+1)-dimensional non-isospectral multi-component Schroedinger equations are obtained. Moreover, new (2+1)-dimensional non-isospectral integrable couplings of the resulting AKNS equations are constructed by enlarging the associated matrix spectral problem.
An integral equation representation approach for valuing Russian options with a finite time horizon
NASA Astrophysics Data System (ADS)
Jeon, Junkee; Han, Heejae; Kim, Hyeonuk; Kang, Myungjoo
2016-07-01
In this paper, we first describe a general solution for the inhomogeneous Black-Scholes partial differential equation with mixed boundary conditions using Mellin transform techniques. Since Russian options with a finite time horizon are usually formulated into the inhomogeneous free-boundary Black-Scholes partial differential equation with a mixed boundary condition, we apply our method to Russian options and derive an integral equation satisfied by Russian options with a finite time horizon. Furthermore, we present some numerical solutions and plots of the integral equation using recursive integration methods and demonstrate the computational accuracy and efficiency of our method compared to other competing approaches.
Numerical solutions to ill-posed and well-posed impedance boundary condition integral equations
NASA Astrophysics Data System (ADS)
Rogers, J. R.
1983-11-01
Exterior scattering from a three-dimensional impedance body can be formulated in terms of various integral equations derived from the Leontovich impedance boundary condition (IBC). The electric and magnetic field integral equations are ill-posed because they theoretically admit spurious solutions at the frequencies of interior perfect conductor cavity resonances. A combined field formulation is well-posed because it does not allow the spurious solutions. This report outlines the derivation of IBC integral equations and describes a procedure for constructing moment-method solutions for bodies of revolution. Numerical results for scattering from impedance spheres are presented which contrast the stability and accuracy of solutions to the ill-posed equations with those of the well-posed equation. The results show that numerical solutions for exterior scattering to the electric and magnetic field integral equations can be severely contaminated by spurious resonant solutions regardless of whether the surface impedance of the body is lossy or lossless.
NASA Astrophysics Data System (ADS)
Hu, Yanxia; Yang, Xiaozhong
2006-08-01
A method for obtaining first integrals and integrating factors of n-th order autonomous systems is proposed. The search for first integrals and integrating factors can be reduced to the search for a class of invariant manifolds of the systems. Finally, the proposed method is applied to Euler-Poisson equations (gyroscope system), and the fourth first integral of the system in general Kovalevskaya case can be obtained.
A path-integral Langevin equation treatment of low-temperature doped helium clusters
NASA Astrophysics Data System (ADS)
Ing, Christopher; Hinsen, Konrad; Yang, Jing; Zeng, Toby; Li, Hui; Roy, Pierre-Nicholas
2012-06-01
We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)], 10.1063/1.3489925 sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of HeN-CO2 clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], 10.1002/(SICI)1096-987X(20000130)21:2<79::AID-JCC1>3.0.CO;2-B, an open-source molecular simulation package.
NASA Technical Reports Server (NTRS)
Walker, K. P.; Freed, A. D.
1991-01-01
New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.
Modern Integral Equation Techniques for Quantum Reactive Scattering Theory.
NASA Astrophysics Data System (ADS)
Auerbach, Scott Michael
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D + H_2 to H _2/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H + H_2 state resolved integral cross sections sigma_{v^' j^ ',vj}(E) for the transitions (v = 0, j = 0) to (v^' = 1,j^ ' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence. To facilitate quantum calculations on more complex reactive systems, we develop a new method to compute the energy Green's function with absorbing boundary conditions (ABC), for use in calculating the cumulative reaction probability. The method is an iterative technique to compute the inverse of a non-Hermitian matrix which is based on Fourier transforming time dependent dynamics, and which requires very little core memory. The Hamiltonian is evaluated in a sinc-function based discrete variable representation (DVR) which we argue may often be superior to the fast Fourier transform method for reactive scattering. We apply the resulting power series Green's function to the benchmark collinear H + H_2 system over the energy range 3.37 to 1.27 eV. The convergence of the power series is stable at all energies, and is accelerated by the use of a stronger absorbing potential. The practicality of computing the ABC-DVR Green's function in a polynomial of the Hamiltonian is
Integrability of the Kruskal--Zabusky Discrete Equation by Multiscale Expansion
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.
Non-integrability of the fourth Painlevé equation in the Liouville-Arnold sense
NASA Astrophysics Data System (ADS)
Stoyanova, Tsvetana
2014-05-01
In this paper we are concerned with the integrability of the fourth Painlevé equation (PIV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlevé equation with parameters a = m, b = -2(1 + 2n + m) where m, n \\in { Z} , is not integrable in the Liouville-Arnold sense by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and the Ziglin-Ramis-Morales-Ruiz-Simó method yield the non-integrability results.
On the collocation methods for singular integral equations with Hilbert kernel
NASA Astrophysics Data System (ADS)
Du, Jinyuan
2009-06-01
In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods of numerical solutions of singular integral equations with Hilbert kernel is given. Under the framework, it is very simple and obvious to obtain the coincidence theorem of collocation methods, then the existence and convergence for constructing approximate solutions are also given based on the coincidence theorem.
NASA Astrophysics Data System (ADS)
Urai, Janos L.; Kukla, Peter A.
2015-04-01
The growing importance of salt in the energy, subsurface storage, and chemical and food industries also increases the challenges with prediction of geometries, kinematics, stress and transport in salt. This requires an approach, which integrates a broader range of knowledge than is traditionally available in the different scientific and engineering disciplines. We aim to provide a starting point for a more integrated understanding of salt, by presenting an overview of the state of the art in a wide range of salt-related topics, from (i) the formation and metamorphism of evaporites, (ii) rheology and transport properties, (iii) salt tectonics and basin evolution, (iv) internal structure of evaporites, (v) fluid flow through salt, to (vi) salt engineering. With selected case studies we show how integration of these domains of knowledge can bring better predictions of (i) sediment architecture and reservoir distribution, (ii) internal structure of salt for optimized drilling and better cavern design, (iii) reliable long-term predictions of deformations and fluid flow in subsurface storage. A fully integrated workflow is based on geomechanical models, which include all laboratory and natural observations and links macro- and micro-scale studies. We present emerging concepts for (i) the initiation dynamics of halokinesis, (ii) the rheology and deformation of the evaporites by brittle and ductile processes, (iii) the coupling of processes in evaporites and the under- and overburden, and (iv) the impact of the layered evaporite rheology on the structural evolution.
Review of Integrated Noise Model (INM) Equations and Processes
NASA Technical Reports Server (NTRS)
Shepherd, Kevin P. (Technical Monitor); Forsyth, David W.; Gulding, John; DiPardo, Joseph
2003-01-01
The FAA's Integrated Noise Model (INM) relies on the methods of the SAE AIR-1845 'Procedure for the Calculation of Airplane Noise in the Vicinity of Airports' issued in 1986. Simplifying assumptions for aerodynamics and noise calculation were made in the SAE standard and the INM based on the limited computing power commonly available then. The key objectives of this study are 1) to test some of those assumptions against Boeing source data, and 2) to automate the manufacturer's methods of data development to enable the maintenance of a consistent INM database over time. These new automated tools were used to generate INM database submissions for six airplane types :737-700 (CFM56-7 24K), 767-400ER (CF6-80C2BF), 777-300 (Trent 892), 717-200 (BR7 15), 757-300 (RR535E4B), and the 737-800 (CFM56-7 26K).
NASA Astrophysics Data System (ADS)
Chen, Duan; Cai, Wei; Zinser, Brian; Cho, Min Hyung
2016-09-01
In this paper, we develop an accurate and efficient Nyström volume integral equation (VIE) method for the Maxwell equations for a large number of 3-D scatterers. The Cauchy Principal Values that arise from the VIE are computed accurately using a finite size exclusion volume together with explicit correction integrals consisting of removable singularities. Also, the hyper-singular integrals are computed using interpolated quadrature formulae with tensor-product quadrature nodes for cubes, spheres and cylinders, that are frequently encountered in the design of meta-materials. The resulting Nyström VIE method is shown to have high accuracy with a small number of collocation points and demonstrates p-convergence for computing the electromagnetic scattering of these objects. Numerical calculations of multiple scatterers of cubic, spherical, and cylindrical shapes validate the efficiency and accuracy of the proposed method.
Bifurcations of traveling wave solutions for an integrable equation
Li Jibin; Qiao Zhijun
2010-04-15
This paper deals with the following equation m{sub t}=(1/2)(1/m{sup k}){sub xxx}-(1/2)(1/m{sup k}){sub x}, which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-(1/2),(1/2),2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions.
Phase Behavior of Active Swimmers in Depletants: Molecular Dynamics and Integral Equation Theory
NASA Astrophysics Data System (ADS)
Das, Subir K.; Egorov, Sergei A.; Trefz, Benjamin; Virnau, Peter; Binder, Kurt
2014-05-01
We study the structure and phase behavior of a binary mixture where one of the components is self-propelling in nature. The interparticle interactions in the system are taken from the Asakura-Oosawa model for colloid-polymer mixtures for which the phase diagram is known. In the current model version, the colloid particles are made active using the Vicsek model for self-propelling particles. The resultant active system is studied by molecular dynamics methods and integral equation theory. Both methods produce results consistent with each other and demonstrate that the Vicsek model-based activity facilitates phase separation, thus, broadening the coexistence region.
Phase behavior of active swimmers in depletants: molecular dynamics and integral equation theory.
Das, Subir K; Egorov, Sergei A; Trefz, Benjamin; Virnau, Peter; Binder, Kurt
2014-05-16
We study the structure and phase behavior of a binary mixture where one of the components is self-propelling in nature. The interparticle interactions in the system are taken from the Asakura-Oosawa model for colloid-polymer mixtures for which the phase diagram is known. In the current model version, the colloid particles are made active using the Vicsek model for self-propelling particles. The resultant active system is studied by molecular dynamics methods and integral equation theory. Both methods produce results consistent with each other and demonstrate that the Vicsek model-based activity facilitates phase separation, thus, broadening the coexistence region. PMID:24877969
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.
Integration of CAS in the Didactics of Differential Equations.
ERIC Educational Resources Information Center
Balderas Puga, Angel
In this paper are described some features of the intensive use of math software, primarily DERIVE, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that included not only course contents, but also the sequence of introduction to various topics and…
Brugarino, Tommaso; Sciacca, Michele
2010-09-15
In this paper, we investigate the integrability of an inhomogeneous nonlinear Schroedinger equation, which has several applications in many branches of physics, as in Bose-Einstein condensates and fiber optics. The main issue deals with Painleve property (PP) and Liouville integrability for a nonlinear Schroedinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose-Einstein condensates are proposed (including Bose-Einstein condensates in three-dimensional in cylindrical symmetry).
Daeva, S.G.; Setukha, A.V.
2015-03-10
A numerical method for solving a problem of diffraction of acoustic waves by system of solid and thin objects based on the reduction the problem to a boundary integral equation in which the integral is understood in the sense of finite Hadamard value is proposed. To solve this equation we applied piecewise constant approximations and collocation methods numerical scheme. The difference between the constructed scheme and earlier known is in obtaining approximate analytical expressions to appearing system of linear equations coefficients by separating the main part of the kernel integral operator. The proposed numerical scheme is tested on the solution of the model problem of diffraction of an acoustic wave by inelastic sphere.
Exponential Methods for the Time Integration of Schrödinger Equation
NASA Astrophysics Data System (ADS)
Cano, B.; González-Pachón, A.
2010-09-01
We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schrödinger 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.
Studies of the Ginzburg-Landau equation
Rodriguez, J.D.
1988-01-01
The turbulence problem is the motivation for the study of reduction of phase space dimension in the Ginzburg-Landau equation. Chaotic solutions to this equation provide a turbulence analog. A basis set for the chaotic attractor is derived using the orthogonal decomposition of the correlation matrix. This matrix is computed explicitly at the point of maximal Liapunov dimension in the parameter range under study. The basis set is shown to be optimal in a least squares sense. Galerdin projection is then used to obtain a small set of O.D.E.'s. The case of spatially periodic, even initial data is studied first. Three complex O.D.E.'s were sufficient to reproduce the solution of the full system as given by a 16 point pseudo-spectral Fourier method. The case of homogeneous boundary conditions was studied next. Ten complex O.D.E.'s were required versus 128 for the pseudo-spectral solution. Using power spectra and Poincare sections the reduced systems were shown to reproduce the exact behavior over a wide parameter range. Savings in C.P.U. time of an order of magnitude were attained over pseudo-spectral algorithms. New results on the asymptotic behavior of limit cycle solutions were also obtained. Singular solutions, zero almost everywhere, with strong boundary layer character were found in the limit of large domain size. An infinite hierarchy of subharmonic solutions was shown to exist for the spatially periodic case, and a countable number of fixed point solutions was found for both spatially periodic and homogeneous cases.
Mickens, R.E.
1997-12-12
The major thrust of this proposal was to continue our investigations of so-called non-standard finite-difference schemes as formulated by other authors. These schemes do not follow the standard rules used to model continuous differential equations by discrete difference equations. The two major aspects of this procedure consist of generalizing the definition of the discrete derivative and using a nonlocal model (on the computational grid or lattice) for nonlinear terms that may occur in the differential equations. Our aim was to investigate the construction of nonstandard finite-difference schemes for several classes of ordinary and partial differential equations. These equations are simple enough to be tractable, yet, have enough complexity to be both mathematically and scientifically interesting. It should be noted that all of these equations differential equations model some physical phenomena under an appropriate set of experimental conditions. The major goal of the project was to better understand the process of constructing finite-difference models for differential equations. In particular, it demonstrates the value of using nonstandard finite-difference procedures. A secondary goal was to construct and study a variety of analytical techniques that can be used to investigate the mathematical properties of the obtained difference equations. These mathematical procedures are of interest in their own right and should be a valuable contribution to the mathematics research literature in difference equations. All of the results obtained from the research done under this project have been published in the relevant research/technical journals or submitted for publication. Our expectation is that these results will lead to improved finite difference schemes for the numerical integration of both ordinary and partial differential equations. Section G of the Appendix gives a concise summary of the major results obtained under funding by the grant.
Integrable pair-transition-coupled nonlinear Schrödinger equations
NASA Astrophysics Data System (ADS)
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.
Integrable pair-transition-coupled nonlinear Schrödinger equations.
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system. PMID:26382492
Study of nonlinear waves described by the cubic Schroedinger equation
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
Variational integration for ideal magnetohydrodynamics with built-in advection equations
Zhou, Yao; Burby, J. W.; Bhattacharjee, A.; Qin, Hong
2014-10-15
Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Variational Integration for Ideal MHD with Built-in Advection Equations
Zhou, Yao; Qin, Hong; Burby, J. W.; Bhattacharjee, A.
2014-08-05
Newcomb's Lagrangian for ideal MHD in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture
NASA Astrophysics Data System (ADS)
Davey, K.; Darvizeh, R.
2016-03-01
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.
Transonic airfoil computation using the integral equation with and without embedded Euler domains
NASA Technical Reports Server (NTRS)
Kandil, Osama A.; Hu, Hong
1987-01-01
Two transonic computational schemes which are based on the Integral Equation Formulation of the full potential equation were presented. The first scheme is a Shock Capturing-Shock Fitting (SCSF) scheme which uses the full potential equation throughout with the exception of the shock wave where the Rankine-Hugoniot relations are used to cross and fit the shock. The second scheme is an Integral Equation with Embedded Euler (IEEE) scheme which uses the full potential equation with an embedded region where the Euler equations are used. The two schemes are applied to several transonic airfoil flows and the results were compared with numerous computational results and experimental domains with fine grids. The SCSF-scheme is restricted to flows with weak shock, while the IEEE-scheme can handle strong shocks. Currently, the IEEE scheme is applied to other transonic flows with strong shocks as well as to unsteady pitching oscillations.
Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs.
Magath, Thore; Serebryannikov, Andriy E
2005-11-01
A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals. PMID:16302391
Path Integral Calculation of GREEN’S Function for SCHRÖDINGER Equation in Unitary Gauge
NASA Astrophysics Data System (ADS)
Rozansky, L.
Green’s function of Schrödinger equation is represented as a time-reparametrization invariant path integral. Unitary gauge fixing enables us to get the WKB preexponential factor without calculating determinants of operators containing derivatives.
Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind
Hong, Zhimin; Yan, Zaizai; Yan, Jiao
2014-01-01
In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach. PMID:25072373
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
A comparison of the efficiency of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations is presented. 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 iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient than evaluating the temperature by integrating its time-derivative.
A study of dynamic energy equations for Stirling cycle analysis
NASA Technical Reports Server (NTRS)
Larson, V. H.
1983-01-01
An analytical and computer study of the dynamic energy equations that describe the physical phenomena that occurs in a Stirling cycle engine. The basic problem is set up in terms of a set o hyperbolic partial differential equations. The characteristic lines are determined. The equations are then transformed to ordinary differential equations that are valid along characteristic lines. Computer programs to solve the differential equations and to plot pertinent factors are described.
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.
The solutions of three dimensional Fredholm integral equations using Adomian decomposition method
NASA Astrophysics Data System (ADS)
Almousa, Mohammad
2016-06-01
This paper presents the solutions of three dimensional Fredholm integral equations by using Adomian decomposition method (ADM). Some examples of these types of equations are tested to show the reliability of the technique. The solutions obtained by ADM give an excellent agreement with exact solution.
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
Solution of coupled integral equations for quantum scattering in the presence of complex potentials
Franz, Jan
2015-01-15
In this paper, we present a method to compute solutions of coupled integral equations for quantum scattering problems in the presence of a complex potential. We show how the elastic and absorption cross sections can be obtained from the numerical solution of these equations in the asymptotic region at large radial distances.
On the stability of numerical integration routines for ordinary differential equations.
NASA Technical Reports Server (NTRS)
Glover, K.; Willems, J. C.
1973-01-01
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Jezewski, D. J.
1979-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration
NASA Astrophysics Data System (ADS)
Breev, A. I.; Shapovalov, A. V.
2016-01-01
Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator.
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.
Integrated optics technology study
NASA Technical Reports Server (NTRS)
Chen, B.; Findakly, T.; Innarella, R.
1982-01-01
The status and near term potential of materials and processes available for the fabrication of single mode integrated electro-optical components are discussed. Issues discussed are host material and orientation, waveguide formation, optical loss mechanisms, wavelength selection, polarization effects and control, laser to integrated optics coupling fiber optic waveguides to integrated optics coupling, sources, and detectors. Recommendations of the best materials, technology, and processes for fabrication of integrated optical components for communications and fiber gyro applications are given.
NASA Astrophysics Data System (ADS)
Tancredi, Lorenzo
2015-12-01
Integration by parts identities (IBPs) can be used to express large numbers of apparently different d-dimensional Feynman Integrals in terms of a small subset of so-called master integrals (MIs). Using the IBPs one can moreover show that the MIs fulfil linear systems of coupled differential equations in the external invariants. With the increase in number of loops and external legs, one is left in general with an increasing number of MIs and consequently also with an increasing number of coupled differential equations, which can turn out to be very difficult to solve. In this paper we show how studying the IBPs in fixed integer numbers of dimension d = n with n ∈ N one can extract the information useful to determine a new basis of MIs, whose differential equations decouple as d → n and can therefore be more easily solved as Laurent expansion in (d - n).
NASA Astrophysics Data System (ADS)
Gan, Hin Hark; Eu, Byung Chan
1993-09-01
A recursive integral equation for the intramolecular correlation function of an isolated linear polymer of N bonds is derived from the integral equations presented in the preceding paper. The derivation basically involves limiting the density of the polymer to zero so that polymers do not interact with each other, and thus taking into account the intramolecular part only. The integral equation still has the form of a generalized Percus-Yevick integral equation. The intramolecular correlation function of a polymer of N bonds is recursively generated by means of it from those of polymers of 2, 3,..., (N-1) bonds. The end-to-end distance distribution functions are computed by using the integral equation for various chain lengths, temperatures, and bond lengths in the case of a repulsive soft-sphere potential. Numerical solutions of the recursive integral equation yield universal exponents for the mean square end-to-end distance in two and three dimensions with values which are close to the Flory results: 0.77 and 0.64 vs Flory's values 0.75 and 0.6 for two and three dimensions, respectively. The intramolecular correlation functions computed can be fitted with displaced Gaussian forms. The N dependence of the internal chemical potential is found to saturate after some value of N depending on the ratio of the bond length to the bead radius.
Integrability: mathematical methods for studying solitary waves theory
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2014-03-01
In recent decades, substantial experimental research efforts have been devoted to linear and nonlinear physical phenomena. In particular, studies of integrable nonlinear equations in solitary waves theory have attracted intensive interest from mathematicians, with the principal goal of fostering the development of new methods, and physicists, who are seeking solutions that represent physical phenomena and to form a bridge between mathematical results and scientific structures. The aim for both groups is to build up our current understanding and facilitate future developments, develop more creative results and create new trends in the rapidly developing field of solitary waves. The notion of the integrability of certain partial differential equations occupies an important role in current and future trends, but a unified rigorous definition of the integrability of differential equations still does not exist. For example, an integrable model in the Painlevé sense may not be integrable in the Lax sense. The Painlevé sense indicates that the solution can be represented as a Laurent series in powers of some function that vanishes on an arbitrary surface with the possibility of truncating the Laurent series at finite powers of this function. The concept of Lax pairs introduces another meaning of the notion of integrability. The Lax pair formulates the integrability of nonlinear equation as the compatibility condition of two linear equations. However, it was shown by many researchers that the necessary integrability conditions are the existence of an infinite series of generalized symmetries or conservation laws for the given equation. The existence of multiple soliton solutions often indicates the integrability of the equation but other tests, such as the Painlevé test or the Lax pair, are necessary to confirm the integrability for any equation. In the context of completely integrable equations, studies are flourishing because these equations are able to describe the
Finding linear dependencies in integration-by-parts equations: A Monte Carlo approach
NASA Astrophysics Data System (ADS)
Kant, Philipp
2014-05-01
The reduction of a large number of scalar integrals to a small set of master integrals via Laporta’s algorithm is common practice in multi-loop calculations. It is also a major bottleneck in terms of running time and memory consumption. It involves solving a large set of linear equations where many of the equations are linearly dependent. We propose a simple algorithm that eliminates all linearly dependent equations from a given system, reducing the time and space requirements of a subsequent run of Laporta’s algorithm.
On integrals for some class of ordinary difference equations admitting a Lax representation
NASA Astrophysics Data System (ADS)
Svinin, Andrei K.
2016-03-01
We consider two infinite classes of ordinary difference equations admitting Lax pair representation. Discrete equations in these classes are parameterized by two integers k≥slant 0 and s≥slant k+1. We describe the first integrals for these two classes in terms of special discrete polynomials. We show an equivalence between two difference equations belonging to different classes corresponding to the same pair (k, s). We show that solution spaces {{ N }}sk of different ordinary difference equations with a fixed value of s + k are organized in a chain of inclusions.
NASA Technical Reports Server (NTRS)
Tsinganos, K. C.
1982-01-01
The steady equations of hydromagnetics for the isentropic or nonisentropic flow of an inviscid magnetofluid of high electrical conductivity, with one ignorable coordinate in a general orthogonal system, are treated. Several integrals of the equations are established thereafter reducing them to a scalar, quasi-linear, second order, partial differential equation for the magnetic potential. Simple solutions of this final equation are presented. The result, together with a similar treatment of helically symmetric hydromagnetic flows presented in a subsequent paper, allows a unified and systematic approach to the solution of problems involving steady hydromagnetic fields with a topological invariance in various curvilinear coordinates.
Koga, James
2004-10-01
Usually the motion of an electron under the influence of electromagnetic fields is influenced to a small extent by radiation damping. With the advent of high power high irradiance lasers it has become possible to generate focused laser irradiances where electrons interacting with the laser become highly relativistic over very short time and spatial scales. By focusing petawatt class lasers to very small spot sizes the amount of radiation emitted by electrons can become very large. Resultingly, the damping of the electron motion by the emission of this radiation can become large. In order to study this problem a code is written to solve a set of equations describing the evolution of a strong electromagnetic wave interacting with a single electron. Usually the equation of motion of an electron including radiation damping under the influence of electromagnetic fields is derived from the Lorentz-Dirac equation treating the damping as a perturbation. We use this equation to integrate forward in time and use the Lorentz-Dirac equation to integrate backward in time. We show that for very short wavelength electromagnetic radiation deep in the quantum regime at high irradiances differences between the perturbation equation and Lorentz-Dirac can be seen. However, for electron motion in the classical regime the differences are negligible. For electron motion in the classical regime the first order damping equation is found to be very adequate. PMID:15600540
Thermodynamics and structure of a two-dimensional electrolyte by integral equation theory.
Aupic, Jana; Urbic, Tomaz
2014-05-14
Monte Carlo simulations and integral equation theory were used to predict the thermodynamics and structure of a two-dimensional Coulomb fluid. We checked the possibility that integral equations reproduce Kosterlitz-Thouless and vapor-liquid phase transitions of the electrolyte and critical points. Integral equation theory results were compared to Monte Carlo data and the correctness of selected closure relations was assessed. Among selected closures hypernetted-chain approximation results matched computer simulation data best, but these equations unfortunately break down at temperatures well above the Kosterlitz-Thouless transition. The Kovalenko-Hirata closure produces results even at very low temperatures and densities, but no sign of phase transition was detected. PMID:24832290
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.
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.
Thermodynamics and structure of a two-dimensional electrolyte by integral equation theory
Aupic, Jana; Urbic, Tomaz
2014-05-14
Monte Carlo simulations and integral equation theory were used to predict the thermodynamics and structure of a two-dimensional Coulomb fluid. We checked the possibility that integral equations reproduce Kosterlitz-Thouless and vapor-liquid phase transitions of the electrolyte and critical points. Integral equation theory results were compared to Monte Carlo data and the correctness of selected closure relations was assessed. Among selected closures hypernetted-chain approximation results matched computer simulation data best, but these equations unfortunately break down at temperatures well above the Kosterlitz-Thouless transition. The Kovalenko-Hirata closure produces results even at very low temperatures and densities, but no sign of phase transition was detected.
Integration-free interval doubling for Riccati equation solutions
NASA Technical Reports Server (NTRS)
Bierman, G. J.; Sidhu, G. S.
1976-01-01
Various algorithms are given for the case of constant coefficients. The algorithms are based on two ideas: first, relate the Re solution with general initial conditions to anchored RE solutions; and second, when the coefficients are constant the anchored solutions have a basic shift-invariance property. These ideas are used to construct an integration free superlinearly convergent iterative solution to the algebraic RE. The algorithm, arranged in square-root form, is thought to be numerically stable and competitive with other methods of solving the algebraic RE.
A Collocation Method for Volterra Integral Equations with Diagonal and Boundary Singularities
NASA Astrophysics Data System (ADS)
Kolk, Marek; Pedas, Arvet; Vainikko, Gennadi
2009-08-01
We propose a smoothing technique associated with piecewise polynomial collocation methods for solving linear weakly singular Volterra integral equations of the second kind with kernels which, in addition to a diagonal singularity, may have a singularity at the initial point of the interval of integration.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations is presented. Different integration methods are tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, homogeneous gas-phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications of this result to the development of fast integration techniques for combustion kinetic rate equations are discussed.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations was examined. Different integration methods were tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, and homogeneous gas phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications which result in the development of fast integration techniques for combustion kinetic rate equations are discussed.
Classical integrable systems and soliton equations related to eleven-vertex R-matrix
NASA Astrophysics Data System (ADS)
Levin, A.; Olshanetsky, M.; Zotov, A.
2014-10-01
In our recent paper we suggested a natural construction of the classical relativistic integrable tops in terms of the quantum R-matrices. Here we study the simplest case - the 11-vertex R-matrix and related gl2 rational models. The corresponding top is equivalent to the 2-body Ruijsenaars-Schneider (RS) or the 2-body Calogero-Moser (CM) model depending on its description. We give different descriptions of the integrable tops and use them as building blocks for construction of more complicated integrable systems such as Gaudin models and classical spin chains (periodic and with boundaries). The known relation between the top and CM (or RS) models allows to rewrite the Gaudin models (or the spin chains) in the canonical variables. Then they assume the form of n-particle integrable systems with 2n constants. We also describe the generalization of the top to 1+1 field theories. It allows us to get the Landau-Lifshitz type equation. The latter can be treated as non-trivial deformation of the classical continuous Heisenberg model. In a similar way the deformation of the principal chiral model is described.
The Reduction of Ducted Fan Engine Noise Via A Boundary Integral Equation Method
NASA Technical Reports Server (NTRS)
Tweed, J.; Dunn, M.
1997-01-01
The development of a Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is discussed. The method is motivated by the need for an efficient and versatile computational tool to assist in parametric noise reduction studies. In this research, the work in reference 1 was extended to include passive noise control treatment on the duct interior. The BEM considers the scattering of incident sound generated by spinning point thrust dipoles in a uniform flow field by a thin cylindrical duct. The acoustic field is written as a superposition of spinning modes. Modal coefficients of acoustic pressure are calculated term by term. The BEM theoretical framework is based on Helmholtz potential theory. A boundary value problem is converted to a boundary integral equation formulation with unknown single and double layer densities on the duct wall. After solving for the unknown densities, the acoustic field is easily calculated. The main feature of the BIEM is the ability to compute any portion of the sound field without the need to compute the entire field. Other noise prediction methods such as CFD and Finite Element methods lack this property. Additional BIEM attributes include versatility, ease of use, rapid noise predictions, coupling of propagation and radiation both forward and aft, implementable on midrange personal computers, and valid over a wide range of frequencies.
Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction.
Wu, Yumao; Lu, Ya Yan
2011-06-01
Boundary integral equation methods for diffraction gratings are particularly suitable for gratings with complicated material interfaces but are difficult to implement due to the quasi-periodic Green's function and the singular integrals at the corners. In this paper, the boundary integral equation Neumann-to-Dirichlet map method for in-plane diffraction problems of gratings [Y. Wu and Y. Y. Lu, J. Opt. Soc. Am. A26, 2444 (2009)] is extended to conical diffraction problems. The method uses boundary integral equations to calculate the so-called Neumann-to-Dirichlet maps for homogeneous subdomains of the grating, so that the quasi-periodic Green's functions can be avoided. Since wave field components are coupled on material interfaces with the involvement of tangential derivatives, a least squares polynomial approximation technique is developed to evaluate tangential derivatives along these interfaces for conical diffraction problems. Numerical examples indicate that the method performs equally well for dielectric or metallic gratings. PMID:21643404
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.
Solution of the Bartels-Kwiecinski-Praszalowicz equation via Monte Carlo integration
NASA Astrophysics Data System (ADS)
Chachamis, Grigorios; Sabio Vera, Agustín
2016-08-01
We present a method of solution of the Bartels-Kwiecinski-Praszalowicz (BKP) equation based on the numerical integration of iterated integrals in transverse momentum and rapidity space. As an application, our procedure, which makes use of Monte Carlo integration techniques, is applied to obtain the gluon Green function in the Odderon case at leading order. The same approach can be used for more complicated scenarios.
Integral formulation of shallow-water equations with anisotropic porosity for urban flood modeling
NASA Astrophysics Data System (ADS)
Sanders, Brett F.; Schubert, Jochen E.; Gallegos, Humberto A.
2008-11-01
SummaryAn integral form of the shallow-water equations suitable for urban flood modeling is derived by applying Reynolds transport theorem to a finite control volume encompassing buildings on a flood plain. The effect of buildings on storage and conveyance is modeled with a binary density function i(x,y) that equals unity when (x,y) corresponds to a void, and nil otherwise, and can be measured using remote sensing data such as classified aerial imagery; the effect of buildings on flow resistance is modeled with a drag formulation. Discrete equations are obtained by applying the integral equations to a computational cell and adopting a Godunov-type, piecewise linear distribution of flow variables. The discrete equations include a volumetric porosity ϕ that represents the integral of i over the cell, normalized by the cell area, and an areal porosity ψ that represents the integral of i over an edge of the mesh, normalized by the edge length. The latter is directionally dependent which introduces anisotropy to the shallow-water equations and captures sub-grid preferential flow directions which occur in urban settings due to asymmetric building shapes and spacings and the alignment of buildings along streets. A important implication is that model predictions are necessarily grid dependent; therefore, a mesh design strategy is proposed. First- and second-order accurate numerical methods are presented to solve the discrete equations, and applications are shown for verification and validation purposes including the ability of the model to resolve preferential flow directions.
NASA Astrophysics Data System (ADS)
Pearson, L. W.; Whitaker, R. A.
1991-02-01
The transverse-aperture/integral-equation method provides a means of computation for diffraction coefficients at blunt edges of a broad class of stratified layers, including sheet-anisotropy models for conducting composites. This paper concentrates on the application of the method when the material profile comprises layers of homogeneous, potentially lossy material. The method proceeds from defining an artificial aperture perpendicular to a semiinfinite, planar, stratified region and passing through the terminal edge of the region. An integral equation is formulated over this infinite-extent aperture, and the solution to the integral equation represents the influence of the edge. The kernel in the integral equation is a weighted sum of the Green functions for the respective half spaces lying on either side of the aperture plane. The vector wave equation is separable in each of these half spaces, resulting in Green functions that are expressible analytically. The Green function for the stratified half space is stated in terms of a Sommerfeld-type integral.
Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG.
Rahol, Jussi; Tissari, Satu
2002-03-21
We study the iterative solution of dense linear systems that arise from boundary element discretizations of the electrostatic integral equation in magnetoencephalography (MEG). We show that modern iterative methods can be used to decrease the total computation time by avoiding the time-consuming computation of the LU decomposition of the coefficient matrix. More importantly, the modern iterative methods make it possible to avoid the explicit formation of the coefficient matrix which is needed when a large number of unknowns are used. To study the convergence of iterative solvers we examine the eigenvalue distributions of the coefficient matrices. For the sphere we show how the eigenvalues of the integral operator are approximated by the eigenvalues of the coefficient matrix when the collocation and Galerkin methods are used as discretization methods. The collocation method approximates the eigenvalues of the integral operator directly. The Galerkin method produces a coefficient matrix that needs to be preconditioned in order to maintain optimal convergence speed. With the ILU(0) preconditioner iterative methods converge fast and independent of the number of discretization points for both the collocation and Galerkin approaches. The preconditioner has no significant effect on the total computational time. PMID:11936181
Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG
NASA Astrophysics Data System (ADS)
Rahola, Jussi; Tissari, Satu
2002-03-01
We study the iterative solution of dense linear systems that arise from boundary element discretizations of the electrostatic integral equation in magnetoencephalography (MEG). We show that modern iterative methods can be used to decrease the total computation time by avoiding the time-consuming computation of the LU decomposition of the coefficient matrix. More importantly, the modern iterative methods make it possible to avoid the explicit formation of the coefficient matrix which is needed when a large number of unknowns are used. To study the convergence of iterative solvers we examine the eigenvalue distributions of the coefficient matrices. For the sphere we show how the eigenvalues of the integral operator are approximated by the eigenvalues of the coefficient matrix when the collocation and Galerkin methods are used as discretization methods. The collocation method approximates the eigenvalues of the integral operator directly. The Galerkin method produces a coefficient matrix that needs to be preconditioned in order to maintain optimal convergence speed. With the ILU(0) preconditioner iterative methods converge fast and independent of the number of discretization points for both the collocation and Galerkin approaches. The preconditioner has no significant effect on the total computational time.
NASA Astrophysics Data System (ADS)
Yang, Yunqing; Yan, Zhenya; Malomed, Boris A.
2015-10-01
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems.
Yang, Yunqing; Yan, Zhenya; Malomed, Boris A
2015-10-01
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems. PMID:26520078
ICM: an Integrated Compartment Method for numerically solving partial differential equations
Yeh, G.T.
1981-05-01
An integrated compartment method (ICM) is proposed to construct a set of algebraic equations from a system of partial differential equations. The ICM combines the utility of integral formulation of finite element approach, the simplicity of interpolation of finite difference approximation, and the flexibility of compartment analyses. The integral formulation eases the treatment of boundary conditions, in particular, the Neumann-type boundary conditions. The simplicity of interpolation provides great economy in computation. The flexibility of discretization with irregular compartments of various shapes and sizes offers advantages in resolving complex boundaries enclosing compound regions of interest. The basic procedures of ICM are first to discretize the region of interest into compartments, then to apply three integral theorems of vectors to transform the volume integral to the surface integral, and finally to use interpolation to relate the interfacial values in terms of compartment values to close the system. The Navier-Stokes equations are used as an example of how to derive the corresponding ICM alogrithm for a given set of partial differential equations. Because of the structure of the algorithm, the basic computer program remains the same for cases in one-, two-, or three-dimensional problems.
Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight
NASA Astrophysics Data System (ADS)
Ikonomov, N. R.; Kovacheva, R. K.; Suetin, S. P.
2015-12-01
We obtain Nuttall's integral equation provided that the corresponding complex-valued function σ(x) does not vanish and belongs to the Dini-Lipschitz class. Using this equation, we obtain a complex analogue of Bernshtein's classical asymptotic formulae for polynomials orthogonal on the closed unit interval Δ= \\lbrack -1,1 \\rbrack with respect to a complex-valued weight h(x)=σ(x)/\\sqrt{1-x^2}.
Subprograms for integrating the equations of motion of satellites. FORTRAN 4
NASA Technical Reports Server (NTRS)
Prokhorenko, V. I.
1980-01-01
The subprograms for the formation of the right members of the equations of motion of artificial Earth satellites (AES), integration of systems of differential equations by Adams' method, and the calculation of the values of various functions from the AES parameters of motion are described. These subprograms are written in the FORTRAN 4 language and constitute an essential part of the package of applied programs for the calculation of navigational parameters AES.
Analytical solution of boundary integral equations for 2-D steady linear wave problems
NASA Astrophysics Data System (ADS)
Chuang, J. M.
2005-10-01
Based on the Fourier transform, the analytical solution of boundary integral equations formulated for the complex velocity of a 2-D steady linear surface flow is derived. It has been found that before the radiation condition is imposed, free waves appear both far upstream and downstream. In order to cancel the free waves in far upstream regions, the eigensolution of a specific eigenvalue, which satisfies the homogeneous boundary integral equation, is found and superposed to the analytical solution. An example, a submerged vortex, is used to demonstrate the derived analytical solution. Furthermore, an analytical approach to imposing the radiation condition in the numerical solution of boundary integral equations for 2-D steady linear wave problems is proposed.
Symmetries, Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations
NASA Astrophysics Data System (ADS)
Liu, Han-Ze; Xin, Xiang-Peng
2016-08-01
This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method. Supported by the National Natural Science Foundation of China under Grant Nos. 11171041 and 11505090, Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009, and the doctorial foundation of Liaocheng University under Grant No. 31805
NASA Astrophysics Data System (ADS)
Zhao, Huaqing
There are two major objectives of this thesis work. One is to study theoretically the fracture and fatigue behavior of both homogeneous and functionally graded materials, with or without crack bridging. The other is to further develop the singular integral equation approach in solving mixed boundary value problems. The newly developed functionally graded materials (FGMs) have attracted considerable research interests as candidate materials for structural applications ranging from aerospace to automobile to manufacturing. From the mechanics viewpoint, the unique feature of FGMs is that their resistance to deformation, fracture and damage varies spatially. In order to guide the microstructure selection and the design and performance assessment of components made of functionally graded materials, in this thesis work, a series of theoretical studies has been carried out on the mode I stress intensity factors and crack opening displacements for FGMs with different combinations of geometry and material under various loading conditions, including: (1) a functionally graded layer under uniform strain, far field pure bending and far field axial loading, (2) a functionally graded coating on an infinite substrate under uniform strain, and (3) a functionally graded coating on a finite substrate under uniform strain, far field pure bending and far field axial loading. In solving crack problems in homogeneous and non-homogeneous materials, a very powerful singular integral equation (SEE) method has been developed since 1960s by Erdogan and associates to solve mixed boundary value problems. However, some of the kernel functions developed earlier are incomplete and possibly erroneous. In this thesis work, mode I fracture problems in a homogeneous strip are reformulated and accurate singular Cauchy type kernels are derived. Very good convergence rates and consistency with standard data are achieved. Other kernel functions are subsequently developed for mode I fracture in
Volume integrals associated with the inhomogeneous Helmholtz equation. Part 1: Ellipsoidal region
NASA Technical Reports Server (NTRS)
Fu, L. S.; Mura, T.
1983-01-01
Problems of wave phenomena in fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) to alpha(2), for the case of an ellipsoidal region. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r 4' and r r', where r and r' are distances from the origin to the point of observation and source, respectively. Derivatives of these integrals are easily evaluated. When the wave number approaches zero, the results reduce directly to the potentials of variable densities.
NASA Technical Reports Server (NTRS)
Desmarais, R. N.; Rowe, W. S.
1984-01-01
For the design of active controls to stabilize flight vehicles, which requires the use of unsteady aerodynamics that are valid for arbitrary complex frequencies, algorithms are derived for evaluating the nonelementary part of the kernel of the integral equation that relates unsteady pressure to downwash. This part of the kernel is separated into an infinite limit integral that is evaluated using Bessel and Struve functions and into a finite limit integral that is expanded in series and integrated termwise in closed form. The developed series expansions gave reliable answers for all complex reduced frequencies and executed faster than exponential approximations for many pressure stations.
NASA Astrophysics Data System (ADS)
Gazzillo, Domenico; Munaò, Gianmarco; Prestipino, Santi
2016-06-01
We study a pure fluid of heteronuclear sticky Janus dumbbells, considered to be the result of complete chemical association between unlike species in an initially equimolar mixture of hard spheres (species A) and sticky hard spheres (species B) with different diameters. The B spheres are particles whose attractive surface layer is infinitely thin. Wertheim's two-density integral equations are employed to describe the mixture of AB dumbbells together with unbound A and B monomers. After Baxter factorization, these equations are solved analytically within the associative Percus-Yevick approximation. The limit of complete association is taken at the end. The present paper extends to the more general, heteronuclear case of A and B species with size asymmetry a previous study by Wu and Chiew [J. Chem. Phys. 115, 6641 (2001)], which was restricted to dumbbells with equal monomer diameters. Furthermore, the solution for the Baxter factor correlation functions qi j α β ( r ) is determined here in a fully analytic way, since we have been able to find explicit analytic expressions for all the intervening parameters.
Gazzillo, Domenico; Munaò, Gianmarco; Prestipino, Santi
2016-06-21
We study a pure fluid of heteronuclear sticky Janus dumbbells, considered to be the result of complete chemical association between unlike species in an initially equimolar mixture of hard spheres (species A) and sticky hard spheres (species B) with different diameters. The B spheres are particles whose attractive surface layer is infinitely thin. Wertheim's two-density integral equations are employed to describe the mixture of AB dumbbells together with unbound A and B monomers. After Baxter factorization, these equations are solved analytically within the associative Percus-Yevick approximation. The limit of complete association is taken at the end. The present paper extends to the more general, heteronuclear case of A and B species with size asymmetry a previous study by Wu and Chiew [J. Chem. Phys. 115, 6641 (2001)], which was restricted to dumbbells with equal monomer diameters. Furthermore, the solution for the Baxter factor correlation functions qij (αβ)(r) is determined here in a fully analytic way, since we have been able to find explicit analytic expressions for all the intervening parameters. PMID:27334176
A wavelet-based computational method for solving stochastic Itô–Volterra integral equations
Mohammadi, Fakhrodin
2015-10-01
This paper presents a computational method based on the Chebyshev wavelets for solving stochastic Itô–Volterra integral equations. First, a stochastic operational matrix for the Chebyshev wavelets is presented and a general procedure for forming this matrix is given. Then, the Chebyshev wavelets basis along with this stochastic operational matrix are applied for solving stochastic Itô–Volterra integral equations. Convergence and error analysis of the Chebyshev wavelets basis are investigated. To reveal the accuracy and efficiency of the proposed method some numerical examples are included.
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.
Dynamical Behavior of Solution in Integrable Nonlocal Lakshmanan—Porsezian—Daniel Equation
NASA Astrophysics Data System (ADS)
Liu, Wei; Qiu, De-Qin; Wu, Zhi-Wei; He, Jing-Song
2016-06-01
The integrable nonlocal Lakshmanan—Porsezian—Daniel (LPD) equation which has the higher-order terms (dispersions and nonlinear effects) is first introduced. We demonstrate the integrability of the nonlocal LPD equation, provide its Lax pair, and present its rational soliton solutions and self-potential function by using the degenerate Darboux transformation. From the numerical plots of solutions, the compression effects of the real refractive index profile and the gain-or-loss distribution produced by δ are discussed. Supported by the National Natural Science Foundation of China under Grant No. 11271210 and the K.C. Wong Magna Fund in Ningbo University
NASA Technical Reports Server (NTRS)
Hoffman, David K.; Sharafeddin, Omar; Judson, Richard S.; Kouri, Donald J.
1990-01-01
The time-dependent form of the Lippmann-Schwinger integral equation is used as the basis of several new wave packet propagation schemes. These can be formulated in terms of either the time-dependent wave function or a time-dependent amplitude density. The latter is nonzero only in the region of configuratiaon space for which the potential is nonzero, thereby in principle obviating the necessity of large grids or the use of complex absorbing potentials when resonances cause long collision times (leading, consequently, to long propagation times). Transition amplitudes are obtained in terms of Fourier transforms of the amplitude density from the time to the energy domain. The approach is illustrated by an application to a standard potential scattering model problem where, as in previous studies, the action of the kinetic energy operator is evaluated by fast Fourier transform (FFT) techniques.
NASA Astrophysics Data System (ADS)
Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel
2009-02-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and Theoretical dedicated to the subject of the `SIDE8 International Conference', Sainte-Adéle, Canada, 22-28 June 2008 (http://www.crm.umontreal.ca/SIDE8/index_e.shtml). Participants at that meeting, as well as other researchers working in the field, are invited to submit a research paper to this issue. Editorial policy The Editorial Board has invited Decio Levi, Peter Olver, Zora Thomova and Pavel Winternitz to serve as Guest Editors for the special issue. Their criteria for the acceptance of contributions are as follows. The subject of the paper should relate to the subject of the conference: Ordinary and partial difference equations Analytic difference equations Orthogonal polynomials and special functions Symmetries and reductions Difference geometry Integrable discrete systems on graphs Integrable dynamical mappings Discrete Painlevè equations Singularity confinement Algebraic entropy Complexity and growth of multivalued mapping Representations of affine Weyl groups Quantum mappings Quantum field theory on the space-time lattice All contributions will be refereed and processed according to the usual procedure of the journal. Papers should report original and significant research that has not already been published. Guidelines for preparation of contributions The DEADLINE for contributed papers will be 1 March 2009. This deadline will allow the special issue to appear in October 2009. There is a nominal page limit of 12 printed pages (approximately 7200 words) per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. Further advice on publishing your work in Journal of Physics A: Mathematical and Theoretical may be found at www.iop.org/Journals/jphysa. Contributions to the special issue should if possible be submitted electronically by web upload at www.iop.org/Journals/jphysa, or by email to jphysa
NASA Technical Reports Server (NTRS)
Zuffada, C.; Crisp, D.
1996-01-01
Reliable descriptions of the optical properties of clouds and aerosols are essential for studies of radiative transfer in the terrestrial atmosphere...Here we explore the utility of two approaches for deriving the single scattering optical properties of particles with sharp corners and large axial ratios.
NASA Technical Reports Server (NTRS)
Ghaffari, A.
1971-01-01
Investigation of two cases of integrability of a second-order differential equation describing the projection of an axisymmetric satellite orbit on to a plane perpendicular to the rotation axis. It is demonstrated that for these two cases the integration can be carried out either by quadratures or reduced to a first-order differential equation. Analytical and physical properties are expressed, and it is shown that the equation can be derived from the classical plane eikonal equation of geometric optics.
NASA Astrophysics Data System (ADS)
Jee, SolKeun; Moser, Robert D.
2012-08-01
This study provides a simple moving-grid scheme which is based on a modified conservative form of the incompressible Navier-Stokes equations for flow around a moving rigid body. The modified integral form is conservative and seeks the solution of the absolute velocity. This approach is different from previous conservative differential forms [1-3] whose reference frame is not inertial. Keeping the reference frame being inertial results in simpler mathematical derivation to the governing equation which includes one dyadic product of velocity vectors in the convective term, whereas the previous [2,3] needs to obtain the time derivative with respect to non-inertial frames causing an additional dyadic product in the convective term. The scheme is implemented in a second-order accurate Navier-Stokes solver and maintains the order of the accuracy. After this verification, the scheme is validated for a pitching airfoil with very high frequencies. The simulation results match very well with the experimental results [4,5], including vorticity fields and a net thrust force. This airfoil simulation also provides detailed vortical structures near the trailing edge and time-evolving aerodynamic forces that are used to investigate the mechanism of the thrust force generation and the effects of the trailing edge shape. The developed moving-grid scheme demonstrates its validity for a rapid oscillating motion.
Functional integral derivation of the kinetic equation of two-dimensional point vortices
NASA Astrophysics Data System (ADS)
Fouvry, Jean-Baptiste; Chavanis, Pierre-Henri; Pichon, Christophe
2016-08-01
We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order 1 / N, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 2-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.
NASA Astrophysics Data System (ADS)
Kravtseva, A. K.
2013-04-01
In the paper, existence conditions for Feynman integrals in the sense of analytic continuation of Gaussian integrals with respect to operator arguments are found. A representation of Feynman integrals in the form of Gaussian integrals is also constructed and, finally, the class of evolution equations having solutions representable by Feynman integrals is described.
AN INTEGRAL EQUATION REPRESENTATION OF WIDE-BAND ELECTROMAGNETIC SCATTERING BY THIN SHEETS
An efficient, accurate numerical modeling scheme has been developed, based on the integral equation solution to compute electromagnetic (EM) responses of thin sheets over a wide frequency band. The thin-sheet approach is useful for simulating the EM response of a fracture system ...
A Family of Exponential Fitting Direct Quadrature Methods for Volterra Integral Equations
NASA Astrophysics Data System (ADS)
Cardone, A.; Ferro, M.; Ixaru, L. Gr.; Paternoster, B.
2010-09-01
A new class of direct quadrature methods for the solution of Volterra Integral Equations with periodic solution is illustrated. Such methods are based on an exponential fitting gaussian quadrature formula, whose coefficients depend on the problem parameters, in order to better reproduce the behavior the analytical solution. The construction of the methods is described, together with the analysis of the order of accuracy.
The Transmission Line as a Simple Example for Introducing Integral Equations to Undergraduates
ERIC Educational Resources Information Center
Rothwell, E. J.
2009-01-01
Integral equations are becoming a common means for describing problems in electromagnetics, and so it is important to expose students to methods for their solution. Typically this is done using examples in antennas, scattering, or electrostatics. Unfortunately, many difficult issues arise in the formulation and solution of the associated…
One-loop pentagon integral in d dimensions from differential equations in ɛ-form
NASA Astrophysics Data System (ADS)
Kozlov, Mikhail G.; Lee, Roman N.
2016-02-01
We apply the differential equation technique to the calculation of the one-loop massless diagram with five onshell legs. Using the reduction to ɛ-form, we manage to obtain a simple one-fold integral representation exact in space-time dimensionality. The expansion of the obtained result in ɛ and the analytical continuation to physical regions are discussed.
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.
New integration techniques for chemical kinetic rate equations. II - Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures 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. The comparisons show that LSODE is 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 an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
New integration techniques for chemical kinetic rate equations. 2: Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures 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. The comparisons show that LSODE is 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 an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
NASA Technical Reports Server (NTRS)
Jamnejad, V.; Cwik, T.; Zuffada, C.
1994-01-01
A coupled finite element-combined field integral equation technique was originally developed for solving scattering problems involving inhomogeneous objects of arbitrary shape and large dimensions in wavelength.
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.
Multiple Integration of the Heat-Conduction Equation for a Space Bounded From the Inside
NASA Astrophysics Data System (ADS)
Kot, V. A.
2016-03-01
An N-fold integration of the heat-conduction equation for a space bounded from the inside has been performed using a system of identical equalities with definition of the temperature function by a power polynomial with an exponential factor. It is shown that, in a number of cases, the approximate solutions obtained can be considered as exact because their errors comprise hundredths and thousandths of a percent. The method proposed for N-fold integration represents an alternative to classical integral transformations.
Giurgiutiu, V.; Ionita, A.; Dillard, D.A.; Graffeo, J.K.
1996-12-31
Fracture mechanics analysis of adhesively bonded joints has attracted considerable attention in recent years. A possible approach to the analysis of adhesive layer cracks is to study a brittle adhesive between 2 elastic half-planes representing the substrates. A 2-material 3-region elasticity problem is set up and has to be solved. A modeling technique based on the work of Fleck, Hutchinson, and Suo is used. Two complex potential problems using Muskelishvili`s formulation are set up for the 3-region, 2-material model: (a) a distribution of edge dislocations is employed to simulate the crack and its near field; and (b) a crack-free problem is used to simulate the effect of the external loading applied in the far field. Superposition of the two problems is followed by matching tractions and displacements at the bimaterial boundaries. The Cauchy principal value integral is used to treat the singularities. Imposing the traction-free boundary conditions over the entire crack length yielded a linear system of two integral equations. The parameters of the problem are Dundurs` elastic mismatch coefficients, {alpha} and {beta}, and the ratio c/H representing the geometric position of the crack in the adhesive layer.
Integrable nonlinear Schrödinger equation on simple networks: connection formula at vertices.
Sobirov, Z; Matrasulov, D; Sabirov, K; Sawada, S; Nakamura, K
2010-06-01
We study the case in which the nonlinear Schrödinger equation (NLSE) on simple networks consisting of vertices and bonds has an infinite number of constants of motion and becomes completely integrable just as in the case of a simple one-dimensional (1D) chain. Here the strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The connection formula at vertices obtained from norm and energy conservation rules shows (1) the solution on each bond is a part of the universal (bond-independent) soliton solution of the completely integrable NLSE on the 1D chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule. Under these conditions, we also showed an infinite number of constants of motion. The argument on a branched chain or a primary star graph is generalized to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. As a relevant issue, with use of reflectionless propagation of Zakharov-Shabat's soliton through networks we have obtained the transmission probabilities on the outgoing bonds, which are inversely proportional to the bond-dependent strength of nonlinearity. Numerical evidence is also given to verify the prediction. PMID:20866536
Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices
NASA Astrophysics Data System (ADS)
Sobirov, Z.; Matrasulov, D.; Sabirov, K.; Sawada, S.; Nakamura, K.
2010-06-01
We study the case in which the nonlinear Schrödinger equation (NLSE) on simple networks consisting of vertices and bonds has an infinite number of constants of motion and becomes completely integrable just as in the case of a simple one-dimensional (1D) chain. Here the strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The connection formula at vertices obtained from norm and energy conservation rules shows (1) the solution on each bond is a part of the universal (bond-independent) soliton solution of the completely integrable NLSE on the 1D chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule. Under these conditions, we also showed an infinite number of constants of motion. The argument on a branched chain or a primary star graph is generalized to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. As a relevant issue, with use of reflectionless propagation of Zakharov-Shabat’s soliton through networks we have obtained the transmission probabilities on the outgoing bonds, which are inversely proportional to the bond-dependent strength of nonlinearity. Numerical evidence is also given to verify the prediction.
Integrability and Solutions of the (2 + 1)-dimensional Hunter–Saxton Equation
NASA Astrophysics Data System (ADS)
Cai, Hong-Liu; Qu, Chang-Zheng
2016-04-01
In this paper, the (2 + 1)-dimensional Hunter-Saxton equation is proposed and studied. It is shown that the (2 + 1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations. Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation, a non-isospectral Lax-pair of the (2 + 1)-dimensional Hunter–Saxton equation is derived. In addition, exact singular solutions with a finite number of corners are obtained. Furthermore, the (2 + 1)-dimensional μ-Hunter–Saxton equation is presented, and its exact peaked traveling wave solutions are derived. Supported by National Natural Science Foundation of China under Grant No. 11471174 and NSF of Ningbo under Grant No. 2014A610018
Multi-off-grid methods in multi-step integration of ordinary differential equations
NASA Technical Reports Server (NTRS)
Beaudet, P. R.
1974-01-01
Description of methods of solving first- and second-order systems of differential equations in which all derivatives are evaluated at off-grid locations in order to circumvent the Dahlquist stability limitation on the order of on-grid methods. The proposed multi-off-grid methods require off-grid state predictors for the evaluation of the n derivatives at each step. Progressing forward in time, the off-grid states are predicted using a linear combination of back on-grid state values and off-grid derivative evaluations. A comparison is made between the proposed multi-off-grid methods and the corresponding Adams and Cowell on-grid integration techniques in integrating systems of ordinary differential equations, showing a significant reduction in the error at larger step sizes in the case of the multi-off-grid integrator.
A complete list of conservation laws for non-integrable compacton equations of K(m, m) type
NASA Astrophysics Data System (ADS)
Vodová, Jiřina
2013-03-01
In 1993, P Rosenau and J M Hyman introduced and studied Korteweg-de-Vries-like equations with nonlinear dispersion admitting compacton solutions, u_t+D_x^3(u^n)+D_x(u^m)=0 , m, n > 1, which are known as the K(m, n) equations. In this paper we consider a slightly generalized version of the K(m, n) equations for m = n, namely, u_t=aD_x^3(u^m)+bD_x(u^m) , where m, a, b are arbitrary real numbers. We describe all generalized symmetries and conservation laws thereof for m ≠ -2, -1/2, 0, 1; for these four exceptional values of m the equation in question is either completely integrable (m = -2, -1/2) or linear (m = 0, 1). It turns out that for m ≠ -2, -1/2, 0, 1 there are only three symmetries corresponding to x- and t-translations and scaling of t and u, and four non-trivial conservation laws, one of which expresses the conservation of energy, and the other three are associated with the Casimir functionals of the Hamiltonian operator \\mathfrak{D}=aD_x^3+bD_x admitted by our equation. Our result provides inter alia a rigorous proof of the fact that the K (2, 2) equation has just four conservation laws from the paper of P Rosenau and J M Hyman.
NASA Astrophysics Data System (ADS)
Yla-Oijala, Pasi
Electron multipacting is a serious problem in many rf components operating in vacuum. Multipacting can cause remarkable power losses and heating of the walls. This phenomenon starts if certain resonant conditions for electron trajectories are fulfilled and if the impacted surface has a secondary yield larger than one. In this work new computational methods have been developed which combine the standard trajectory calculations with advanced searching and analyzing methods for multipacting resonances. These methods have been applied to the analysis of electron multipacting in TESLA superconducting cavities and input power couplers with ceramic windows. TESLA is an international linear collider research and development project. Since even small errors in the rf field may destroy the trajectory calculation of a relativistic electron, the electromagnetic fields must be known accurately, especially close to the surfaces. The electromagnetic field computation is carried out by the boundary integral equation method. Due to the singularities of the integral equations, the numerical computations become rather involved, especially when computing the fields near the boundaries. Therefore, in this work special integration techniques and algorithms have been developed. In the axisymmetric geometries the numerical efficiency of various boundary integral equations has been studied.
Ciraolo, Giulio Gargano, Francesco Sciacca, Vincenzo
2013-08-01
We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.
Accurate integral equation theory for the central force model of liquid water and ionic solutions
NASA Astrophysics Data System (ADS)
Ichiye, Toshiko; Haymet, A. D. J.
1988-10-01
The atom-atom pair correlation functions and thermodynamics of the central force model of water, introduced by Lemberg, Stillinger, and Rahman, have been calculated accurately by an integral equation method which incorporates two new developments. First, a rapid new scheme has been used to solve the Ornstein-Zernike equation. This scheme combines the renormalization methods of Allnatt, and Rossky and Friedman with an extension of the trigonometric basis-set solution of Labik and co-workers. Second, by adding approximate ``bridge'' functions to the hypernetted-chain (HNC) integral equation, we have obtained predictions for liquid water in which the hydrogen bond length and number are in good agreement with ``exact'' computer simulations of the same model force laws. In addition, for dilute ionic solutions, the ion-oxygen and ion-hydrogen coordination numbers display both the physically correct stoichiometry and good agreement with earlier simulations. These results represent a measurable improvement over both a previous HNC solution of the central force model and the ex-RISM integral equation solutions for the TIPS and other rigid molecule models of water.
Dynamics of multibody chains in circular orbit: non-integrability of equations of motion
NASA Astrophysics Data System (ADS)
Maciejewski, Andrzej J.; Przybylska, Maria
2016-06-01
This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups.
On the solution of integral equations with a generalized Cauchy kernel
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1987-01-01
A numerical technique is developed analytically to solve a class of singular integral equations occurring in mixed boundary-value problems for nonhomogeneous elastic media with discontinuities. The approach of Kaya and Erdogan (1987) is extended to treat equations with generalized Cauchy kernels, reformulating the boundary-value problems in terms of potentials as the unknown functions. The numerical implementation of the solution is discussed, and results for an epoxy-Al plate with a crack terminating at the interface and loading normal to the crack are presented in tables.
Time transformations and Cowell's method. [for numerical integration of satellite motion equations
NASA Technical Reports Server (NTRS)
Velez, C. E.; Hilinski, S.
1978-01-01
The precise numerical integration of Cowell's equations of satellite motion is frequently performed with an independent variable s defined by an equation of the form dt = cr to the n-th power ds, where t represents time, r the radial distance from the center of attraction, c is a constant, and n is a parameter. This has been primarily motivated by the 'uniformizing' effects of such a transformation resulting in desirable 'analytic' stepsize control for elliptical orbits. This report discusses the 'proper' choice of the parameter n defining the independent variable s for various types of orbits and perturbation models, and develops a criterion for its selection.
NASA Technical Reports Server (NTRS)
Rosenbaum, J. S.
1976-01-01
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
NASA Astrophysics Data System (ADS)
Singh, R. C.
2009-07-01
The effects of quadrupole moments on the phase behaviour of isotropic-nematic transition are studied by using density functional theory for a system of molecules which interact via the Gay-Berne pair potential. The pair correlation functions of isotropic phase, which enter in the theory as input information, are found from the Percus-Yevick integral equation theory. The method used involves an expansion of angle-dependent functions appearing in the integral equations in terms of spherical harmonics and the harmonic coefficients are obtained by an iterative algorithm. All the terms of harmonic coefficients which involve l indices up to less than or equal to six have been considered. The dependence of the accuracy of the results on the number of terms taken in the basis set is explored for both fluids at different densities, temperatures and quadrupole moments. The results have been compared with the available computer simulation results.
On a method for constructing the Lax pairs for nonlinear integrable equations
NASA Astrophysics Data System (ADS)
Habibullin, I. T.; Khakimova, A. R.; Poptsova, M. N.
2016-01-01
We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.
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…
Mukherjee, Abhik Janaki, M. S. Kundu, Anjan
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 corresponding 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.
Schmidt, Rita; Webb, Andrew
2016-01-01
Electrical Properties Tomography (EPT) using MRI is a technique that has been developed to provide a new contrast mechanism for in vivo imaging. Currently the most common method relies on the solution of the homogeneous Helmholtz equation, which has limitations in accurate estimation at tissue interfaces. A new method proposed in this work combines a Maxwell's integral equation representation of the problem, and the use of high permittivity materials (HPM) to control the RF field, in order to reconstruct the electrical properties image. The magnetic field is represented by an integral equation considering each point as a contrast source. This equation can be solved in an inverse method. In this study we use a reference simulation or scout scan of a uniform phantom to provide an initial estimate for the inverse solution, which allows the estimation of the complex permittivity within a single iteration. Incorporating two setups with and without the HPM improves the reconstructed result, especially with respect to the very low electric field in the center of the sample. Electromagnetic simulations of the brain were performed at 3T to generate the B1(+) field maps and reconstruct the electric properties images. The standard deviations of the relative permittivity and conductivity were within 14% and 18%, respectively for a volume consisting of white matter, gray matter and cerebellum. PMID:26679289
NASA Astrophysics Data System (ADS)
Schmidt, Rita; Webb, Andrew
2016-01-01
Electrical Properties Tomography (EPT) using MRI is a technique that has been developed to provide a new contrast mechanism for in vivo imaging. Currently the most common method relies on the solution of the homogeneous Helmholtz equation, which has limitations in accurate estimation at tissue interfaces. A new method proposed in this work combines a Maxwell's integral equation representation of the problem, and the use of high permittivity materials (HPM) to control the RF field, in order to reconstruct the electrical properties image. The magnetic field is represented by an integral equation considering each point as a contrast source. This equation can be solved in an inverse method. In this study we use a reference simulation or scout scan of a uniform phantom to provide an initial estimate for the inverse solution, which allows the estimation of the complex permittivity within a single iteration. Incorporating two setups with and without the HPM improves the reconstructed result, especially with respect to the very low electric field in the center of the sample. Electromagnetic simulations of the brain were performed at 3 T to generate the B1+ field maps and reconstruct the electric properties images. The standard deviations of the relative permittivity and conductivity were within 14% and 18%, respectively for a volume consisting of white matter, gray matter and cerebellum.
Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations
NASA Astrophysics Data System (ADS)
Chvartatskyi, Oleksandr; Dimakis, Aristophanes; Müller-Hoissen, Folkert
2016-08-01
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.
Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations
NASA Astrophysics Data System (ADS)
Chvartatskyi, Oleksandr; Dimakis, Aristophanes; Müller-Hoissen, Folkert
2016-06-01
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.
Integral equation for gauge invariant quark two-point Green's function in QCD
Sazdjian, H.
2008-02-15
Gauge invariant quark two-point Green's functions defined with path-ordered gluon field phase factors along skew-polygonal lines joining the quark to the antiquark are considered. Functional relations between Green's functions with different numbers of path segments are established. An integral equation is obtained for the Green's function defined with a phase factor along a single straight line. The equation implicates an infinite series of two-point Green's functions, having an increasing number of path segments; the related kernels involve Wilson loops with contours corresponding to the skew-polygonal lines of the accompanying Green's function and with functional derivatives along the sides of the contours. The series can be viewed as an expansion in terms of the global number of the functional derivatives of the Wilson loops. The lowest-order kernel, which involves a Wilson loop with two functional derivatives, provides the framework for an approximate resolution of the equation.
Integral-equation approach to the weak-field asymptotic theory of tunneling ionization
NASA Astrophysics Data System (ADS)
Dnestryan, Andrey I.; Tolstikhin, Oleg I.
2016-03-01
An integral equation approach to the weak-field asymptotic theory (WFAT) of tunneling ionization is developed. An integral representation for the exact partial amplitudes of ionization into parabolic channels is derived. The WFAT expansion for the ionization rate follows immediately from this relation. Integral representations for the coefficients in the expansion are obtained. The integrals accumulate where the ionizing orbital has large amplitude and are not sensitive to its behavior in the asymptotic region. Hence, these formulas enable one to reliably calculate the WFAT coefficients even if the orbital is represented by an expansion in Gaussian basis, as is usually the case in standard software packages for electronic structure calculations. This development is expected to greatly simplify the implementation of the WFAT for polyatomic molecules, and thus facilitate its growing applications in strong-field physics.
Error analysis of exponential integrators for oscillatory second-order differential equations
NASA Astrophysics Data System (ADS)
Grimm, Volker; Hochbruck, Marlis
2006-05-01
In this paper, we analyse a family of exponential integrators for second-order differential equations in which high-frequency oscillations in the solution are generated by a linear part. Conditions are given which guarantee that the integrators allow second-order error bounds independent of the product of the step size with the frequencies. Our convergence analysis generalizes known results on the mollified impulse method by García-Archilla, Sanz-Serna and Skeel (1998, SIAM J. Sci. Comput. 30 930-63) and on Gautschi-type exponential integrators (Hairer E, Lubich Ch and Wanner G 2002 Geometric Numerical Integration (Berlin: Springer), Hochbruck M and Lubich Ch 1999 Numer. Math. 83 403-26).
An, Hongli; Fan, Engui; Zhu, Haixing
2015-01-01
The 2+1-dimensional compressible Euler equations are investigated here. A power-type elliptic vortex ansatz is introduced and thereby reduction obtains to an eight-dimensional nonlinear dynamical system. The latter is shown to have an underlying integral Ermakov-Ray-Reid structure of Hamiltonian type. It is of interest to notice that such an integrable Ermakov structure exists not only in the density representations but also in the velocity components. A class of typical elliptical vortex solutions termed pulsrodons corresponding to warm-core eddy theory is isolated and its behavior is simulated. In addition, a Lax pair formulation is constructed and the connection with stationary nonlinear cubic Schrödinger equations is established. PMID:25679730
Two-dimensional time-domain volume integral equations for scattering of inhomogeneous objects
NASA Astrophysics Data System (ADS)
Wang, Jianguo; Fan, Ruyu
2003-08-01
This paper proposes a time-domain volume integral equation based method for analyzing the transient scattering from a two-dimensional inhomogeneous cylinder by invoking the volume equivalence principle for both the transverse magnetic and electric cases. The cylinder is discretized into triangular cells, and the electric flux is chosen as the unknown. For the transverse magnetic case, the electric flux is defined on the surfaces of the triangles. For the transverse electric case, because of the electric charges induced inside and on the surface of the cylinder, the electric flux is defined on the edges of the triangles, and expanded in space in terms of two-dimensional surface roof-top basis functions. The time-domain volume integral equation is solved by using a marching-on-in-time scheme. Numerical results obtained using this method are in excellent agreement with the data obtained using the finite-difference time-domain method.
Global integral gradient bounds for quasilinear equations below or near the natural exponent
NASA Astrophysics Data System (ADS)
Phuc, Nguyen Cong
2014-10-01
We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón-Zygmund type estimates below the natural exponent are also obtained for -superharmonic functions in the whole space ℝ n . This answers a question raised in our earlier work (On Calderón-Zygmund theory for p- and -superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.
Seismic response of laterally inhomogeneous geological region by boundary integral equations
NASA Astrophysics Data System (ADS)
Parvanova, S.; Dineva, P.; Fontara, I.-K.; Wuttke, F.
2015-07-01
The proposed study deals with synthesis of seismograms by BIEM (boundary integral equation method) taking into account all three base components-seismic source, wave path and local region of interest. Consider a laterally inhomogeneous geological profile situated in a half-plane with non-parallel layers. Seismic load is time-harmonic or transient in time. It is presented by incident SH wave or wave radiating from an embedded line seismic source. Two types of lateral inhomogeneities with arbitrary shape and located in the inhomogeneous half-plane are considered: (i) free-surface relief as a canyon or a hill; (ii) alluvial basin with properties different from those of the layered half-plane. The computational tool is BIEM based on the frequency-dependent elastodynamic fundamental solutions. A relation between displacements and tractions along the free surface and arbitrary interface of the soil stratum is derived, which is applicable for arbitrary geometry of the interfaces between soil layers. Validation and convergence study is presented. All simulations reveal the sensitivity of the synthetic seismic signals on the type and characteristics of the seismic time-harmonic or transient load, on the wave path inhomogeneity and on the specific geotechnical properties of the local geological region.
Romá, Federico; Cugliandolo, Leticia F; Lozano, Gustavo S
2014-08-01
We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy. PMID:25215839
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Honwah, Tam
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz–Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
A three dimensional integral equation approach for fluids under confinement: Argon in zeolites.
Lomba, Enrique; Bores, Cecilia; Sánchez-Gil, Vicente; Noya, Eva G
2015-10-28
In this work, we explore the ability of an inhomogeneous integral equation approach to provide a full three dimensional description of simple fluids under conditions of confinement in porous media. Explicitly, we will consider the case of argon adsorbed into silicalite-1, silicalite-2, and an all-silica analogue of faujasite, with a porous structure composed of linear (and zig-zag in the case of silicalite-1) channels of 5-8 Å diameter. The equation is based on the three dimensional Ornstein-Zernike approximation proposed by Beglov and Roux [J. Chem. Phys. 103, 360 (1995)] in combination with the use of an approximate fluid-fluid direct correlation function furnished by the replica Ornstein-Zernike equation with a hypernetted chain closure. Comparison with the results of grand canonical Monte Carlo/molecular dynamics simulations evidences that the theory provides an accurate description for the three dimensional density distribution of the adsorbed fluid, both at the level of density profiles and bidimensional density maps across representative sections of the porous material. In the case of very tight confinement (silicalite-1 and silicalite-2), solutions at low temperatures could not be found due to convergence difficulties, but for faujasite, which presents substantially larger channels, temperatures as low as 77 K are accessible to the integral equation. The overall results indicate that the theoretical approximation can be an excellent tool to characterize the microscopic adsorption behavior of porous materials. PMID:26520539
A three dimensional integral equation approach for fluids under confinement: Argon in zeolites
NASA Astrophysics Data System (ADS)
Lomba, Enrique; Bores, Cecilia; Sánchez-Gil, Vicente; Noya, Eva G.
2015-10-01
In this work, we explore the ability of an inhomogeneous integral equation approach to provide a full three dimensional description of simple fluids under conditions of confinement in porous media. Explicitly, we will consider the case of argon adsorbed into silicalite-1, silicalite-2, and an all-silica analogue of faujasite, with a porous structure composed of linear (and zig-zag in the case of silicalite-1) channels of 5-8 Å diameter. The equation is based on the three dimensional Ornstein-Zernike approximation proposed by Beglov and Roux [J. Chem. Phys. 103, 360 (1995)] in combination with the use of an approximate fluid-fluid direct correlation function furnished by the replica Ornstein-Zernike equation with a hypernetted chain closure. Comparison with the results of grand canonical Monte Carlo/molecular dynamics simulations evidences that the theory provides an accurate description for the three dimensional density distribution of the adsorbed fluid, both at the level of density profiles and bidimensional density maps across representative sections of the porous material. In the case of very tight confinement (silicalite-1 and silicalite-2), solutions at low temperatures could not be found due to convergence difficulties, but for faujasite, which presents substantially larger channels, temperatures as low as 77 K are accessible to the integral equation. The overall results indicate that the theoretical approximation can be an excellent tool to characterize the microscopic adsorption behavior of porous materials.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2016-04-01
Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection-diffusion-reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.
The use of time-domain integral equations in electromagnetism problems
NASA Astrophysics Data System (ADS)
Berthon, A.; Vallet, E.
1984-10-01
The application of time-domain integral equations to the analysis of the scattering and propagation of EM fields is surveyed. Explicit iterative time-stepping procedures are developed for such problems as simple-conductor obstacles, one-dimensional obstacles, orifices, small obstacles, and media of finite conductivity (interface fields, reflection factors, and shielding problems). Numerical results for problems involving the interaction of a dipole field with various cylindrical obstacles are presented in graphs and diagrams.
NASA Technical Reports Server (NTRS)
Jones, J. E.; Richmond, J. H.
1974-01-01
An integral equation formulation is applied to predict pitch- and roll-plane radiation patterns of a thin VHF/UHF (very high frequency/ultra high frequency) annular slot communications antenna operating at several locations in the nose region of the space shuttle orbiter. Digital computer programs used to compute radiation patterns are given and the use of the programs is illustrated. Experimental verification of computed patterns is given from measurements made on 1/35-scale models of the orbiter.
Integral equation for a strip coil antenna located on a dielectric cylinder
NASA Astrophysics Data System (ADS)
Dementyev, A. N.; Klyuev, D. S.; Shatrov, S. A.
2016-01-01
The problem about the distribution of the surface current density in a narrow circular strip antenna as an infinitely thin perfectly conducting ribbon folded in a circle and positioned on the surface of a dielectric cylinder is reduced to a one-dimensional integral equation (IE). A method for solving the obtained IE is proposed. Complex distributions of the azimuthal component of the surface current density over the circular conductor are presented for different values of the dielectric permittivity of the cylinder.
Implicit integration of the time-dependent Ginzburg-Landau equations of superconductivity.
Gunter, D. O.; Kaper, H. G.; Leaf, G. K.; Mathematics and Computer Science
2002-05-17
This article is concerned with the integration of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. Four algorithms, ranging from fully explicit to nonlinearly implicit, are presented and evaluated for stability, accuracy, and compute time. The benchmark problem for the evaluation is the equilibration of a vortex configuration in a superconductor that is embedded in a thin insulator and subject to an applied magnetic field.
Parameter estimation for boundary value problems by integral equations of the second kind
NASA Technical Reports Server (NTRS)
Kojima, Fumio
1988-01-01
This paper is concerned with the parameter estimation for boundary integral equations of the second kind. The parameter estimation technique through use of the spline collocation method is proposed. Based on the compactness assumption imposed on the parameter space, the convergence analysis for the numerical method of parameter estimation is discussed. The results obtained here are applied to a boundary parameter estimation for 2-D elliptic systems.
NASA Technical Reports Server (NTRS)
Hu, Fang Q.; Pizzo, Michelle E.; Nark, Douglas M.
2016-01-01
Based on the time domain boundary integral equation formulation of the linear convective wave equation, a computational tool dubbed Time Domain Fast Acoustic Scattering Toolkit (TD-FAST) has recently been under development. The time domain approach has a distinct advantage that the solutions at all frequencies are obtained in a single computation. In this paper, the formulation of the integral equation, as well as its stabilization by the Burton-Miller type reformulation, is extended to cases of a constant mean flow in an arbitrary direction. In addition, a "Source Surface" is also introduced in the formulation that can be employed to encapsulate regions of noise sources and to facilitate coupling with CFD simulations. This is particularly useful for applications where the noise sources are not easily described by analytical source terms. Numerical examples are presented to assess the accuracy of the formulation, including a computation of noise shielding by a thin barrier motivated by recent Historical Baseline F31A31 open rotor noise shielding experiments. Furthermore, spatial resolution requirements of the time domain boundary element method are also assessed using point per wavelength metrics. It is found that, using only constant basis functions and high-order quadrature for surface integration, relative errors of less than 2% may be obtained when the surface spatial resolution is 5 points-per-wavelength (PPW) or 25 points-per-wavelength squared (PPW2).
Numerical analysis of composite STEEL-CONCRETE SECTIONS using integral equation of Volterra
NASA Astrophysics Data System (ADS)
Partov, Doncho; Kantchev, Vesselin
2011-09-01
The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann — Volterra for the concrete part. On the basis of the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time "t", two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated. The creep functions is suggested by the model CEB MC90-99 and the "ACI 209R-92 model. The elastic modulus of concrete Ec(t) is assumed to be constant in time `t'. The obtained results from the both models are compared.
Numerical analysis of composite STEEL-CONCRETE SECTIONS using integral equation of Volterra
NASA Astrophysics Data System (ADS)
Partov, Doncho; Kantchev, Vesselin
2011-09-01
The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann — Volterra for the concrete part. On the basis of the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time "t", two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated. The creep functions is suggested by the model CEB MC90-99 and the "ACI 209R-92 model. The elastic modulus of concrete E c (t) is assumed to be constant in time `t'. The obtained results from the both models are compared.
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.
NASA Astrophysics Data System (ADS)
Ghiner, A. V.; Surdutovich, G. I.
1994-07-01
An approach using the generalized method of integral equations by substitution of the variables in the integral equation is applied to two- and quasi-two-dimensional systems. As a result, the connection between the integral and Maxwell equations as well as an extinction theorem for this case are established. The technique developed may be applied to any composite medium with a columnlike mesostructure. By use of the elementary cylinder radiator (``mesoscopic atom'') concept we reduce the problem of finding the optical properties of such media to the calculation of the susceptibility of a dense two-dimensional gas. The calculated optical anisotropy depends dramatically not only on the concentration but also on the form of the inclusions (mesostructure). Our calculations of the dielectric permittivity tensor for a two-dimensional composite medium with wire mesostructure show excellent agreement with the experimental measurements of the long-wavelength dielectric constants for two orthogonal polarizations in a photonic crystal made of dielectric rods [W. M. Robertson et al., J. Opt. Soc. Am. B 10, 322 (1993)].
Solvation effects on chemical shifts by embedded cluster integral equation theory.
Frach, Roland; Kast, Stefan M
2014-12-11
The accurate computational prediction of nuclear magnetic resonance (NMR) parameters like chemical shifts represents a challenge if the species studied is immersed in strongly polarizing environments such as water. Common approaches to treating a solvent in the form of, e.g., the polarizable continuum model (PCM) ignore strong directional interactions such as H-bonds to the solvent which can have substantial impact on magnetic shieldings. We here present a computational methodology that accounts for atomic-level solvent effects on NMR parameters by extending the embedded cluster reference interaction site model (EC-RISM) integral equation theory to the prediction of chemical shifts of N-methylacetamide (NMA) in aqueous solution. We examine the influence of various so-called closure approximations of the underlying three-dimensional RISM theory as well as the impact of basis set size and different treatment of electrostatic solute-solvent interactions. We find considerable and systematic improvement over reference PCM and gas phase calculations. A smaller basis set in combination with a simple point charge model already yields good performance which can be further improved by employing exact electrostatic quantum-mechanical solute-solvent interaction energies. A larger basis set benefits more significantly from exact over point charge electrostatics, which can be related to differences of the solvent's charge distribution. PMID:25377116
NASA Astrophysics Data System (ADS)
Kalogiratou, Z.; Monovasilis, Th.; Psihoyios, G.; Simos, T. E.
2014-03-01
In this work we review single step methods of the Runge-Kutta type with special properties. Among them are methods specially tuned to integrate problems that exhibit a pronounced oscillatory character and such problems arise often in celestial mechanics and quantum mechanics. Symplectic methods, exponentially and trigonometrically fitted methods, minimum phase-lag and phase-fitted methods are presented. These are Runge-Kutta, Runge-Kutta-Nyström and Partitioned Runge-Kutta methods. The theory of constructing such methods is given as well as several specific methods. In order to present the performance of the methods we have tested 58 methods from all categories. We consider the two dimensional harmonic oscillator, the two body problem, the pendulum problem and the orbital problem studied by Stiefel and Bettis. Also we have tested the methods on the computation of the eigenvalues of the one dimensional time independent Schrödinger equation with the harmonic oscillator, the doubly anharmonic oscillator and the exponential potentials.
NASA Astrophysics Data System (ADS)
Schmidt, Matthew; Constable, Steve; Ing, Christopher; Roy, Pierre-Nicholas
2014-06-01
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, ψT, equal to unity. The present paper assesses the effectiveness of three different trial wavefunctions on three isotopes of hydrogen for 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.
Schmidt, Matthew; Constable, Steve; Ing, Christopher; Roy, Pierre-Nicholas
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 for 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.
Computational attributes of the integral form of the equation of transfer
NASA Technical Reports Server (NTRS)
Frankel, J. I.
1991-01-01
Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. This paper, departs from classical thought and presents an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. The approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.
Classical integrability for beta-ensembles and general Fokker-Planck equations
Rumanov, Igor
2015-01-15
Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here, we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g., there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system–a closed system of two nonlinear partial differential equations (PDEs) of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.
NASA Astrophysics Data System (ADS)
Dault, Daniel Lawrence
The moment method is the predominant approach for the solution of electromagnetic boundary integral equations. Traditional moment method discretizations rely on the projection of solution currents onto basis sets that must satisfy strict continuity properties to model physical currents. The choice of basis sets is further restricted by the tight coupling of traditional functional descriptions to the underlying geometrical approximation of the scattering or radiating body. As a result, the choice of approximation function spaces and geometry discretizations for a given boundary integral equation is significantly limited. A quasi-meshless partition of unity based method called the Generalized Method of Moments (GMM) was recently introduced to overcome some of these limitations. The GMM partition of unity scheme affords automatic continuity of solution currents, and therefore permits the use of a much wider range of basis functions than traditional moment methods. However, prior to the work in this thesis, GMM was limited in practical applicability because it was only formulated for a few geometry types, could not be accurately applied to arbitrary scatterers, e.g. those with mixtures of geometrical features, and was not amenable to traditional acceleration methodologies that would permit its application to electrically large problems. The primary contribution of this thesis is to introduce several new GMM formulations that significantly expand the capabilities of the method to make it a practical, broadly applicable approach for solving boundary integral equations and overcoming the limitations inherent in traditional moment method discretizations. Additionally, several of the topics covered address continuing open problems in electromagnetic boundary integral equations with applicability beyond GMM. The work comprises five broad contributions. The first is a new GMM formulation capable of mixing both GMM-type basis sets and traditional basis sets in the same
White, J.; Phillips, J.R.; Korsmeyer, T.
1994-12-31
Mixed first- and second-kind surface integral equations with (1/r) and {partial_derivative}/{partial_derivative} (1/r) kernels are generated by a variety of three-dimensional engineering problems. For such problems, Nystroem type algorithms can not be used directly, but an expansion for the unknown, rather than for the entire integrand, can be assumed and the product of the singular kernal and the unknown integrated analytically. Combining such an approach with a Galerkin or collocation scheme for computing the expansion coefficients is a general approach, but generates dense matrix problems. Recently developed fast algorithms for solving these dense matrix problems have been based on multipole-accelerated iterative methods, in which the fast multipole algorithm is used to rapidly compute the matrix-vector products in a Krylov-subspace based iterative method. Another approach to rapidly computing the dense matrix-vector products associated with discretized integral equations follows more along the lines of a multigrid algorithm, and involves projecting the surface unknowns onto a regular grid, then computing using the grid, and finally interpolating the results from the regular grid back to the surfaces. Here, the authors describe a precorrectted-FFT approach which can replace the fast multipole algorithm for accelerating the dense matrix-vector product associated with discretized potential integral equations. The precorrected-FFT method, described below, is an order n log(n) algorithm, and is asymptotically slower than the order n fast multipole algorithm. However, initial experimental results indicate the method may have a significant constant factor advantage for a variety of engineering problems.
Closed-form integrator for the quaternion (euler angle) kinematics equations
NASA Technical Reports Server (NTRS)
Whitmore, Stephen A. (Inventor)
2000-01-01
The invention is embodied in a method of integrating kinematics equations for updating a set of vehicle attitude angles of a vehicle using 3-dimensional angular velocities of the vehicle, which includes computing an integrating factor matrix from quantities corresponding to the 3-dimensional angular velocities, computing a total integrated angular rate from the quantities corresponding to a 3-dimensional angular velocities, computing a state transition matrix as a sum of (a) a first complementary function of the total integrated angular rate and (b) the integrating factor matrix multiplied by a second complementary function of the total integrated angular rate, and updating the set of vehicle attitude angles using the state transition matrix. Preferably, the method further includes computing a quanternion vector from the quantities corresponding to the 3-dimensional angular velocities, in which case the updating of the set of vehicle attitude angles using the state transition matrix is carried out by (a) updating the quanternion vector by multiplying the quanternion vector by the state transition matrix to produce an updated quanternion vector and (b) computing an updated set of vehicle attitude angles from the updated quanternion vector. The first and second trigonometric functions are complementary, such as a sine and a cosine. The quantities corresponding to the 3-dimensional angular velocities include respective averages of the 3-dimensional angular velocities over plural time frames. The updating of the quanternion vector preserves the norm of the vector, whereby the updated set of vehicle attitude angles are virtually error-free.
A study of orbital instability using first integrals
NASA Astrophysics Data System (ADS)
Brium, A. Z.
1989-12-01
A method is proposed for studying the orbital instability of periodic solutions to normal systems of ordinary differential equations. In accordance with the approach proposed here, the Liapunov function is constructed from the first integrals of the equations of perturbed motion and from the scalar product of the orbital motion velocity and the perturbation vector. The discussion includes pendulum motions in a central Newtonian field, pendulum motions of the Kovalevskaya gyroscope, and the Bobylev-Steklov case.
Orientation-dependent integral equation theory for a two-dimensional model of water
NASA Astrophysics Data System (ADS)
Urbič, T.; Vlachy, V.; Kalyuzhnyi, Yu. V.; Dill, K. A.
2003-03-01
We develop an integral equation theory that applies to strongly associating orientation-dependent liquids, such as water. In an earlier treatment, we developed a Wertheim integral equation theory (IET) that we tested against NPT Monte Carlo simulations of the two-dimensional Mercedes Benz model of water. The main approximation in the earlier calculation was an orientational averaging in the multidensity Ornstein-Zernike equation. Here we improve the theory by explicit introduction of an orientation dependence in the IET, based upon expanding the two-particle angular correlation function in orthogonal basis functions. We find that the new orientation-dependent IET (ODIET) yields a considerable improvement of the predicted structure of water, when compared to the Monte Carlo simulations. In particular, ODIET predicts more long-range order than the original IET, with hexagonal symmetry, as expected for the hydrogen bonded ice in this model. The new theoretical approximation still errs in some subtle properties; for example, it does not predict liquid water's density maximum with temperature or the negative thermal expansion coefficient.
Analytical integrability and physical solutions of d-KdV equation
Karmakar, P.K.; Dwivedi, C.B.
2006-03-15
A new idea of electron inertia-induced ion sound wave excitation for transonic plasma equilibrium has already been reported. In such unstable plasma equilibrium, a linear source driven Korteweg-de Vries (d-KdV) equation describes the nonlinear ion sound wave propagation behavior. By numerical techniques, two distinct classes of solution (soliton and oscillatory shocklike structures) are obtained. Present contribution deals with the systematic methodological efforts to find out its (d-KdV) analytical solutions. As a first step, we apply the Painleve method to test whether the derived d-KdV equation is analytically integrable or not. We find that the derived d-KdV equation is indeed analytically integrable since it satisfies Painleve property. Hirota's bilinearization method and the modified sine-Gordon method (also termed as sine-cosine method) are used to derive the analytical results. Perturbative technique is also applied to find out quasistationary solutions. A few graphical plots are provided to offer a glimpse of the structural profiles obtained by different methods applied. It is conjectured that these solutions may open a new scope of acoustic spectroscopy in plasma hydrodynamics.
Advances in numerical solutions to integral equations in liquid state theory
NASA Astrophysics Data System (ADS)
Howard, Jesse J.
Solvent effects play a vital role in the accurate description of the free energy profile for solution phase chemical and structural processes. The inclusion of solvent effects in any meaningful theoretical model however, has proven to be a formidable task. Generally, methods involving Poisson-Boltzmann (PB) theory and molecular dynamic (MD) simulations are used, but they either fail to accurately describe the solvent effects or require an exhaustive computation effort to overcome sampling problems. An alternative to these methods are the integral equations (IEs) of liquid state theory which have become more widely applicable due to recent advancements in the theory of interaction site fluids and the numerical methods to solve the equations. In this work a new numerical method is developed based on a Newton-type scheme coupled with Picard/MDIIS routines. To extend the range of these numerical methods to large-scale data systems, the size of the Jacobian is reduced using basis functions, and the Newton steps are calculated using a GMRes solver. The method is then applied to calculate solutions to the 3D reference interaction site model (RISM) IEs of statistical mechanics, which are derived from first principles, for a solute model of a pair of parallel graphene plates at various separations in pure water. The 3D IEs are then extended to electrostatic models using an exact treatment of the long-range Coulomb interactions for negatively charged walls and DNA duplexes in aqueous electrolyte solutions to calculate the density profiles and solution thermodynamics. It is found that the 3D-IEs provide a qualitative description of the density distributions of the solvent species when compared to MD results, but at a much reduced computational effort in comparison to MD simulations. The thermodynamics of the solvated systems are also qualitatively reproduced by the IE results. The findings of this work show the IEs to be a valuable tool for the study and prediction of
NASA Technical Reports Server (NTRS)
Clarke, R.; Lintereur, L.; Bahm, C.
2016-01-01
A desire for more complete documentation of the National Aeronautics and Space Administration (NASA) Armstrong Flight Research Center (AFRC), Edwards, California legacy code used in the core simulation has led to this e ort to fully document the oblate Earth six-degree-of-freedom equations of motion and integration algorithm. The authors of this report have taken much of the earlier work of the simulation engineering group and used it as a jumping-o point for this report. The largest addition this report makes is that each element of the equations of motion is traced back to first principles and at no point is the reader forced to take an equation on faith alone. There are no discoveries of previously unknown principles contained in this report; this report is a collection and presentation of textbook principles. The value of this report is that those textbook principles are herein documented in standard nomenclature that matches the form of the computer code DERIVC. Previous handwritten notes are much of the backbone of this work, however, in almost every area, derivations are explicitly shown to assure the reader that the equations which make up the oblate Earth version of the computer routine, DERIVC, are correct.
New integration techniques for chemical kinetic rate equations. I - Efficiency comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the efficiency of several recently developed numerical techniques for solving chemical kinetic rate equations is presented. The solution procedures examined include two general-purpose codes, EPISODE and LSODE, developed as multipurpose differential equation solvers, and three specialzed codes, CHEMEQ, CREK1D, and GCKP84, developed specifically for chemical kinetics. The efficiency comparison is made by applying these codes to two practical combustion kinetics problems. Both problems describe adiabatic, constant-pressure, gas-phase chemical reactions and include all three combustion regimes: induction, heat release, and equilibration. The comparison shows that LSODE is the fastest routine currently available for solving chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for temperature can be significantly faster than evaluation of the temperature by integration of its time derivative. Significant increases in computational speed are realized by updating the reaction rate constants 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 presented and is shown to result in increased computational speed.
NASA Astrophysics Data System (ADS)
Peñaloza M, Marcos A.
1999-01-01
The problem defined in the title has partially been addressed by various studies, in a complicated manner, without providing sufficient details or in ways that are to some extent confusing. For these reasons the basic equation governing the operation of the cell-reciprocal integrating nephelometer (CRIN) has been deduced by a comprehensible and didactic approach in this work. A comparison of this equation with the respective one for the cell-direct integrating nephelometer (CDIN) has been undertaken. An introductory review analysis of the equation and its first and second corrections, due to the so-called truncation error and light extinction error, respectively, has been performed. An indication of how they can, in theory, be minimized is given. The essential CRIN-calibration procedure has been described with special emphasis on the convenient use of a monodisperse, non-absorbing and spherical aerosol as a calibration substance instead of using toxic and environmentally damaging gases. However, in this respect, some comments on fundamental technical problems and possible errors arising in generating, sampling and operating an aerosol of this kind for calibration purposes have been added. For the various aspects analysed in this work it is shown that the CRIN-calibration procedure is a very difficult task to accomplish, especially when an aerosol is used.
A hybrid method for solving time-domain integral equations in transient scattering
NASA Astrophysics Data System (ADS)
Tijhuis, A. G.; Wiemans, R.; Kuester, E. F.
A new hybrid method is proposed for the numerical solution of integral equations describing transient scattering problems. The basic idea behind the method is to first discretize in space, and then solve the resulting system of linear time-domain equations by carrying out a temporal Laplace transformation. This approach combines the efficiency of the marching-on-in-time method with the stability and the accuracy of frequency-domain techniques. Numerical results are presented for the scattering of E-polarized, pulsed waves by a one-dimensionally inhomogeneous, lossy dielectric slab located in between two homogeneous, lossless dielectric half-spaces, and by a radially inhomogeneous, lossy dielectric circular cylinder embedded in vacuum. For these problems, the hybrid method turns out to be more powerful than existing solution techniques.
Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1M Unknowns
NASA Technical Reports Server (NTRS)
Shaeffer, John
2008-01-01
Matrix methods for solving integral equations via direct solve LU factorization are presently limited to weeks to months of very expensive supercomputer time for problems sizes of several hundred thousand unknowns. This report presents matrix LU factor solutions for electromagnetic scattering problems for problem sizes to one million unknowns with thousands of right hand sides that run in mere days on PC level hardware. This EM solution is accomplished by utilizing the numerical low rank nature of spatially blocked unknowns using the Adaptive Cross Approximation for compressing the rank deficient blocks of the system Z matrix, the L and U factors, the right hand side forcing function and the final current solution. This compressed matrix solution is applied to a frequency domain EM solution of Maxwell's equations using standard Method of Moments approach. Compressed matrix storage and operations count leads to orders of magnitude reduction in memory and run time.
NASA Astrophysics Data System (ADS)
Napora, Jolanta
2000-10-01
A given Riccati equation, as is well known, can be naturally reduced to a system of nonlinear evolution equations on an infinite-dimensional functional manifold with Cauchy-Goursat initial data. We describe the Lie algebraic reduction procedure of nonlocal type for this infinite-dimensional dynamical system upon the set of critical points of an invariant Lagrangian functional. As one of our main results, we show that the reduced dynamical system generates the completely integrable Hamiltonian flow on this submanifold with respect to the canonical symplectic structure upon it. The above also makes it possible to find effectively its finite-dimensional Lax type representation via both the well known Moser type reduction procedure and the dual momentum mapping scheme on some matrix manifold.
Erguel, Ozguer; Guerel, Levent
2008-12-01
We present a novel stabilization procedure for accurate surface formulations of electromagnetic scattering problems involving three-dimensional dielectric objects with arbitrarily low contrasts. Conventional surface integral equations provide inaccurate results for the scattered fields when the contrast of the object is low, i.e., when the electromagnetic material parameters of the scatterer and the host medium are close to each other. We propose a stabilization procedure involving the extraction of nonradiating currents and rearrangement of the right-hand side of the equations using fictitious incident fields. Then, only the radiating currents are solved to calculate the scattered fields accurately. This technique can easily be applied to the existing implementations of conventional formulations, it requires negligible extra computational cost, and it is also appropriate for the solution of large problems with the multilevel fast multipole algorithm. We show that the stabilization leads to robust formulations that are valid even for the solutions of extremely low-contrast objects.
NASA Astrophysics Data System (ADS)
Dimakis, Aristophanes; Müller-Hoissen, Folkert
2013-02-01
We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D-2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings).
Matrix equilibration in method of moment solutions of surface integral equations
NASA Astrophysics Data System (ADS)
Kolundzija, Branko M.; Kostic, Milan M.
2014-12-01
Basic theory of matrix equilibration is presented, relating it to other techniques for decreasing the condition number of matrix equations obtained by the method of moments (MOM) applied to surface integral equations (SIEs). It is shown that matrix equilibration is a general technique that can be used for both (1) balancing field and source quantities in SIEs, which is used to decrease the condition number in the case of SIEs of mixed type and high contrast in material properties, and (2) scaling basis and test functions in MOM, which is used to decrease the condition number in the case of higher-order bases and patches of different sizes. In particular, it is demonstrated that a combination of such balancing and scaling can be performed using simple matrix equilibration based on magnitudes of diagonal elements and 2-norms of rows/columns of the MOM matrix.
A molecular site-site integral equation that yields the dielectric constant
NASA Astrophysics Data System (ADS)
Dyer, Kippi M.; Perkyns, John S.; Stell, George; Pettitt, B. Montgomery
2008-09-01
Our recent derivation [K. M. Dyer et al., J. Chem. Phys. 127, 194506 (2007)] of a diagrammatically proper, site-site, integral equation theory using molecular angular expansions is extended to polar fluids. With the addition of atomic site charges we take advantage of the formal long-ranged potential field cancellations before renormalization to generate a set of numerically stable equations. Results for calculations in a minimal (spherical) angular basis set are presented for the radial distribution function, the first dipolar (110) projection, and the dielectric constant for two model diatomic systems. All results, when compared to experiment and simulation, are a significant quantitative and qualitative improvement over previous site-site theories. More importantly, the dielectric constant is not trivial and close to simulation and experiment.
Iteration methods and algorithms for dielectric-resonator integral equations of the field
NASA Astrophysics Data System (ADS)
Davidovich, M. V.; Stefyuk, Yu. V.
2010-10-01
Quasi-eigenmodes of open cylindrical and rectangular dielectric resonators (DRs) are determined by the method of iterative solution of the volume integral and integro-differential equations with corresponding functionals. New forms of equations and iteration algorithms for the nonlinear input of the desired complex parameter are proposed. Frequencies and Q-factors of the H01δ and H011 modes of a cylindrical DR and the H mode of a rectangular DR for the uniform and nonuniform cases are obtained numerically. The influence of a thin semiconductor layer located at the ends of the DR and irradiated by high-power laser pulses on the frequencies and Q-factors of the DR modes is examined. It is shown that an up to ten or more percent tuning of resonant frequencies can be reached by transformation of a low conducting state to a high conducting state.
Phase integral approximation for coupled ordinary differential equations of the Schroedinger type
Skorupski, Andrzej A.
2008-05-15
Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u{sub j}{sup ''}(x)+{sigma}{sub k=1}{sup N}R{sub jk}(x)u{sub k}(x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[{identical_to}(R{sub jk}(x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u{sup ''}(x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph].
Integrability: mathematical methods for studying solitary waves theory
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2014-03-01
In recent decades, substantial experimental research efforts have been devoted to linear and nonlinear physical phenomena. In particular, studies of integrable nonlinear equations in solitary waves theory have attracted intensive interest from mathematicians, with the principal goal of fostering the development of new methods, and physicists, who are seeking solutions that represent physical phenomena and to form a bridge between mathematical results and scientific structures. The aim for both groups is to build up our current understanding and facilitate future developments, develop more creative results and create new trends in the rapidly developing field of solitary waves. The notion of the integrability of certain partial differential equations occupies an important role in current and future trends, but a unified rigorous definition of the integrability of differential equations still does not exist. For example, an integrable model in the Painlevé sense may not be integrable in the Lax sense. The Painlevé sense indicates that the solution can be represented as a Laurent series in powers of some function that vanishes on an arbitrary surface with the possibility of truncating the Laurent series at finite powers of this function. The concept of Lax pairs introduces another meaning of the notion of integrability. The Lax pair formulates the integrability of nonlinear equation as the compatibility condition of two linear equations. However, it was shown by many researchers that the necessary integrability conditions are the existence of an infinite series of generalized symmetries or conservation laws for the given equation. The existence of multiple soliton solutions often indicates the integrability of the equation but other tests, such as the Painlevé test or the Lax pair, are necessary to confirm the integrability for any equation. In the context of completely integrable equations, studies are flourishing because these equations are able to describe the
Orbit determination based on meteor observations using numerical integration of equations of motion
NASA Astrophysics Data System (ADS)
Dmitriev, Vasily; Lupovka, Valery; Gritsevich, Maria
2015-11-01
Recently, there has been a worldwide proliferation of instruments and networks dedicated to observing meteors, including airborne and future space-based monitoring systems . There has been a corresponding rapid rise in high quality data accumulating annually. In this paper, we present a method embodied in the open-source software program "Meteor Toolkit", which can effectively and accurately process these data in an automated mode and discover the pre-impact orbit and possibly the origin or parent body of a meteoroid or asteroid. The required input parameters are the topocentric pre-atmospheric velocity vector and the coordinates of the atmospheric entry point of the meteoroid, i.e. the beginning point of visual path of a meteor, in an Earth centered-Earth fixed coordinate system, the International Terrestrial Reference Frame (ITRF). Our method is based on strict coordinate transformation from the ITRF to an inertial reference frame and on numerical integration of the equations of motion for a perturbed two-body problem. Basic accelerations perturbing a meteoroid's orbit and their influence on the orbital elements are also studied and demonstrated. Our method is then compared with several published studies that utilized variations of a traditional analytical technique, the zenith attraction method, which corrects for the direction of the meteor's trajectory and its apparent velocity due to Earth's gravity. We then demonstrate the proposed technique on new observational data obtained from the Finnish Fireball Network (FFN) as well as on simulated data. In addition, we propose a method of analysis of error propagation, based on general rule of covariance transformation.
Fedorov, Yuri E-mail: Chara.Pantazi@upc.edu; Pantazi, Chara E-mail: Chara.Pantazi@upc.edu
2014-03-15
We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard–Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard–Fuchs equations can be used to study various properties of the actions.
Nonlinear inversion of the integral equation to estimate ocean wave spectra from HF radar
NASA Astrophysics Data System (ADS)
Hisaki, Yukiharu
1996-01-01
Since all ocean wave components contribute to the second-order scattering of a high-frequency radio wave by the sea surface, it is theoretically possible to estimate the ocean wave spectrum from first- and second-order scattering in the Doppler spectrum measured with an HF ocean radar. To extract the wave spectral information, however, it is necessary to solve a nonlinear integral equation. This paper describes in detail how to solve the nonlinear integral equation without linearization or approximation. We show that the problem of solving the nonlinear integral equation can be converted into a nonlinear optimization problem. An algorithm to find the optimal solution is described. Examples of the algorithm applied to simulated data and measured data are shown. The wave frequency spectrum can be estimated even if the Doppler spectrum is available in only a single direction. In this case, however, the solution of the two-dimensional wavenumber spectrum tends to converge to a spectrum that is symmetrical to the beam direction. Even if the wave spectrum is dominant in a single direction, the solution may give two peaks in the wavenumber spectrum. One of them is the true peak and the other is the mirror image of it with respect to the beam direction. This ambiguity can be avoided by using Doppler spectra measured in at least two different directions. Although there is still some room for improvement in the practical application of this method, it can be applied to estimate the wave directional spectrum up to a rather high frequency, or Bragg frequency.
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
NASA Astrophysics Data System (ADS)
Castro-Ramos, Jorge; Alejandro Juárez-Reyes, Salvador; Marcelino-Aranda, Mariana; Ortega-Vidals, Paula; Silva-Ortigoza, Gilberto; Silva-Ortigoza, Ramón; Suárez-Xique, Román
2015-01-01
In this work we assume that we have two given optical media with constant refraction indexes, which are separated by an arbitrary refracting surface. In one of the optical media we place a point light source at an arbitrary position. The aim of this work is to use a particular complete integral of the eikonal equation and Huygens’ principle to obtain the refraction and reflection laws. We remark that this complete integral associates a new point light source with each light ray that arrives at the refracting surface. This means that by using only this complete integral it is not possible to determine the direction of propagation of the refracted light rays; the direction of propagation is obtained by imposing two extra conditions on the complete integral which are equivalent to Huygens’ principle (in two dimensions, only one condition is needed). Finally, we establish the connection between the complete integral used here and that derived by using the k-function procedure introduced by Stavroudis, which works with plane wavefronts instead of spherical ones.
Chremmos, Ioannis D; Uzunoglu, Nikolaos K
2004-05-01
The excitation of a whispering gallery resonator by a surface wave guided in a dielectric slab is analyzed with a rigorous volume-integral-equation approach. The analysis is based on the Green's function concept and the application of the entire-domain Galerkin technique through expansion of the electric field in the resonator in terms of cylindrical wave functions. The algorithm developed yields highly accurate results for the transmission and reflection coefficients in the waveguide. The radiated far field is computed, and the effect of the excitation of a whispering gallery mode on the radiation pattern is studied. PMID:15139438
Geometrical integration of Landau Lifshitz Gilbert equation based on the mid-point rule
NASA Astrophysics Data System (ADS)
d'Aquino, Massimiliano; Serpico, Claudio; Miano, Giovanni
2005-11-01
Landau-Lifshitz-Gilbert (LLG) equation is the fundamental equation to describe magnetization vector field dynamics in microscale and nanoscale magnetic systems. This equation is highly nonlinear in nature and, for this reason, it is generally solved by using numerical techniques. In this paper, the mid-point rule time-stepping technique is applied to the numerical time integration of LLG equation and the relevant properties of the numerical scheme are discussed. The mid-point rule is an unconditionally stable and second order accurate scheme which preserves the fundamental geometrical properties of LLG dynamics. First, it exactly preserves the LLG property of conserving the magnetization magnitude at each spatial location. Second, for constant in time applied fields, it preserves the LLG Lyapunov structure, namely the fact that the free energy is a decreasing function of time. In addition, in the case of zero damping, the mid-point rule preserves the conservation of the system free energy. The above preservation properties are unconditionally valid, i.e. they are fulfilled for any value of the time-step. Finally, the LLG hamiltonian structure in the case of zero damping is preserved up to the third order terms with respect to the time-step. The main difficulty related to this scheme is the necessity of solving a large system of globally coupled nonlinear equations. This problem has been circumvented by using special and reasonably fast quasi-Newton iterative technique. The proposed numerical scheme is then tested on the standard micromagnetic problem no. 4. In the numerical computations, the spatial discretization is obtained by finite difference technique and the magnetostatic field is computed through the Fast Fourier Transform method.
NASA Astrophysics Data System (ADS)
Zhou, Yaoqi; Stell, George
1988-12-01
Integral equations that yield the charge and density profiles are derived for a Donnan system, in which an ionic solution is separated into two regions by a semipermeable membrane (SPM) or a spherical semipermeable vesicle (SPV). These equations are obtained from the Ornstein-Zernike (OZ) equation. We show how quantitative results can be obtained from either the mean spherical approximation (MSA) closure or the hypernetted-chain (HNC) closure for profiles. Use is made of bulk-correlation input obtained by means of the Debye-Hückel approximation, the MSA approximation, or the HNC approximation. The resulting approximations will be referred as MSA/DH, HNC/DH, MSA/MSA, etc. The system on which we focus contains three charged hard-sphere species: cation, anion, and a large ion (a protein or polymer ion) separated by a plane SPM, through which the large ion cannot pass, and to one side of which all large ions are confined, or a spherical SPV, outside of which the large ions are confined. Analytical expressions for the bulk density ratio between the two sides of a plane membrane as well as the membrane potential in various approximations are obtained. Results obtained from these expresssions are compared with the results obtained by equating electrochemical potentials. A new contact-value theorem is provided for the plane SPM system. Analytical solutions for the charge profile and the potential profile in the MSA/DH approximation are obtained. It turns out that results obtained in the HNC/DH approximation are exactly the same as those obtained by using 1D nonlinear Poisson-Boltzmann equations if the repulsive cores of the macroions are neglected.
NASA Technical Reports Server (NTRS)
Bednarcyk, Brett A.; Aboudi, Jacob; Arnold, Steven M.
2006-01-01
The radial return and Mendelson methods for integrating the equations of classical plasticity, which appear independently in the literature, are shown to be identical. Both methods are presented in detail as are the specifics of their algorithmic implementation. Results illustrate the methods' equivalence across a range of conditions and address the question of when the methods require iteration in order for the plastic state to remain on the yield surface. FORTRAN code implementations of the radial return and Mendelson methods are provided in the appendix.
Yanovitskii, E.G.
1981-01-01
The general invariance principle (GIP) for arbitrary plane inhomogeneous atmospheres is formulated on the basis of ideas contained in (V. V. Ivanov, Sov. Astron. 19, 137 (1975)). All the known invariance relations follow as particular cases from the GIP. The problem of diffuse light reflection by a semi-infinite atmosphere and the Milne problem are analyzed in detail. The existence of a number of integrals, quadratic with respect to intensity, of the transfer equation is shown, the majority of which are invariant relative to optical depth.
NASA Astrophysics Data System (ADS)
Colombo, V.; Ravetto, P.; Sumini, M.
1988-08-01
An approximate determination of the critical eigenvalue of the neutron transport equation in integral form, within both the one speed and energy multigroup models, for a homogeneous medium, is achieved by means of a variational technique. The space asymptotic solutions for both the direct and adjoint problems are used as trial functions. A variational procedure is also developed and numerically exploited within the Fourier transformed domain, where noticeable theoretical features can be demonstrated. It is evidenced that excellent results can be obtained with little computational effort, and a set of critical calculations in plane geometry is presented and discussed.
Colombo, V.; Ravetto, P.; Sumini, M.
1988-08-01
An approximate determination of the critical eigenvalue of the neutron transport equation in integral form, within both the one speed and energy multigroup models, for a homogeneous medium, is achieved by means of a variational technique. The space asymptotic solutions for both the direct and adjoint problems are used as trial functions. A variational procedure is also developed and numerically exploited within the Fourier transformed domain, where noticeable theoretical features can be demonstrated. It is evidenced that excellent results can be obtained with little computational effort, and a set of critical calculations in plane geometry is presented and discussed. copyright 1988 Academic Press, Inc.
NASA Astrophysics Data System (ADS)
Artoun, Ojenie; David-Rus, Diana; Emmett, Matthew; Fishman, Lou; Fital, Sandra; Hogan, Chad; Lim, Jisun; Lushi, Enkeleida; Marinov, Vesselin
2006-05-01
In this report we summarize an extension of Fourier analysis for the solution of the wave equation with a non-constant coefficient corresponding to an inhomogeneous medium. The underlying physics of the problem is exploited to link pseudodifferential operators and phase space path integrals to obtain a marching algorithm that incorporates the backward scattering into the evolution of the wave. This allows us to successfully apply single-sweep, one-way marching methods in inherently two-way environments, which was not achieved before through other methods for this problem.
NASA Technical Reports Server (NTRS)
Madsen, Niel K.
1992-01-01
Several new discrete surface integral (DSI) methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general nonorthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve divergence or charge. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) these methods allow more accurate modeling of non-rectangular structures and objects because the traditional stair-stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.
NASA Technical Reports Server (NTRS)
Yates, E. Carson, Jr.
1990-01-01
Progress in the development of computational methods for steady and unsteady aerodynamics has perennially paced advancements in aeroelastic analysis and design capabilities. Since these capabilities are of growing importance in the analysis and design of high-performance aircraft, considerable effort has been directed toward the development of appropriate aerodynamic methodology. The contributions to those efforts from the integral-equations research program at the NASA Langley Research Center is reviewed. Specifically, the current scope, progress, and plans for research and development for inviscid and viscous flows are discussed, and example applications are shown in order to highlight the generality, versatility, and attractive features of this methodology.
Integrating matrix solution of the hybrid state vector equations for beam vibration
NASA Technical Reports Server (NTRS)
Lehman, L. L.
1982-01-01
A simple, versatile, and efficient computational technique has been developed for dynamic analysis of linear elastic beam and rod type of structures. Moreover, the method provides a rather general solution approach for two-point boundary value problems that are described by a single independent spatial variable. For structural problems, the method is implemented by a mixed state vector formulation of the differential equations, combined with an integrating matrix solution procedure. Highly accurate solutions are easily achieved with this approach. Example solutions are given for beam vibration problems including discontinuous stiffness and mass parameters, elastic restraint boundary conditions, concentrated inertia loading, and rigid body modes
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.
NASA Technical Reports Server (NTRS)
Fymat, A. L.
1975-01-01
The determination of the microstructure, chemical nature, and dynamical evolution of scattering particulates in the atmosphere is considered. A description is given of indirect sampling techniques which can circumvent most of the difficulties associated with direct sampling techniques, taking into account methods based on scattering, extinction, and diffraction of an incident light beam. Approaches for reconstructing the particulate size distribution from the direct and the scattered radiation are discussed. A new method is proposed for determining the chemical composition of the particulates and attention is given to the relevance of methods of solution involving first kind Fredholm integral equations.
NASA Technical Reports Server (NTRS)
Sharafeddin, Omar A.; Judson, Richard S.; Kouri, Donald J.; Hoffman, David K.
1990-01-01
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
Integrated nonthermal treatment system study
Biagi, C.; Bahar, D.; Teheranian, B.; Vetromile, J.; Quapp, W.J.; Bechtold, T.; Brown, B.; Schwinkendorf, W.; Swartz, G.
1997-01-01
This report presents the results of a study of nonthermal treatment technologies. The study consisted of a systematic assessment of five nonthermal treatment alternatives. The treatment alternatives consist of widely varying technologies for safely destroying the hazardous organic components, reducing the volume, and preparing for final disposal of the contact-handled mixed low-level waste (MLLW) currently stored in the US Department of Energy complex. The alternatives considered were innovative nonthermal treatments for organic liquids and sludges, process residue, soil and debris. Vacuum desorption or various washing approaches are considered for treatment of soil, residue and debris. Organic destruction methods include mediated electrochemical oxidation, catalytic wet oxidation, and acid digestion. Other methods studied included stabilization technologies and mercury separation of treatment residues. This study is a companion to the integrated thermal treatment study which examined 19 alternatives for thermal treatment of MLLW waste. The quantities and physical and chemical compositions of the input waste are based on the inventory database developed by the US Department of Energy. The Integrated Nonthermal Treatment Systems (INTS) systems were evaluated using the same waste input (2,927 pounds per hour) as the Integrated Thermal Treatment Systems (ITTS). 48 refs., 68 figs., 37 tabs.
Kamon, M.; Phillips, J.R.
1994-12-31
In this paper techniques are presented for preconditioning equations generated by discretizing constrained vector integral equations associated with magnetoquasistatic analysis. Standard preconditioning approaches often fail on these problems. The authors present a specialized preconditioning technique and prove convergence bounds independent of the constraint equations and electromagnetic excitation frequency. Computational results from analyzing several electronic packaging examples are given to demonstrate that the new preconditioning approach can sometimes reduce the number of GMRES iterations by more than an order of magnitude.
Yangian symmetries and integrability of the Dirac equation with spin symmetry
Xu, Lei; Jing, Jian; Yuan, Zi-Gang; Kong, Ling-Bao; Long, Zheng-Wen
2013-02-15
We show that a Yangian symmetry, namely, Y(su(2)), exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2)) explicitly and get the energy spectrum of this model from the representation theory for Y(su(2)). We also show that this model is integrable, from RTT relations. - Highlights: Black-Right-Pointing-Pointer The Y(sl(2)) symmetry is found in a model, and the generators of Y(sl(2)) are constructed from the so(4) Lie algebra. Black-Right-Pointing-Pointer The energy spectrum is derived on the basis of the representation theory for Y(sl(2)). Black-Right-Pointing-Pointer The integrability of this model is proved from the RTT relation.
On the Regularity Set and Angular Integrability for the Navier-Stokes Equation
NASA Astrophysics Data System (ADS)
D'Ancona, Piero; Lucà, Renato
2016-09-01
We investigate the size of the regular set for suitable weak solutions of the Navier-Stokes equation, in the sense of Caffarelli-Kohn-Nirenberg (Commun Pure Appl Math 35:771-831, 1982). We consider initial data in weighted Lebesgue spaces with mixed radial-angular integrability, and we prove that the regular set increases if the data have higher angular integrability, invading the whole half space {\\{t > 0\\}} in an appropriate limit. In particular, we obtain that if the {L2} norm with weight {|x|^{-frac12}} of the data tends to 0, the regular set invades {\\{t > 0\\}}; this result improves Theorem D of Caffarelli et al. (Commun Pure Appl Math 35:771-831, 1982).
NASA Astrophysics Data System (ADS)
Bakaleinikov, L. A.; Flegontova, E. Yu.; Ender, A. Ya.; Ender, I. A.
2016-04-01
A recurrence procedure for a sequential construction of kernels G_{{l_1},{l_2}}^l ( c, c 1, c 2) appearing upon the expansion of a nonlinear collision integral of the Boltzmann equation in spherical harmonics is developed. The starting kernel for this procedure is kernel G 0,0 0 ( c, c 1, c 2) of the collision integral for the distribution function isotropic with respect to the velocities. Using the recurrence procedure, a set of kernels G_{{l_1},{l_2}}^{ + l} ( c, c 1, c 2) for a gas consisting of hard spheres and Maxwellian molecules is constructed. It is shown that the resultant kernels exhibit similarity and symmetry properties and satisfy the relations following from the conservation laws.
Energy Science and Technology Software Center (ESTSC)
2014-06-01
ARKode is part of a software family called SUNDIALS: SUite of Nonlinear and Differential/ALgebraic equation Solvers [1]. The ARKode solver library provides an adaptive-step time integration package for stiff, nonstiff and multi-rate systems of ordinary differential equations (ODEs) using Runge Kutta methods [2].
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Taflove, Allen
1992-01-01
The initial results for femtosecond electromagnetic soliton propagation and collision obtained from first principles, i.e., by a direct time integration of Maxwell's equations are reported. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit the modeling of 2D and 3D optical soliton propagation, scattering, and switching from the full-vector Maxwell's equations.
NASA Astrophysics Data System (ADS)
Macomber, B.; Woollands, R. M.; Probe, A.; Younes, A.; Bai, X.; Junkins, J.
2013-09-01
Modified Chebyshev Picard Iteration (MCPI) is an iterative numerical method for approximating solutions of linear or non-linear Ordinary Differential Equations (ODEs) to obtain time histories of system state trajectories. Unlike other step-by-step differential equation solvers, the Runge-Kutta family of numerical integrators for example, MCPI approximates long arcs of the state trajectory with an iterative path approximation approach, and is ideally suited to parallel computation. Orthogonal Chebyshev Polynomials are used as basis functions during each path iteration; the integrations of the Picard iteration are then done analytically. Due to the orthogonality of the Chebyshev basis functions, the least square approximations are computed without matrix inversion; the coefficients are computed robustly from discrete inner products. As a consequence of discrete sampling and weighting adopted for the inner product definition, Runge phenomena errors are minimized near the ends of the approximation intervals. The MCPI algorithm utilizes a vector-matrix framework for computational efficiency. Additionally, all Chebyshev coefficients and integrand function evaluations are independent, meaning they can be simultaneously computed in parallel for further decreased computational cost. Over an order of magnitude speedup from traditional methods is achieved in serial processing, and an additional order of magnitude is achievable in parallel architectures. This paper presents a new MCPI library, a modular toolset designed to allow MCPI to be easily applied to a wide variety of ODE systems. Library users will not have to concern themselves with the underlying mathematics behind the MCPI method. Inputs are the boundary conditions of the dynamical system, the integrand function governing system behavior, and the desired time interval of integration, and the output is a time history of the system states over the interval of interest. Examples from the field of astrodynamics are
Higher-order time integration of Coulomb collisions in a plasma using Langevin equations
Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; Rosin, M. S.; Ricketson, L. F.
2013-02-08
The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt^{1/2})] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.
Solving systems of phaseless equations via Kaczmarz methods: a proof of concept study
NASA Astrophysics Data System (ADS)
Wei, Ke
2015-12-01
We study the Kaczmarz methods for solving systems of phaseless equations, i.e., the generalized phase retrieval problem. The methods extend the Kaczmarz methods for solving systems of linear equations by integrating a phase selection heuristic in each iteration and overall have the same per iteration computational complexity. Extensive empirical performance comparisons establish the computational advantages of the Kaczmarz methods over other state-of-the-art phase retrieval algorithms both in terms of the number of measurements needed for successful recovery and in terms of computation time. Preliminary convergence analysis is presented for the randomized Kaczmarz methods.
Uysal, Ismail E; Arda Ülkü, H; Bağci, Hakan
2016-09-01
Transient electromagnetic interactions on plasmonic nanostructures are analyzed by solving the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) surface integral equation (SIE). Equivalent (unknown) electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, are expanded using Rao-Wilton-Glisson and polynomial basis functions in space and time, respectively. Inserting this expansion into the PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations that is solved for the current expansion coefficients by a marching on-in-time (MOT) scheme. The resulting MOT-PMCHWT-SIE solver calls for computation of additional convolutions between the temporal basis function and the plasmonic medium's permittivity and Green function. This computation is carried out with almost no additional cost and without changing the computational complexity of the solver. Time-domain samples of the permittivity and the Green function required by these convolutions are obtained from their frequency-domain samples using a fast relaxed vector fitting algorithm. Numerical results demonstrate the accuracy and applicability of the proposed MOT-PMCHWT solver. PMID:27607496
Integrating the quantum Hamilton-Jacobi equations by wavefront expansion and phase space analysis
NASA Astrophysics Data System (ADS)
Bittner, Eric R.; Wyatt, Robert E.
2000-11-01
In this paper we report upon our computational methodology for numerically integrating the quantum Hamilton-Jacobi equations using hydrodynamic trajectories. Our method builds upon the moving least squares method developed by Lopreore and Wyatt [Phys. Rev. Lett. 82, 5190 (1999)] in which Lagrangian fluid elements representing probability volume elements of the wave function evolve under Newtonian equations of motion which include a nonlocal quantum force. This quantum force, which depends upon the third derivative of the quantum density, ρ, can vary rapidly in x and become singular in the presence of nodal points. Here, we present a new approach for performing quantum trajectory calculations which does not involve calculating the quantum force directly, but uses the wavefront to calculate the velocity field using mv=∇S, where S/ℏ is the argument of the wave function ψ. Additional numerical stability is gained by performing local gauge transformations to remove oscillatory components of the wave function. Finally, we use a dynamical Rayleigh-Ritz approach to derive ancillary equations-of-motion for the spatial derivatives of ρ, S, and v. The methodologies described herein dramatically improve the long time stability and accuracy of the quantum trajectory approach even in the presence of nodes. The method is applied to both barrier crossing and tunneling systems. We also compare our results to semiclassical based descriptions of barrier tunneling.
Method to solve integral equations of the first kind with an approximate input.
Efros, Victor D
2012-07-01
Techniques are proposed for solving integral equations of the first kind with an input known not precisely. The requirement that the solution sought for includes a given number of maxima and minima is imposed. It is shown that when the deviation of the approximate input from the true one is sufficiently small and some additional conditions are fulfilled the method leads to an approximate solution that is necessarily close to the true solution. No regularization is required in the present approach. Requirements on features of the solution at integration limits are also imposed. The problem is treated with the help of an ansatz proposed for the derivative of the solution. The ansatz is the most general one compatible with the above mentioned requirements. The techniques are tested with exactly solvable examples. Inversions of the Lorentz, Stieltjes, and Laplace integral transforms are performed, and very satisfactory results are obtained. The method is useful, in particular, for the calculation of quantum-mechanical reaction amplitudes and inclusive spectra of perturbation-induced reactions in the framework of the integral transform approach. PMID:23005560
Time integration algorithms for the two-dimensional Euler equations on unstructured meshes
NASA Technical Reports Server (NTRS)
Slack, David C.; Whitaker, D. L.; Walters, Robert W.
1994-01-01
Explicit and implicit time integration algorithms for the two-dimensional Euler equations on unstructured grids are presented. Both cell-centered and cell-vertex finite volume upwind schemes utilizing Roe's approximate Riemann solver are developed. For the cell-vertex scheme, a four-stage Runge-Kutta time integration, a fourstage Runge-Kutta time integration with implicit residual averaging, a point Jacobi method, a symmetric point Gauss-Seidel method and two methods utilizing preconditioned sparse matrix solvers are presented. For the cell-centered scheme, a Runge-Kutta scheme, an implicit tridiagonal relaxation scheme modeled after line Gauss-Seidel, a fully implicit lower-upper (LU) decomposition, and a hybrid scheme utilizing both Runge-Kutta and LU methods are presented. A reverse Cuthill-McKee renumbering scheme is employed for the direct solver to decrease CPU time by reducing the fill of the Jacobian matrix. A comparison of the various time integration schemes is made for both first-order and higher order accurate solutions using several mesh sizes, higher order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The results obtained for a transonic flow over a circular arc suggest that the preconditioned sparse matrix solvers perform better than the other methods as the number of elements in the mesh increases.
NASA Astrophysics Data System (ADS)
Li, Jie; Dault, Daniel; Liu, Beibei; Tong, Yiying; Shanker, Balasubramaniam
2016-08-01
The analysis of electromagnetic scattering has long been performed on a discrete representation of the geometry. This representation is typically continuous but not differentiable. The need to define physical quantities on this geometric representation has led to development of sets of basis functions that need to satisfy constraints at the boundaries of the elements/tessellations (viz., continuity of normal or tangential components across element boundaries). For electromagnetics, these result in either curl/div-conforming basis sets. The geometric representation used for analysis is in stark contrast with that used for design, wherein the surface representation is higher order differentiable. Using this representation for both geometry and physics on geometry has several advantages, and is elucidated in Hughes et al. (2005) [7]. Until now, a bulk of the literature on isogeometric methods have been limited to solid mechanics, with some effort to create NURBS based basis functions for electromagnetic analysis. In this paper, we present the first complete isogeometry solution methodology for the electric field integral equation as applied to simply connected structures. This paper systematically proceeds through surface representation using subdivision, definition of vector basis functions on this surface, to fidelity in the solution of integral equations. We also present techniques to stabilize the solution at low frequencies, and impose a Calderón preconditioner. Several results presented serve to validate the proposed approach as well as demonstrate some of its capabilities.
Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations
Wannamaker, P.E.; Hohmann, G.W.
1982-01-01
An algorithm based on the method of integral equations has been developed to simulate the electromagnetic response of 3-D bodies in layered earths. The inhomogeneities are replaced mathematically by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric dyadic Green's function appropriate to a layered earth and is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic dyadic Green's functions over the scattering currents. Efficient evaluation of the dyadic Green's functions is a major consideration in reducing computation time. It is found that tabulation/interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green's functions is preferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate 3-D bodies with responses over 2-D bodies of identical cross section using plane wave incident fields is the only check available on our solution. Agreement is excellent; however, the length that a 3-D body must have before departures between 2-D transverse electric and corresponding 3-D signatures are insignificant depends strongly on the layering. The 2-D transverse magnetic and corresponding 3-D calculations agree closely regardless of the layered host.
NASA Astrophysics Data System (ADS)
Vaneeva, Olena; Sophocleous, Christodoulos; Popovych, Roman; Boyko, Vyacheslav; Damianou, Pantelis
2015-06-01
The Seventh International Workshop "Group Analysis of Differential Equations and Integrable Systems" (GADEIS-VII) took place at Flamingo Beach Hotel, Larnaca, Cyprus during the period June 15-19, 2014. Fifty nine scientists from nineteen countries participated in the Workshop, and forty one lectures were presented. The Workshop topics ranged from theoretical developments of group analysis of differential equations, hypersymplectic structures, theory of Lie algebras, integrability and superintegrability to their applications in various fields. The Series of Workshops is a joint initiative by the Department of Mathematics and Statistics, University of Cyprus, and the Department of Applied Research of the Institute of Mathematics, National Academy of Sciences, Ukraine. The Workshops evolved from close collaboration among Cypriot and Ukrainian scientists. The first three meetings were held at the Athalassa campus of the University of Cyprus (October 27, 2005, September 25-28, 2006, and October 4-5, 2007). The fourth (October 26-30, 2008), the fifth (June 6-10, 2010) and the sixth (June 17-21, 2012) meetings were held at the coastal resort of Protaras. We would like to thank all the authors who have published papers in the Proceedings. All of the papers have been reviewed by at least two independent referees. We express our appreciation of the care taken by the referees. Their constructive suggestions have improved most of the papers. The importance of peer review in the maintenance of high standards of scientific research can never be overstated. Olena Vaneeva, Christodoulos Sophocleous, Roman Popovych, Vyacheslav Boyko, Pantelis Damianou
Three-body-continuum Coulomb problem using a compact-kernel-integral-equation approach
NASA Astrophysics Data System (ADS)
Silenou Mengoue, M.
2013-02-01
We present an approach associated with the Jacobi matrix method to calculate a three-body wave function that describes the double continuum of an atomic two-electron system. In this approach, a symmetrized product of two Coulomb waves is used to describe the asymptotic wave function, while a smooth cutoff function is introduced to the dielectronic potential that enters its integral part in order to have a compact kernel of the corresponding Lippmann-Schwinger-type equation to be solved. As an application, the integral equation for the (e-,e-,He2+) system is solved numerically; the fully fivefold differential cross sections (FDCSs) for (e,3e) processes in helium are presented within the first-order Born approximation. The calculation is performed for a coplanar geometry in which the incident electron is fast (˜6 keV) and for a symmetric energy sharing between both slow ejected electrons at excess energy of 20 eV. The experimental and theoretical FDCSs agree satisfactorily both in shape and in magnitude. Full convergence in terms of the basis size is reached and presented.
NASA Astrophysics Data System (ADS)
Chang, Ruinan
The ability to have a good understanding of and to manipulate electromagnetic elds has been increasingly important for many hardware technologies. There is a strong need for advanced numeric algorithms that yield fast and accuracy controllable solvers for electromagnetic and micromagnetic simulations. The first part of the dissertation presents methods constituting the core of the high-performance simulator FastMag. FastMag derives its high speed from three aspects. First, it leverages the state-of-the-art graphics processing unit computational architectures, which can be hundreds of times faster than a single central processing unit. Moreover, ecient and and accurate implementations of numeric quadrature was invoked. Thirdly, we provide an analytic method for Jacobian vector products. Some advanced features are provided in FastMag. Quadratic basis functions are used to provide better accuracy. Hexahedral elements were also implemented because they are more accurate, consume less memory. The second part of the dissertation is devoted to electromagnetic scattering problems. We developed new algorithms that signicantly improved the traditional methods. First of all, potential volume integral equations were implemented, where the potential quantities (vector and scalar potential). Another important contribution of this disertation is quadrilateral barycentric basis functions (QBBFs). The QBBFs can serve as a fundamental block for primary basis functions (PBFs) and dual basis functions (DBFs). The PBFs and DBFs, when applied in combination into traditional electric and magnetic eld integral equations (EFIE and MFIE), give rise to accurate and robust results. Moreover, the DBFs make the famous Calderon preconditioner multiplicative.
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.
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.
CALL FOR PAPERS: Special Issue on `Geometric Numerical Integration of Differential Equations'
NASA Astrophysics Data System (ADS)
Quispel, G. R. W.; McLachlan, R. I.
2005-02-01
This is a call for contributions to a special issue of Journal of Physics A: Mathematical and General entitled `Geometric Numerical Integration of Differential Equations'. This issue should be a repository for high quality original work. We are interested in having the topic interpreted broadly, that is, to include contributions dealing with symplectic or multisymplectic integration; volume-preserving integration; symmetry-preserving integration; integrators that preserve first integrals, Lyapunov functions, or dissipation; exponential integrators; integrators for highly oscillatory systems; Lie-group integrators, etc. Papers on geometric integration of both ODEs and PDEs will be considered, as well as application to molecular-scale integration, celestial mechanics, particle accelerators, fluid flows, population models, epidemiological models and/or any other areas of science. We believe that this issue is timely, and hope that it will stimulate further development of this new and exciting field. The Editorial Board has invited G R W Quispel and R I McLachlan to serve as Guest Editors for the special issue. Their criteria for acceptance of contributions are the following: • The subject of the paper should relate to geometric numerical integration in the sense described above. • Contributions will be refereed and processed according to the usual procedure of the journal. • Papers should be original; reviews of a work published elsewhere will not be accepted. The guidelines for the preparation of contributions are as follows: • The DEADLINE for submission of contributions is 1 September 2005. This deadline will allow the special issue to appear in late 2005 or early 2006. • There is a strict page limit of 16 printed pages (approximately 9600 words) per contribution. For papers exceeding this limit, the Guest Editors reserve the right to request a reduction in length. Further advice on publishing your work in Journal of Physics A: Mathematical and General
ERIC Educational Resources Information Center
Santos-George, Arlene A.
2012-01-01
This dissertation empirically tested Tinto's student integration theory through structural equation modeling using a national sample of 2,847 first-time entering community college students. Tinto theorized that the more academically and socially integrated a student is to the college environment, the more likely the student will persist…
NASA Astrophysics Data System (ADS)
Singh, Ram Chandra; Ram, Jokhan
2006-09-01
A closure for the pair-correlation functions of molecular fluids is described in which the hypernetted-chain and the Percus-Yevick approximations are “mixed” as a function of interparticle separation. An adjustable parameter α in the mixing function is used to enforce thermodynamic consistency, by which it is meant that identical results are obtained when the equations of state are calculated via the virial and compressibility routes, respectively. The mixed integral equation for the pair-correlation functions has been solved for two model fluids: (i) a fluid of the hard Gaussian overlap model, and (ii) a fluid the molecules of which interact via a modified Gay-Berne model potential. For the modified Gay-Berne fluid we have slightly modified the original Gay-Berne potential to study the effect of attraction on hard core systems. The pair-correlation functions of the isotropic phase which enter in the density-functional theory as input informations have been calculated from the integral equation theories for these model fluids. We have used two different versions of the density-functional theory known as the second order and modified weighted-density-functional theory to locate the isotropic-nematic (I-N) transitions and calculate the values of transition parameters for the hard Gaussian overlap and modified Gay-Berne model fluids. We have compared our results with those of computer simulations wherever they are available. We find that the density-functional theory is good to study the I-N transition in molecular fluids if the values of the pair-correlation functions in the isotropic phase are accurately known.
Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps
Yuan Jianhua; Lu Yayan Antoine, Xavier
2008-04-20
Efficient numerical methods for analyzing photonic crystals (PhCs) can be developed using the Dirichlet-to-Neumann (DtN) maps of the unit cells. The DtN map is an operator that takes the wave field on the boundary of a unit cell to its normal derivative. In frequency domain calculations for band structures and transmission spectra of finite PhCs, the DtN maps allow us to reduce the computation to the boundaries of the unit cells. For two-dimensional (2D) PhCs with unit cells containing circular cylinders, the DtN maps can be constructed from analytic solutions (the cylindrical waves). In this paper, we develop a boundary integral equation method for computing DtN maps of general unit cells containing cylinders with arbitrary cross sections. The DtN map method is used to analyze band structures for 2D PhCs with elliptic and other cylinders.
Solution of the WFNDEC 2015 eddy current benchmark with surface integral equation method
NASA Astrophysics Data System (ADS)
Demaldent, Edouard; Miorelli, Roberto; Reboud, Christophe; Theodoulidis, Theodoros
2016-02-01
In this paper, a numerical solution of WFNDEC 2015 eddy current benchmark is presented. In particular, the Surface Integral Equation (SIE) method has been employed for numerically solving the benchmark problem. The SIE method represent an effective and efficient alternative to standard numerical solver like Finite Element Method (FEM) when electromagnetic fields need to be calculated in problems involving homogeneous media. The formulation of SIE method allows to properly solve the electromagnetic problem by meshing the surface of the media instead to the complete media volume as done in FEM. The surface meshing enables to describe the problem with a smaller number of unknowns with respect to FEM. This property is directly translated in an obvious gain in terms of CPU time efficiency.
NASA Astrophysics Data System (ADS)
Abas, Z. Abal; Salleh, S.; Basari, A. S. Hassan; Ibrahim, Nuzulha Khilwani
2010-11-01
A conceptual model of integrating the sensor network and the radiative heat transfer equation is developed and presented in this paper. The idea is to present possible deployment of sensor networks in the Ethylene furnace so that valuable input in the form of boundary value can be generated in order to produce intensity distribution and heat flux distribution. Once the location of sensor deployment has been recommended, the mesh at the physical space between the furnace wall and the reactor tube is constructed. The paper concentrates only at 2D model with only 1 U-bend reactor tube in the ethylene furnace as an initial phase of constructing a complete simulation in real furnace design.
On the derivation of variational integrators for the rotating shallow-water equations
NASA Astrophysics Data System (ADS)
Bauer, Werner; Gay-Balmaz, François
2016-04-01
We present a structure-preserving discretization of the rotating shallow-water equations. This novel numerical scheme is based on a finite dimensional approximation of the group of diffeomorphisms and is derived via a discrete version of the Euler-Poincaré variational formulation of rotating compressible fluids. The resulting variational integrator, currently derived for regular triangular meshes, provides the first successful derivation and implementation of a compressible two-dimensional model by this discrete variational principle. We illustrate on various test cases that this variationally derived scheme exhibits excellent long term energy behavior, shows a second order convergence rate in space, and respects conservation laws such as geostrophic balance and mass conservation.
The lambda-scheme. [for numerical integration of Euler equation of compressible gas flow
NASA Technical Reports Server (NTRS)
Moretti, G.
1979-01-01
A method for integrating the Euler equations of gas dynamics for compressible flows in any hyperbolic case is presented. This method is applied to the Mach number distribution over a stretch of an infinite duct having a variable cross section, and to the distribution in a channel opening into a vacuum with the Mach number equalling 1.04. An example of the ability of this method to handle two-dimensional unsteady flows is shown using the steady shock-and-isobars pattern reached asymptotically about an ablated blunt body with a free stream Mach number equalling 12. A final example is presented where the technique is applied to a three-dimensional steady supersonic flow, with a Mach number of 2 and an angle of attack of 5 deg.
NASA Technical Reports Server (NTRS)
Womble, M. E.; Potter, J. E.
1975-01-01
A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.
Efficient solution of liquid state integral equations using the Newton-GMRES algorithm
NASA Astrophysics Data System (ADS)
Booth, Michael J.; Schlijper, A. G.; Scales, L. E.; Haymet, A. D. J.
1999-06-01
We present examples of the accurate, robust and efficient solution of Ornstein-Zernike type integral equations which describe the structure of both homogeneous and inhomogeneous fluids. In this work we use the Newton-GMRES algorithm as implemented in the public-domain nonlinear Krylov solvers NKSOL [ P. Brown, Y. Saad, SIAM J. Sci. Stat. Comput. 11 (1990) 450] and NITSOL [ M. Pernice, H.F. Walker, SIAM J. Sci. Comput. 19 (1998) 302]. We compare and contrast this method with more traditional approaches in the literature, using Picard iteration (successive-substitution) and hybrid Newton-Raphson and Picard methods, and a recent vector extrapolation method [ H.H.H. Homeier, S. Rast, H. Krienke, Comput. Phys. Commun. 92 (1995) 188]. We find that both the performance and ease of implementation of these nonlinear solvers recommend them for the solution of this class of problem.
Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Wright, O. C.
2016-05-01
An effective integration method based on the classical solution of the Jacobi inversion problem, using Kleinian ultra-elliptic functions and Riemann theta functions, is presented for the quasi-periodic two-phase solutions of the focusing cubic nonlinear Schrödinger equation. Each two-phase solution with real quasi-periods forms a two-real-dimensional torus, modulo a circle of complex-phase factors, expressed as a ratio of theta functions associated with the Riemann surface of the invariant spectral curve. The initial conditions of the Dirichlet eigenvalues satisfy reality conditions which are explicitly parametrized by two physically-meaningful real variables: the squared modulus and a scalar multiple of the wavenumber. Simple new formulas for the maximum modulus and the minimum modulus are obtained in terms of the imaginary parts of the branch points of the Riemann surface.
Boundary integral equation method calculations of surface regression effects in flame spreading
NASA Technical Reports Server (NTRS)
Altenkirch, R. A.; Rezayat, M.; Eichhorn, R.; Rizzo, F. J.
1982-01-01
A solid-phase conduction problem that is a modified version of one that has been treated previously in the literature and is applicable to flame spreading over a pyrolyzing fuel is solved using a boundary integral equation (BIE) method. Results are compared to surface temperature measurements that can be found in the literature. In addition, the heat conducted through the solid forward of the flame, the heat transfer responsible for sustaining the flame, is also computed in terms of the Peclet number based on a heated layer depth using the BIE method and approximate methods based on asymptotic expansions. Agreement between computed and experimental results is quite good as is agreement between the BIE and the approximate results.
Shafii, Mohammad Ali Meidianti, Rahma Wildian, Fitriyani, Dian; Tongkukut, Seni H. J.; Arkundato, Artoto
2014-09-30
Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation.
Numerical study of fractional nonlinear Schrödinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
Anthropometric equations for studying body fat in pregnant women.
Paxton, A; Lederman, S A; Heymsfield, S B; Wang, J; Thornton, J C; Pierson, R N
1998-01-01
Anthropometric data from 200 pregnant women were used to estimate body fat at gestation weeks 14 and 37 and changes in body fat from week 14 to week 37 with four formulas from the literature. The resulting estimates were evaluated against the estimation of fat by a four-compartment model that determined fat from weight, total body water, bone mineral mass, and body density. The estimates of fat by existing anthropometric models were statistically different from those by the four-compartment model in both early and late pregnancy. Most importantly, the change in body fat estimated by the anthropometric models (all > 4 kg) was considerably higher than that estimated by the four-compartment model (3.3 kg). Two new anthropometric equations were developed, both of which used the four-compartment model as the reference method. The equation for predicting change in fat mass from week 14 to 37 of pregnancy was as follows: 0.77 (change in weight, kg)+ 0.07 (change in thigh skinfold thickness, mm)-6.13 (r2 = 0.73). The equation for determining fat (kg) at term was as follows: 0.40 (weight at week 37, kg)+ 0.16 (biceps skinfold thickness at week 37, mm) + 0.15 (thigh skinfold thickness at week 37, mm)-0.09 (wrist circumference at week 37. mm)+ 0.10 (prepregnancy weight.kg)-6.56 (r2 = 0.89). Both equations were derived on a randomly selected half of the total sample and validated on the remaining half. Both equations were found to be valid for use in studying pregnant women with different prepregnancy body mass indexes, different gestational weight gains, different ethnicities, and different socioeconomic status. PMID:9440383
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.
NASA Technical Reports Server (NTRS)
Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.
Solutions to Kuessner's integral equation in unsteady flow using local basis functions
NASA Technical Reports Server (NTRS)
Fromme, J. A.; Halstead, D. W.
1975-01-01
The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.
A CNN-based approach to integrate the 3-D turbolent diffusion equation
NASA Astrophysics Data System (ADS)
Nunnari, G.
2003-04-01
The paper deals with the integration of the 3-D turbulent diffusion equation. This problem is relevant in several application fields including fluid dynamics, air/water pollution, volcanic ash emissions and industrial hazard assessment. As it is well known numerical solution of such a kind of equation is very time consuming even by using modern digital computers and this represents a short-coming for on-line applications. To overcome this drawback a Cellular Neural Network Approach is proposed in this paper. CNN's proposed by Chua and Yang in 1988 are massive parallel analog non-linear circuits with local interconnections between the computing elements that allow very fast distributed computations. Nowadays several producers of semiconductors such as SGS-Thomson are producing on chip CNN's so that their massive use for heavy computing applications is expected in the near future. In the paper the methodological background of the proposed approach will be outlined. Further some results both in terms of accuracy and computation time will be presented also in comparison with traditional three-dimensional computation schemes. Some results obtained to model 3-D pollution problems in the industrial area of Siracusa (Italy), characterised by a large concentration of petrol-chemical plants, will be presented.
A multigrid integral equation method for large-scale models with inhomogeneous backgrounds
NASA Astrophysics Data System (ADS)
Endo, Masashi; Čuma, Martin; Zhdanov, Michael S.
2008-12-01
We present a multigrid integral equation (IE) method for three-dimensional (3D) electromagnetic (EM) field computations in large-scale models with inhomogeneous background conductivity (IBC). This method combines the advantages of the iterative IBC IE method and the multigrid quasi-linear (MGQL) approximation. The new EM modelling method solves the corresponding systems of linear equations within the domains of anomalous conductivity, Da, and inhomogeneous background conductivity, Db, separately on coarse grids. The observed EM fields in the receivers are computed using grids with fine discretization. The developed MGQL IBC IE method can also be applied iteratively by taking into account the return effect of the anomalous field inside the domain of the background inhomogeneity Db, and vice versa. The iterative process described above is continued until we reach the required accuracy of the EM field calculations in both domains, Da and Db. The method was tested for modelling the marine controlled-source electromagnetic field for complex geoelectrical structures with hydrocarbon petroleum reservoirs and a rough sea-bottom bathymetry.
A hybrid boundary-integral/thin-sheet equation for subduction modeling
NASA Astrophysics Data System (ADS)
Xu, Bingrui; Ribe, Neil M.
2016-06-01
Subducting oceanic lithosphere is an example of a thin sheet-like object whose characteristic lateral dimension greatly exceeds its thickness. Here we exploit this property to derive a new hybrid boundary-integral/thin sheet (BITS) representation of subduction that combines in a single equation all the forces acting on the sheet: gravity, internal resistance to bending and stretching, and the tractions exerted by the ambient mantle. For simplicity, we limit ourselves to two dimensions. We solve the BITS equations using a discrete Lagrangian approach in which the sheet is represented by a set of vertices connected by edges. Instantaneous solutions for the sinking speed of a slab attached to a trailing flat sheet obey a scaling law of the form V/VStokes = fct(St), where VStokes is a characteristic Stokes sinking speed and St is the sheet's flexural stiffness. Time-dependent solutions for the evolution of the sheet's shape and thickness show that these are controlled by the viscosity ratio between the sheet and its surroundings. An important advantage of the BITS approach is the possibility of generalizing the sheet's rheology, either to a viscosity that varies along the sheet or to a non-Newtonian shear-thinning rheology.
Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation
Bokhari, Ashfaque H.; Zaman, F. D.; Mahomed, F. M.
2010-05-15
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.
Efficient solution of time-domain boundary integral equations arising in sound-hard scattering
NASA Astrophysics Data System (ADS)
Veit, Alexander; Merta, Michal; Zapletal, Jan; Lukáš, Dalibor
2016-08-01
We consider the efficient numerical solution of the three-dimensional wave equation with Neumann boundary conditions via time-domain boundary integral equations. A space-time Galerkin method with $C^\\infty$-smooth, compactly supported basis functions in time and piecewise polynomial basis functions in space is employed. We discuss the structure of the system matrix and its efficient parallel assembly. Different preconditioning strategies for the solution of the arising systems with block Hessenberg matrices are proposed and investigated numerically. Furthermore, a C++ implementation parallelized by OpenMP and MPI in shared and distributed memory, respectively, is presented. The code is part of the boundary element library BEM4I. Results of numerical experiments including convergence and scalability tests up to a thousand cores on a cluster are provided. The presented implementation shows good parallel scalability of the system matrix assembly. Moreover, the proposed algebraic preconditioner in combination with the FGMRES solver leads to a significant reduction of the computational time.
Wavelets in the solution of the volume integral equation: Application to eddy current modeling
Wang, B.; Moulder, J.C.; Basart, J.P.
1997-05-01
There is growing interest in the applications of wavelets as basis functions in solutions of integral equations, especially in the area of electromagnetic field problems. In this article we apply a wavelet expansion to the solution of the three-dimensional eddy current modeling problem based on the volume integral method. Although this method shows promise for eddy current modeling of three-dimensional flaws, it is restricted by the computing power required to solve a large linear system. In this article we show that applying a wavelet basis to the volume integral method can dramatically reduce the size of the linear system to be solved. In our approach, the unknown total field is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function that is often referred to as the mother wavelet. The wavelet expansion can adaptively fit itself to the total field distribution by distributing the localized functions near the flaw boundary, where the field change is large, and the more spatially diffused functions over the interior of the flaw where the total field tends to be smooth. The approach is thus best suited to modeling large three-dimensional flaws where the large number of elements used in the volume integral method requires extremely large memory space and computational capacity. The feasibility of the wavelet method is discussed in the context of the physical nature of eddy-current modeling problems. Numerical examples using both Haar wavelets and Daubechies compactly supported wavelets with periodic extension are given. The results of the wavelet method are also compared with experimental results from a cylindrical flat-bottom hole in an aluminum plate. These numerical examples and comparisons indicate that the wavelet method can greatly reduce the numerical complexity of the problem with negligible loss in accuracy. {copyright} {ital 1997 American Institute of Physics.}
NASA Astrophysics Data System (ADS)
Tarhini, Rana
2015-12-01
In this paper, we study a nonlocal degenerate parabolic equation of order α + 2 for α ∈ (0, 2). The equation is a generalization of the one arising in the modeling of hydraulic fractures studied by Imbert and Mellet in 2011. Using the same approach, we prove the existence of solutions for this equation for 0 < α < 2 and for nonnegative initial data satisfying appropriate assumptions. The main difference is the compactness results due to different Sobolev embeddings. Furthermore, for α > 1, we construct a nonnegative solution for nonnegative initial data under weaker assumptions.
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.
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.
NASA Astrophysics Data System (ADS)
Xie, Guizhong; Zhang, Jianming; Huang, Cheng; Lu, Chenjun; Li, Guangyao
2014-04-01
This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.
NASA Astrophysics Data System (ADS)
Zhang, Tian-Tian; Ma, Pan-Li; Xu, Mei-Juan; Zhang, Xing-Yong; Tian, Shou-Fu
2015-05-01
In this paper, a (3+1)-dimensional generalized variable-coefficients Kadomtsev-Petviashvili (gvcKP) equation is proposed, which describes many nonlinear phenomena in fluid dynamics and plasma physics. By a very natural way, the integrable constraint conditions on the variable coefficients are presented to investigate the integrabilities of the gvcKP equation. Based on the generalized Bell's polynomials, we succinctly obtain its bilinear representations, bilinear Bäcklund transformation and Lax pair, respectively. Furthermore, by virtue of the binary Bell polynomial form, the infinite conservation laws of the equation are found with explicit recursion formulas as well by using its Lax equations via algebraic and differential manipulation. In addition, by using the Hirota bilinear method, its N-soliton solutions are also obtained.
NASA Astrophysics Data System (ADS)
Fishman, Louis
2000-11-01
The role of mathematical modeling in the physical sciences will be briefly addressed. Examples will focus on computational acoustics, with applications to underwater sound propagation, electromagnetic modeling, optics, and seismic inversion. Direct and inverse wave propagation problems in both the time and frequency domains will be considered. Focusing on fixed-frequency (elliptic) wave propagation problems, the usual, two-way, partial differential equation formulation will be exactly reformulated, in a well-posed manner, as a one-way (marching) problem. This is advantageous for both direct and inverse considerations, as well as stochastic modeling problems. The reformulation will require the introduction of pseudodifferential operators and their accompanying phase space analysis (calculus), in addition to path integral representations for the fundamental solutions and their subsequent computational algorithms. Unlike the more traditional, purely numerical applications of, for example, finite-difference and finite-element methods, this approach, in effect, writes the exact, or, more generally, the asymptotically correct, answer as a functional integral and, subsequently, computes it directly. The overall computational philosophy is to combine analysis, asymptotics, and numerical methods to attack complicated, real-world problems. Exact and asymptotic analysis will stress the complementary nature of the direct and inverse formulations, as well as indicating the explicit structural connections between the time- and frequency-domain solutions.
NASA Technical Reports Server (NTRS)
Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.
2002-01-01
The rapid increase in available computational power over the last decade has enabled higher resolution flow simulations and more widespread use of unstructured grid methods for complex geometries. While much of this effort has been focused on steady-state calculations in the aerodynamics community, the need to accurately predict off-design conditions, which may involve substantial amounts of flow separation, points to the need to efficiently simulate unsteady flow fields. Accurate unsteady flow simulations can easily require several orders of magnitude more computational effort than a corresponding steady-state simulation. For this reason, techniques for improving the efficiency of unsteady flow simulations are required in order to make such calculations feasible in the foreseeable future. The purpose of this work is to investigate possible reductions in computer time due to the choice of an efficient time-integration scheme from a series of schemes differing in the order of time-accuracy, and by the use of more efficient techniques to solve the nonlinear equations which arise while using implicit time-integration schemes. This investigation is carried out in the context of a two-dimensional unstructured mesh laminar Navier-Stokes solver.
NASA Astrophysics Data System (ADS)
Wang, Shyh-Wei; Guo, Shuang-Fa
1998-01-01
New techniques for more accurate and efficient simulation of ion implantations by a stepwise numerical integration of the Boltzmann transport equation (BTE) have been developed in this work. Instead of using uniform energy grid, a non-uniform grid is employed to construct the momentum distribution matrix. A more accurate simulation result is obtained for heavy ions implanted into silicon. In the same time, rather than utilizing the conventional Lindhard, Nielsen and Schoitt (LNS) approximation, an exact evaluation of the integrals involving the nuclear differential scattering cross-section (dσn=2πp dp) is proposed. The impact parameter p as a function of ion energy E and scattering angle φ is obtained by solving the magic formula iteratively and an interpolation techniques is devised during the simulation process. The simulation time using exact evaluation is about 3.5 times faster than that using the Littmark and Ziegler (LZ) spline fitted cross-section function for phosphorus implantation into silicon.
Integrated technology wing design study
NASA Technical Reports Server (NTRS)
Hays, A. P.; Beck, W. E.; Morita, W. H.; Penrose, B. J.; Skarshaug, R. E.; Wainfan, B. S.
1984-01-01
The technology development costs and associated benefits in applying advanced technology associated with the design of a new wing for a new or derivative trijet with a capacity for 350 passengers and maximum range of 8519 km, entering service in 1990 were studied. The areas of technology are: (1) airfoil technology; (2) planform parameters; (3) high lift; (4) pitch active control system; (5) all electric systems; (6) E to 3rd power propulsion; (7) airframe/propulsion integration; (8) graphite/epoxy composites; (9) advanced aluminum alloys; (10) titanium alloys; and (11) silicon carbide/aluminum composites. These technologies were applied to the reference aircraft configuration. Payoffs were determined for block fuel reductions and net value of technology. These technologies are ranked for the ratio of net value of technology (NVT) to technology development costs.
NASA Technical Reports Server (NTRS)
Banyukevich, A.; Ziolkovski, K.
1975-01-01
A number of hybrid methods for solving Cauchy problems are described on the basis of an evaluation of advantages of single and multiple-point numerical integration methods. The selection criterion is the principle of minimizing computer time. The methods discussed include the Nordsieck method, the Bulirsch-Stoer extrapolation method, and the method of recursive Taylor-Steffensen power series.
NASA Technical Reports Server (NTRS)
Gottlieb, D.; Turkel, E.
1980-01-01
New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.
Neumann type integrable reduction for nonlinear evolution equations in 1+1 and 2+1 dimensions
NASA Astrophysics Data System (ADS)
Chen, Jinbing
2009-12-01
A family of new Neumann type systems is given in view of the nonlinearization technique, realizing the variable separation of the modified Jaulent-Miodek hierarchy and a new coupled modified Kadomtsev-Petviashvili equation on the symplectic submanifold TSN -1. By two Casimir functions and a special solution of the Lenard eigenvalue equation, we deduce the Lax-Moser matrix of the Neumann type systems that yields integrals of motion and the constrained Hamiltonians whose vector fields are tangent to TSN -1. Based on the Dirac-Poisson bracket and a Lax equation on TSN -1, a new systematic way is proposed to prove the Liouville integrability of a family of Neumann type systems synchronously. The Dirac-Poisson bracket and the generating function are used to reveal the explicit relation between infinite dimensional integrable systems and Neumann type systems, and then we point out that compatible solutions of Neumann type systems yield the finite parametric solutions of 1+1 and 2+1 dimensional integrable nonlinear evolution equations and the finite-gap solutions of the Novikov equation.
Linear complexity integral-equation based methods for large-scale electromagnetic analysis
NASA Astrophysics Data System (ADS)
Chai, Wenwen
In general, to solve problems with N parameters, the optimal computational complexity is linear complexity O( N). However, for most computational electromagnetic methods, the complexity is higher than O(N). In this work, we introduced and further developed the H - and H2 -matrix based mathematical framework to break the computational barrier of existing integral-equation (IE)-based methods for large-scale electromagnetic analysis. Our significant contributions include the first-time dense matrix inversion and LU factorization of O(N) complexity for large-scale 3-D circuit extraction and a fast direct integral equation solver that outperforms existing direct solvers for large-scale electrodynamic analysis having millions of unknowns and ˜100 wavelengths. The major contributions of this work are: (1) Direct Matrix Solution of Linear Complexity for 3-D Integrated Circuit (IC) and Package Extraction • O(N) complexity dense matrix inversion and LU factorization algorithms and their applications to capacitance extraction and impedance extraction of large-scale 3-D circuits • O(N) direct matrix solution of highly irregular matrices consisting of both dense and sparse matrix blocks arising from full-wave analysis of general 3-D circuits with lossy conductors in multiple dielectrics. (2) Fast H - and H2 -Based IE Solvers for Large-Scale Electrodynamic Analysis • theoretical proof on the error bounded low-rank representation of electrodynamic integral operators • fast H2 -based iterative solver with O(N) computational cost and controlled accuracy from small to tens of wavelengths • fast H -based direct solver with computational cost minimized based on accuracy • Findings on how to reduce the complexity of H - and H2 -based methods for electrodynamic analysis, which are also applicable to many other fast IE solvers. (3) Fast Algorithms for Accelerating H - and H2 -Based Iterative and Direct Solvers • Optimal H -based representation and its applications from
Integral equation theory of the structure and thermodynamics of polymer blends
NASA Astrophysics Data System (ADS)
Schweizer, Kenneth S.; Curro, John G.
1989-10-01
Our recently developed RISM integral equation theory of the structure and thermodynamics of homopolymer melts is generalized to polymer mixtures. The mean spherical approximation (MSA) closure to the generalized Ornstein-Zernike equations is employed, in conjunction with the neglect of explicit chain end effects and the assumption of ideality of intramolecular structure. The theory is developed in detail for binary blends, and the random phase approximation (RPA) form for concentration fluctuation scattering is rigorously obtained by enforcing incompressibility. A microscopic, wave vector-dependent expression for the effective chi parameter measured in small angle neutron scattering (SANS) experiments is derived in terms of the species-dependent direct correlation functions of the blend. The effective chi parameter is found to depend, in general, on thermodynamic state, intermolecular forces, intramolecular structure, degree of polymerization, and global architecture. The relationship between the mean field Flory-Huggins expression for the free energy of mixing and our RISM-MSA theory is determined, along with general analytical connections between the chi parameter and intermolecular pair correlations in the liquid. Detailed numerical applications to athermal and isotopic chain polymer blend models are presented for both the chi parameter and the structure. For athermal blends a negative, concentration-dependent chi parameter is found which decreases with density, structural asymmetry, and increases with molecular weight. For isotopic blends, the effective (positive) chi parameter is found to be strongly renormalized downward from its mean field enthalpic value by long range fluctuations in monomer concentration induced by polymeric connectivity and excluded volume. Both the renormalization and composition dependence of the chi parameter increase with chain length and proximity to the spinodal instability. The critical temperature is found to be proportional to
NASA Astrophysics Data System (ADS)
Fedotova, M. V.; Kruchinin, S. E.
2012-12-01
The structural parameters of glycine zwitterion in water were studied by means of the integral equation method in the framework of the RISM approximation. According to calculations, five water molecules are located in the nearest environment of the -NH{3/+} group, and two of them are the H-bonded with this group. At the same time, six water molecules are located in the nearest environment of the -COO- group, and three of them are the H-bonded with this group. The average number of water molecules in the first hydration shell of -CH2 group is four. It has been shown that the probability of hydrogen bond formation between water molecules and the hydrogen atom H1 of the -NH{3/+} group is low, and there is no H-bonding between water molecules and the nitrogen atom the -NH{3/+} group.
NASA Astrophysics Data System (ADS)
Ying Huang, Shao; Wu, Bae-Ian; Foong, Shaohui
2013-01-01
Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) surface integral equation method is applied for the first time to accurately estimate the surface-enhanced Raman scattering (SERS) enhancement factor distribution for arbitrary nanoparticles and nano-aggregates. It is the first time in literature that the distributions of SERS enhancement factors of nanoparticles of a large variety are reported. It is shown that not every SERS substrate exhibits a long-tail distribution as a dimer consisting of two spheres in close proximity. Generic methods are proposed to evaluate the performance of nanoparticles on SERS substrates. A cumulative distribution is proposed to examine the contributions of hot and warm spots around the nanoparticles. It is used to identify the importance of warm spots on a SERS substrate. A parameter q is proposed to describe the likelihood of a randomly positioned molecule that can be activated. This study provides guidance and insights for the optimization of SERS substrate fabrication techniques.
Manenkov, A B; Latsas, G P; Tigelis, L G
2001-12-01
We study the problem of the scattering of the first TM guided mode from an abruptly ended strongly asymmetrical slab waveguide by an improved iteration technique, which is based on the integral equation method with "accelerating" parameters. We demonstrate that the values of these parameters are related to the variational principle, and we save approximately 1-2 iterations compared with the case in which these parameters are not employed. The tangential electric-field distribution on the terminal plane, the reflection coefficient of the first TM guided mode, and the far-field radiation pattern are computed. Furthermore, a simple technique based on the Aitken extrapolation procedure is employed for faster computation of the higher-order solutions of the reflection coefficient. Numerical results are presented for several cases of abruptly ended waveguides, including systems with variational profile, while special attention is given to the far-field radiation pattern rotation and its explanation. PMID:11760208
Lee, Yong Woo; Lee, Duck Joo
2014-12-01
Kirchhoff's formula for the convective wave equation is derived using the generalized function theory. The generalized convective wave equation for a stationary surface is obtained, and the integral formulation, the convective Kirchhoff's formula, is derived. The formula has a similar form to the classical Kirchhoff's formula, but an additional term appears due to a moving medium effect. For convenience, the additional term is manipulated to a final form as the classical Kirchhoff's formula. The frequency domain boundary integral can be obtained from the current time domain boundary integral form. The derived formula is verified by comparison with the analytic solution of source in the uniform flow. The formula is also utilized as a boundary integral equation. Time domain boundary element method (BEM) analysis using the boundary integral equation is conducted, and the results show good agreement with the analytical solution. The formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain. PMID:25480045
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.
Yu, Zhang; Zhang, Yufeng
2009-01-30
Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified KdV lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of solition equations. PMID:20119478
Tunç, Cemil; Tunç, Osman
2016-01-01
In this paper, certain system of linear homogeneous differential equations of second-order is considered. By using integral inequalities, some new criteria for bounded and [Formula: see text]-solutions, upper bounds for values of improper integrals of the solutions and their derivatives are established to the considered system. The obtained results in this paper are considered as extension to the results obtained by Kroopnick (2014) [1]. An example is given to illustrate the obtained results. PMID:26843982
NASA Astrophysics Data System (ADS)
Zinser, Brian
We present two distinct mathematical models where high-order integral equations are applied to electromagnetic problems. The first problem is to find the electric potential in and around ion channels and Janus particles. The second problem is to find the electromagnetic scattering caused by a set of simple geometric objects. In biology, we consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. A boundary element method (BEM) for the Poisson-Boltzmann equation based on Muller's hyper-singular second kind integral equation formulation is used to accurately compute electrostatic potentials. The proposed BEM gives O(1) condition numbers and we show that the second order basis converges faster and is more accurate than the first order basis. For solar cells, we develop a Nystrom volume integral equation (VIE) method for calculating the electromagnetic scattering according to the Maxwell equations. The Cauchy principal values (CPVs) that arise from the VIE are computed using a finite size exclusion volume with explicit correction integrals. Outside the exclusion, the hyper-singular integrals are computed using an interpolated quadrature formulae with tensor-product quadrature nodes. We considered cubes, rectangles, cylinders, spheres, and ellipsoids. As the new quadrature weights are pre-calculated and tabulated, the integrals are calculated efficiently at runtime. Simulations with many scatterers demonstrate the efficiency of the interpolated quadrature formulae. We also demonstrate that the resulting VIE has high accuracy and p-convergence.
Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang
2014-11-15
We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested.
Hybrid two-chain simulation and integral equation theory : application to polyethylene liquids.
Huimin Li, David T. Wu; Curro, John G.; McCoy, John Dwane
2006-02-01
We present results from a hybrid simulation and integral equation approach to the calculation of polymer melt properties. The simulation consists of explicit Monte Carlo (MC) sampling of two polymer molecules, where the effect of the surrounding chains is accounted for by an HNC solvation potential. The solvation potential is determined from the Polymer Reference Interaction Site Model (PRISM) as a functional of the pair correlation function from simulation. This hybrid two-chain MC-PRISM approach was carried out on liquids of polyethylene chains of 24 and 66 CH{sub 2} units. The results are compared with MD simulation and self-consistent PRISM-PY theory under the same conditions, revealing that the two-chain calculation is close to MD, and able to overcome the defects of the PRISM-PY closure and predict more accurate structures of the liquid at both short and long range. The direct correlation function, for instance, has a tail at longer range which is consistent with MD simulation and avoids the short-range assumptions in PRISM-PY theory. As a result, the self-consistent two-chain MC-PRISM calculation predicts an isothermal compressibility closer to the MD results.
Efficient 3D/1D self-consistent integral-equation analysis of ICRH antennae
NASA Astrophysics Data System (ADS)
Maggiora, R.; Vecchi, G.; Lancellotti, V.; Kyrytsya, V.
2004-08-01
This work presents a comprehensive account of the theory and implementation of a method for the self-consistent numerical analysis of plasma-facing ion-cyclotron resonance heating (ICRH) antenna arrays. The method is based on the integral-equation formulation of the boundary-value problem, solved via a weighted-residual scheme. The antenna geometry (including Faraday shield bars and a recess box) is fairly general and three-dimensional (3D), and the plasma is in the one-dimensional (1D) 'slab' approximation; finite-Larmor radius effects, as well as plasma density and temperature gradients, are considered. Feeding via the voltages in the access coaxial lines is self-consistently accounted throughout and the impedance or scattering matrix of the antenna array obtained therefrom. The problem is formulated in both the dual space (physical) and spectral (wavenumber) domains, which allows the extraction and simple handling of the terms that slow the convergence in the spectral domain usually employed. This paper includes validation tests of the developed code against measured data, both in vacuo and in the presence of plasma. An example of application to a complex geometry is also given.
Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation.
Nakamura, K; Sobirov, Z A; Matrasulov, D U; Sawada, S
2011-08-01
We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one [Sobirov et al., Phys. Rev. E 81, 066602 (2010)] on the continuum NLSE to the discrete case. We find (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds. PMID:21929130
Diagonal Ising susceptibility: elliptic integrals, modular forms and Calabi-Yau equations
NASA Astrophysics Data System (ADS)
Assis, M.; Boukraa, S.; Hassani, S.; van Hoeij, M.; Maillard, J.-M.; McCoy, B. M.
2012-02-01
We give the exact expressions of the partial susceptibilities χ(3)d and χ(4)d for the diagonal susceptibility of the Ising model in terms of modular forms and Calabi-Yau ODEs, and more specifically, 3F2([1/3, 2/3, 3/2], [1, 1] z) and 4F3([1/2, 1/2, 1/2, 1/2], [1, 1, 1] z) hypergeometric functions. By solving the connection problems we analytically compute the behavior at all finite singular points for χ(3)d and χ(4)d. We also give new results for χ(5)d. We see, in particular, the emergence of a remarkable order-6 operator, which is such that its symmetric square has a rational solution. These new exact results indicate that the linear differential operators occurring in the n-fold integrals of the Ising model are not only ‘derived from geometry’ (globally nilpotent), but actually correspond to ‘special geometry’ (homomorphic to their formal adjoint). This raises the question of seeing if these ‘special geometry’ Ising operators are ‘special’ ones, reducing, in fact systematically, to (selected, k-balanced, ...) q + 1Fq hypergeometric functions, or correspond to the more general solutions of Calabi-Yau equations.
NASA Astrophysics Data System (ADS)
Shen, Yongxing; Barnett, David M.; Pinsky, Peter M.
2008-02-01
Kelvin probe force microscopy (KPFM) is designed for measuring the tip-sample contact potential differences by probing the sample surface, measuring the electrostatic interaction, and adjusting a feedback circuit. However, for the case of a dielectric (insulating) sample, the contact potential difference may be ill defined, and the KPFM probe may be sensing electrostatic interactions with a certain distribution of sample trapped charges or dipoles, leading to difficulty in interpreting the images. We have proposed a general framework based on boundary integral equations for simulating the KPFM image based on the knowledge about the sample charge distributions (forward problem) and a deconvolution algorithm solving for the trapped charges on the surface from an image (inverse problem). The forward problem is a classical potential problem, which can be efficiently solved using the boundary element method. Nevertheless, the inverse problem is ill posed due to data incompleteness. For some special cases, we have developed deconvolution algorithms based on the forward problem solution. As an example, this algorithm is applied to process the KPFM image of a gadolinia-doped ceria thin film to solve for its surface charge density, which is a more relevant quantity for samples of this kind than the contact potential difference (normally only defined for conductive samples) values contained in the raw image.
Coupled integral equations for sound propagation above a hard ground surface with trench cuttings.
Wang, Gong Li; Chew, Weng Cho; White, Michael J
2006-09-01
A set of coupled integral equations is formulated for the investigation of sound propagation from an infinitesimal harmonic line source above a hard ground surface corrugated with cuttings. Two half-space Green's functions are employed in the formulation. The first one defined for the upper half space is used to reduce the problem size and eliminate the edge effect resulting from the boundary truncation; the other one for the lower half space is to simplify the representation of the Neumann-Dirichlet map. As a result, the unknowns are only distributed over the corrugated part of the surface, which leads to substantial reduction in the size of the final linear system. The computational complexity of the Neumann-Dirichlet map is also reduced. The method is used to analyze the behavior of sound propagation above textured surfaces the impedance of which is expectedly altered. The effects of number and opening of trench cuttings, and the effect of source height are investigated. The conclusions drawn can be used for reference in a practical problem of mitigating gun blast noise. PMID:17004443
NASA Technical Reports Server (NTRS)
Gedney, Stephen D.; Lansing, Faiza
1994-01-01
It has been found that the Discrete Integral Equation (DSI)technique is a highly effective technique for the analysis of microwave circuits and devices [1,2]. The DSI is much more robust than the traditional Finite Difference Time Domain (FDTD) method in a number of ways.
ERIC Educational Resources Information Center
Cheung, Mike W.-L.
2008-01-01
Meta-analysis and structural equation modeling (SEM) are two important statistical methods in the behavioral, social, and medical sciences. They are generally treated as two unrelated topics in the literature. The present article proposes a model to integrate fixed-, random-, and mixed-effects meta-analyses into the SEM framework. By applying an…
Heydari, M.H.; Hooshmandasl, M.R.; Cattani, C.; Maalek Ghaini, F.M.
2015-02-15
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. Error 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.
NASA Technical Reports Server (NTRS)
Bates, J. R.; Semazzi, F. H. M.; Higgins, R. W.; Barros, Saulo R. M.
1990-01-01
A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.
Numerical study of the semiclassical limit of the Davey-Stewartson II equations
NASA Astrophysics Data System (ADS)
Klein, C.; Roidot, K.
2014-09-01
We present the first detailed numerical study of the semiclassical limit of the Davey-Stewartson II equations both for the focusing and the defocusing variant. We concentrate on rapidly decreasing initial data with a single hump. The formal limit of these equations for vanishing semiclassical parameter ɛ, the semiclassical equations, is numerically integrated up to the formation of a shock. The use of parallelized algorithms allows one to determine the critical time tc and the critical solution for these 2 + 1-dimensional shocks. It is shown that the solutions generically break in isolated points similarly to the case of the 1 + 1-dimensional cubic nonlinear Schrödinger equation, i.e., cubic singularities in the defocusing case and square root singularities in the focusing case. For small values of ɛ, the full Davey-Stewartson II equations are integrated for the same initial data up to the critical time tc. The scaling in ɛ of the difference between these solutions is found to be the same as in the 1 + 1 dimensional case, proportional to ɛ2/7 for the defocusing case and proportional to ɛ2/5 in the focusing case. We document the Davey-Stewartson II solutions for small ɛ for times much larger than the critical time tc. It is shown that zones of rapid modulated oscillations are formed near the shocks of the solutions to the semiclassical equations. For smaller ɛ, the oscillatory zones become smaller and more sharply delimited to lens-shaped regions. Rapid oscillations are also found in the focusing case for initial data where the singularities of the solution to the semiclassical equations do not coincide. If these singularities do coincide, which happens when the initial data are symmetric with respect to an interchange of the spatial coordinates, no such zone is observed. Instead the initial hump develops into a blow-up of the L∞ norm of the solution. We study the dependence of the blow-up time on the semiclassical parameter ɛ.
NASA Astrophysics Data System (ADS)
Chen, Jinbing
2010-08-01
Each soliton equation in the Korteweg-de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN -1 of the unit sphere SN -1. The Lax-Moser matrix for the Neumann systems of degree N -1 is deduced in view of the Mckean-Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax-Moser matrix, the constrained Hamiltonians in the sense of Dirac-Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN -1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN -1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.
NASA Astrophysics Data System (ADS)
Djebali, Smaïl; Sahnoun, Zahira
This paper is devoted to establishing new variants of some nonlinear alternatives of Leray-Schauder and Krasnosel'skij type involving the weak topology of Banach spaces. The De Blasi measure of weak noncompactness is used. An application to solving a nonlinear Hammerstein integral equation in L spaces is given. Our results complement recent ones in [K. Latrach, M.A. Taoudi, A. Zeghal, Some fixed point theorems of the Schauder and the Krasnosel'skij type and application to nonlinear transport equations, J. Differential Equations 221 (2006) 256-2710] and [K. Latrach, M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L spaces, Nonlinear Anal. 66 (2007) 2325-2333].
Yoshikawa, Kohji; Umemura, Masayuki; Yoshida, Naoki
2013-01-10
We present a scheme for numerical simulations of collisionless self-gravitating systems which directly integrates the Vlasov-Poisson equations in six-dimensional phase space. Using the results from a suite of large-scale numerical simulations, we demonstrate that the present scheme can simulate collisionless self-gravitating systems properly. The integration scheme is based on the positive flux conservation method recently developed in plasma physics. We test the accuracy of our code by performing several test calculations, including the stability of King spheres, the gravitational instability, and the Landau damping. We show that the mass and the energy are accurately conserved for all the test cases we study. The results are in good agreement with linear theory predictions and/or analytic solutions. The distribution function keeps the property of positivity and remains non-oscillatory. The largest simulations are run on 64{sup 6} grids. The computation speed scales well with the number of processors, and thus our code performs efficiently on massively parallel supercomputers.
Shewmon, A D
2001-10-01
The mainstream rationale for equating "brain death" (BD) with death is that the brain confers integrative unity upon the body, transforming it from a mere collection of organs and tissues to an "organism as a whole." In support of this conclusion, the impressive list of the brain's myriad integrative functions is often cited. Upon closer examination, and after operational definition of terms, however, one discovers that most integrative functions of the brain are actually not somatically integrating, and, conversely, most integrative functions of the body are not brain-mediated. With respect to organism-level vitality, the brain's role is more modulatory than constitutive, enhancing the quality and survival potential of a presupposedly living organism. Integrative unity of a complex organism is an inherently nonlocalizable, holistic feature involving the mutual interaction among all the parts, not a top-down coordination imposed by one part upon a passive multiplicity of other parts. Loss of somatic integrative unity is not a physiologically tenable rationale for equating BD with death of the organism as a whole. PMID:11588655
Theoretical and numerical studies of nonlinear shell equations
NASA Astrophysics Data System (ADS)
Hermann, M.; Kaiser, D.; Schröder, M.
1999-07-01
We study the solution field M of a parameter dependent nonlinear two-point boundary value problem presented by Troger and Steindl [H. Troger, A. Steindl, Nonlinear Stability and Bifurcation Theory, Springer, Wien, New York, 1991]. This problem models the buckling of a thin-walled spherical shell under a uniform external static pressure. The boundary value problem is formulated as an abstract operator equation T( x, λ)=0 in appropriate Banach spaces. By exploiting the equivariance of T, we obtain detailed informations about the structure of M. These theoretical results are used to compute efficiently interesting parts of M with numerical standard techniques. Bifurcation diagrams, a stability diagram and pictures of deformed shells are presented.
Using wavelets to solve the Burgers equation: A comparative study
Schult, R.L.; Wyld, H.W. )
1992-12-15
The Burgers equation is solved for Reynolds numbers [approx lt]8000 in a representation using coarse-scale scaling functions and a subset of the wavelets at finer scales of resolution. Situations are studied in which the solution develops a shocklike discontinuity. Extra wavelets are kept for several levels of higher resolution in the neighborhood of this discontinuity. Algorithms are presented for the calculation of matrix elements of first- and second-derivative operators and a useful product operation in this truncated wavelet basis. The time evolution of the system is followed using an implicit time-stepping computer code. An adaptive algorithm is presented which allows the code to follow a moving shock front in a system with periodic boundary conditions.
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
NASA Astrophysics Data System (ADS)
Dou, Nicholas G.; Minnich, Austin J.
2016-01-01
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
The integrated Michaelis-Menten rate equation: déjà vu or vu jàdé?
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. PMID:22630075
Study of coupled nonlinear partial differential equations for finding exact analytical solutions
Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.
2015-01-01
Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256
NASA Technical Reports Server (NTRS)
Joseph, Rose M.; Hagness, Susan C.; Taflove, Allen
1991-01-01
The initial results for femtosecond pulse propagation and scattering interactions for a Lorentz medium obtained by a direct time integration of Maxwell's equations are reported. The computational approach provides reflection coefficients accurate to better than 6 parts in 10,000 over the frequency range of dc to 3 x 10 to the 16th Hz for a single 0.2-fs Gaussian pulse incident upon a Lorentz-medium half-space. New results for Sommerfeld and Brillouin precursors are shown and compared with previous analyses. The present approach is robust and permits 2D and 3D electromagnetic pulse propagation directly from the full-vector Maxwell's equations.
NASA Astrophysics Data System (ADS)
Euler, Marianna; Euler, Norbert; Wolf, Thomas
2012-10-01
Recently, Holm and Ivanov, proposed and studied a class of multi-component generalizations of the Camassa-Holm equations [D. D. Holm and R. I. Ivanov, Multi-component generalizations of the CH equation: geometrical aspects, peakons and numerical examples, J. Phys A: Math. Theor.43 (2010) 492001 (20pp)]. We consider two of those systems, denoted by Holm and Ivanov by CH(2,1) and CH(2,2), and report a class of integrating factors and its corresponding conservation laws for these two systems. In particular, we obtain the complete set of first-order integrating factors for the systems in Cauchy-Kovalevskaya form and evaluate the corresponding sets of conservation laws for CH(2,1) and CH(2,2).
Sourcing for Parameter Estimation and Study of Logistic Differential Equation
ERIC Educational Resources Information Center
Winkel, Brian J.
2012-01-01
This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…
Xie, G.; Li, J.; Majer, E.; Zuo, D.
1998-07-01
This paper describes a new 3D parallel GILD electromagnetic (EM) modeling and nonlinear inversion algorithm. The algorithm consists of: (a) a new magnetic integral equation instead of the electric integral equation to solve the electromagnetic forward modeling and inverse problem; (b) a collocation finite element method for solving the magnetic integral and a Galerkin finite element method for the magnetic differential equations; (c) a nonlinear regularizing optimization method to make the inversion stable and of high resolution; and (d) a new parallel 3D modeling and inversion using a global integral and local differential domain decomposition technique (GILD). The new 3D nonlinear electromagnetic inversion has been tested with synthetic data and field data. The authors obtained very good imaging for the synthetic data and reasonable subsurface EM imaging for the field data. The parallel algorithm has high parallel efficiency over 90% and can be a parallel solver for elliptic, parabolic, and hyperbolic modeling and inversion. The parallel GILD algorithm can be extended to develop a high resolution and large scale seismic and hydrology modeling and inversion in the massively parallel computer.
NASA Astrophysics Data System (ADS)
Kashirin, A. A.; Smagin, S. I.; Taltykina, M. Yu.
2016-04-01
Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure.
Parallel numerical integration of Maxwell's full-vector equations in nonlinear focusing media
NASA Astrophysics Data System (ADS)
Bennett, Paul Murray
Maxwell's equations governing the evolution of ultrashort intense coherent pulses of light in a nonlinear focusing dielectric are presented. A discretization of this model using Kane Yee's grid is presented. Initial and boundary conditions are derived, and a serial finite difference algorithm using Yee's grid with the initial and boundary conditions is given. A parallelization of the serial algorithm to more aptly handle the large computational size is performed, and speedup and efficiency results of the parallel program are presented. The parallel code is first used to study the effect of the focusing nonlinearity upon dispersionless pulse propagation. Indications are given of the development of shocks on the optical carrier wave and upon the pulse envelope. The parallel code is then used to study the effect of varying the focusing of the light by varying the intensity as a way to compensate linear dispersion. Blow-up of the pulse in finite propagation distance is demonstrated, and the dependence of the blow-up position upon the intensity of the light is presented. Optical saturation is considered to counter blow-up of intense pulses. Finally, the parallel code is used to study the evolution of intense ultrashort optical pulses in a model featuring nonlinear dispersion, focusing, and optical saturation.
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
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.
NASA Astrophysics Data System (ADS)
Norman, Matthew Ross
The social need for realistic atmospheric simulation in weather prediction, climate change attribution, seasonal forecasting, and climate projection is great. To obtain realistic simulations, we need more physical processes included in the model with greater fidelity and finer spatial resolution. Spatial resolution primarily drives the need for computational resources because reducing the model grid spacing by a factor f requires f 4 times more computation (assuming 3-D refinement). This compute power comes from large parallel machines with 10,000s of separate nodes and accelerators such as graphics processing units (GPUs) making efficiency a complicated problem. Efficiency parallel integration algorithms need low internode communication, minimal synchronization, large time steps, and clustered computation. To this end, we propose new characteristics-based methods for the atmospheric dynamical equations with these properties in mind. These schemes are capable of simulating at a large CFL time step in only one stage of computations, needing only one copy of the state variables. They are implemented in a 2-D non-hydrostatic compressible equation set in an x-z (horizontal-vertical) Cartesian plane to simulate buoyancy-driven flows such as rising thermals and internal gravity waves. The schemes are implemented to run on CPU and multi-GPU architectures using Nvidia's CUDA (Compute Unified Device Architecture) language to test relative efficiency. Even with- out memory tuning, the GPU code showed roughly 2.5x (5x) better performance per Watt. With optimization, this could increase by an order of magnitude. The methods can use any spatial interpolant, so two major formulations are proposed and tested. One uses WENO interpolants which are pre-computed, and the other uses standard polynomials and computes them on-the-fly. The advantage of on-the-fly calculations is a significant reduction in the volume of data communicated to and from the GPU's slow global memory. In some
NASA Astrophysics Data System (ADS)
Kitanine, N.; Maillet, J. M.; Niccoli, G.
2014-05-01
We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q-function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.
A comparative study of different ferrofluid constitutive equations.
NASA Astrophysics Data System (ADS)
Kaloni, Purna
2011-11-01
Ferrofluids are stable colloidal suspensions of fine ferromagnetic monodomain nanoparticles in a non-conducting carrier fluid. The particles are coated with a surfacant to avoid agglomeration and coagulation.Brownian motion keeps the nanoparticles from settling under gravity. In recent years these fluids have found several applications including in liquid seals in rotary shafts for vacuum system and in hard disk drives of personal computers, in cooling and damping of loud speakers, in shock absorbers and in biomedical applications. A continuum description of ferrofluids was initiated by Neuringer and Rosensweig but the theory had some limitations. In subsequent years,several authors have proposed generalization of the above theory.Some of these are based upon the internal particle rotation concept, some are phemonological, some are based upon a thermodynamic framework, some employ statistical approach and some have used the dynamic mean field approach. The results based upon these theories ane in early stages and inconclusive. Our purpose is, first, to critically examine the basic foundations of these equations and then study the pedictions obtained in all the theories related to an experimental as well as a theoretical study.
A two-equation integral model for particle transport in renewal statistical media
Zuchuat, O.; Sanchez, R.
1995-12-31
The authors consider the problem of particle transport including scattering in renewal statistical media. The general description of this problem leads to an infinite hierarchy of equations. A new closure scheme is developed to obtain a more tractable set of equations. Numerical results in planar geometry are given which compare the predictions of this new closure with exact benchmark results as well as with a previous model available in the literature. The development of the new closure and the comparisons the authors make underline the importance of having a physical basis in the elaboration of closure schemes for the hierarchy of equations describing the transport of particle with collisions in stochastic mixtures.
NASA Astrophysics Data System (ADS)
Xu, Gui-qiong; Deng, Shu-fang
2016-06-01
In this article, we apply the singularity structure analysis to test an extended 2+1-dimensional fifth-order KdV equation for integrability. It is proven that the generalized equation passes the Painlevé test for integrability only in three distinct cases. Two of those cases are in agreement with the known results, and a new integrable equation is first given. Then, for the new integrable equation, we employ the Bell polynomial method to construct its bilinear forms, bilinear Bäcklund transformation, Lax pair, and infinite conversation laws systematically. The N-soliton solutions of this new integrable equation are derived, and the propagations and collisions of multiple solitons are shown by graphs.
Stamatakos, G S; Yova, D; Uzunoglu, N K
1997-09-01
A novel mathematical model of light scattering by an oriented monodisperse system of triaxial dielectric ellipsoids of complex index of refraction is presented. It is based on an integral equation solution to the scattering of a plane electromagnetic wave by a single triaxial dielectric ellipsoid. Both the position and the orientation of a single representative scatterer in a given coordinate system are considered arbitrary. A Monte Carlo simulation is developed to reproduce the diffraction pattern of a population of aligned ellipsoids. As an example of practical importance, light scattering by a population of erythrocytes subjected to intense shear stress is modeled. Agreement with experimental observations and the anomalous diffraction theory is illustrated. Thus a novel check of the electromagnetic basis of ektacytometry is provided. Furthermore, the versatility of the integral equation method, particularly in the advent of parallel processing systems, is demonstrated. PMID:18259511
Sasorov, P. V.; Fomin, I. V.
2015-06-15
The collision integral in the kinetic equation for a rarefied spin-polarized gas of fermions (electrons) is derived. The collisions between these fermions and the collisions with much heavier particles (ions) forming a randomly located stationary background (gas) are taken into account. An important new circumstance is that the particle-particle scattering amplitude is not assumed to be small, which could be obtained, for example, in the first Born approximation. The derived collision integral can be used in the kinetic equation, including that for a relatively cold rarefied spin-polarized plasma with a characteristic electron energy below α{sup 2}m{sub e}c{sup 2}, where α is the fine-structure constant.
NASA Astrophysics Data System (ADS)
Zhou, Yaoqi; Stell, George
1989-09-01
The system of a fluid in the presence of a spherical semipermeable vesicle (SPV) with the freely mobile nonpermeating species inside the vesicle is investigated via an integral-equation approach. This system can be used to model certain feature of a biological cell, permeable to simple ions, in which solute proteins inside the cell are unable to permeate its walls. As an illustrative example of the use of our integral equations, the analytical solution for density profiles in the mean-spherical approximation/Debye-Hückel approximation (MSA/DH) is obtained, where the MSA is used to obtain the density profiles near a membrane and the DH approximation to obtain the bulk pair correlation functions. A method which applies to nonmobile protein fixed inside a cell is also considered.
NASA Technical Reports Server (NTRS)
Chao, W. C.
1982-01-01
With appropriate modifications, a recently proposed explicit-multiple-time-step scheme (EMTSS) is incorporated into the UCLA model. In this scheme, the linearized terms in the governing equations that generate the gravity waves are split into different vertical modes. Each mode is integrated with an optimal time step, and at periodic intervals these modes are recombined. The other terms are integrated with a time step dictated by the CFL condition for low-frequency waves. This large time step requires a special modification of the advective terms in the polar region to maintain stability. Test runs for 72 h show that EMTSS is a stable, efficient and accurate scheme.
NASA Technical Reports Server (NTRS)
Joseph, Rose M.; Goorjian, Peter M.; Taflove, Allen
1993-01-01
We present what are to our knowledge first-time calculations from vector nonlinear Maxwell's equations of femtosecond soliton propagation and scattering, including carrier waves, in two-dimensional dielectric waveguides. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and the nonlinear convolution accounts for two quantum effects, the Kerr and Raman interactions. By retaining the optical carrier, the new method solves for fundamental quantities - optical electric and magnetic fields in space and time - rather than a nonphysical envelope function. It has the potential to provide an unprecedented two- and three-dimensional modeling capability for millimeter-scale integrated-optical circuits with submicrometer engineered inhomogeneities.
Some results on the integral transforms and applications to differential equations
Eltayeb, Hassan; Kilicman, Adem
2010-11-11
In this paper we give some remark about the relationship between Sumudu and Laplace transforms, further; for the comparison purpose, we apply both transforms to solve partial differential equations to see the differences and similarities.
On testing a subroutine for the numerical integration of ordinary differential equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1973-01-01
This paper discusses how to numerically test a subroutine for the solution of ordinary differential equations. Results obtained with a variable order Adams method are given for eleven simple test cases.-
NASA Astrophysics Data System (ADS)
Zieniuk, Eugeniusz; Kapturczak, Marta; Sawicki, Dominik
2016-06-01
In solving of boundary value problems the shapes of the boundary can be modelled by the curves widely used in computer graphics. In parametric integral equations system (PIES) such curves are directly included into the mathematical formalism. Its simplify the way of definition and modification of the shape of the boundary. Until now in PIES the B-spline, Bézier and Hermite curves were used. Recent developments in the computer graphics paid our attention, therefore we implemented in PIES possibility of defining the shape of boundary using the NURBS curves. The curves will allow us to modeling different shapes more precisely. In this paper we will compare PIES solutions (with applied NURBS) with the solutions existing in the literature.
Study of polytropes with generalized polytropic equation of state
NASA Astrophysics Data System (ADS)
Azam, M.; Mardan, S. A.; Noureen, I.; Rehman, M. A.
2016-06-01
The aim of this paper is to discuss the theory of Newtonian and relativistic polytropes with a generalized polytropic equation of state. For this purpose, we formulated the general framework to discuss the physical properties of polytropes with an anisotropic inner fluid distribution under conformally flat condition in the presence of charge. We investigate the stability of these polytropes in the vicinity of a generalized polytropic equation through the Tolman mass. It is concluded that one of the derived models is physically acceptable.
Penalized differential pathway analysis of integrative oncogenomics studies.
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. PMID:24552967
NASA Astrophysics Data System (ADS)
Hirose, Masanobu; Miyake, Masayasu; Takada, Jun-Ichi; Arai, Ikuo
1999-01-01
We have derived a new form of the integral equation formulation of the measured equation of invariance (IE-MEI). The new formulation clarifies the existence of a relationship between scattered electric and magnetic fields at consecutive nodes in the IE-MEI and indicates that the relationship in a problem for a perfect electric conductor (PEC) holds for a problem with arbitrary materials. In a scattering problem of a two-dimensional cylinder with an impedance boundary condition (IBC), every matrix in the IE-MEI is a band-like sparse matrix. That is, the solution process in the IE-MEI with an IBC is the same as that for a PEC. Therefore the IE-MEI with an IBC has the same merits of the IE-MEI for a PEC: The more efficient computation can be achieved with the smaller memory than those of the method of moments (MOM). The IE-MEI with an IBC is validated by numerical examples for a circular cylinder and a square cylinder by comparison with a combined field MOM that satisfies exact boundary conditions. Numerical examples show that the IE-MEI with an IBC is applicable to the case where the generalized skin depth is less than half the width of a scatterer.
Neural network training by integration of adjoint systems of equations forward in time
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)
1992-01-01
A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically, it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved, but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. The trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.
Neural Network Training by Integration of Adjoint Systems of Equations Forward in Time
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)
1999-01-01
A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically. it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved. but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. Tbc trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.
Boundary integral equation analysis for steady thermoelastic problems using thermoelastic potential
Koizumi, T.; Shibuya, T.; Kurokawa, K.; Tsuji, T.; Takakuda, K.
1988-01-01
In a boundary integral formulation for a thermoelastic problem, the temperature change is treated as an equivalent body force. Therefore, even in the case of zero body force, the formulation involves volume integrals. In the present paper, it is proved that the introduction of the thermoelastic potential succeeds in eliminating the volume integrals in the three-dimensional formulation. The formulations are transformed from Cartesian coordinates into axisymmetric coordinates. All the surface integrals are replaced by line integrals along the boundary of the axisymmetric domain. By using the above formulation, the deformation and stress distributions of a bonded cylinder subjected to a uniform temperature change are analyzed numerically. 7 references.
Integrating Anthropology in Elementary Social Studies.
ERIC Educational Resources Information Center
Zachlod, Michelle
2000-01-01
Discusses how anthropology can be integrated into the social studies classroom focusing on second and fifth grade levels. Demonstrates how different subject areas can be integrated with anthropology, such as history, geography, science, mathematics, and art. Covers topics such as foods, American Indian folklore, moonsticks, and myths and legends.…
Our Town Integrated Studies: A Resource.
ERIC Educational Resources Information Center
North Carolina State Dept. of Public Education, Raleigh.
This integrated state curriculum guide was developed by North Carolina fourth grade teachers, principals, and supervisors during a workshop which explored methods of integrating curriculum objectives from multiple instructional areas by using the community as both a resource and a subject of study and by introducing the concept of webbing, an…
Integrability and exact solutions of the nonautonomous mixed mKdV-sinh-Gordon equation
NASA Astrophysics Data System (ADS)
Yong, Xuelin; Wang, Hui; Gao, Jianwei
2014-07-01
In this paper, a nonautonomous mixed mKdV-sinh-Gordon equation with one arbitrary time-dependent variable coefficient is discussed in detail. It is proved that the equation passes the Painlevé test in the case of positive and negative resonances, respectively. Furthermore, a dependent variable transformation is introduced to get its bilinear form. Then, soliton, negaton, positon and interaction solutions are introduced by means of the Wronskian representation. Velocities are found to depend on the time-dependent variable coefficient appearing in the equation and this leads to a wide range of interesting behaviours. The singularities and asymptotic estimate of these solutions are discussed. At last, the superposition formulae for these solutions are also constructed.
An implicit fast Fourier transform method for integration of the time dependent Schrodinger equation
Riley, M.E.; Ritchie, A.B.
1997-12-31
One finds that the conventional exponentiated split operator procedure is subject to difficulties when solving the time-dependent Schrodinger equation for Coulombic systems. By rearranging the kinetic and potential energy terms in the temporal propagator of the finite difference equations, one can find a propagation algorithm for three dimensions that looks much like the Crank-Nicholson and alternating direction implicit methods for one- and two-space-dimensional partial differential equations. The authors report investigations of this novel implicit split operator procedure. The results look promising for a purely numerical approach to certain electron quantum mechanical problems. A charge exchange calculation is presented as an example of the power of the method.
Eigenvalue Expansion Approach to Study Bio-Heat Equation
NASA Astrophysics Data System (ADS)
Khanday, M. A.; Nazir, Khalid
2016-07-01
A mathematical model based on Pennes bio-heat equation was formulated to estimate temperature profiles at peripheral regions of human body. The heat processes due to diffusion, perfusion and metabolic pathways were considered to establish the second-order partial differential equation together with initial and boundary conditions. The model was solved using eigenvalue method and the numerical values of the physiological parameters were used to understand the thermal disturbance on the biological tissues. The results were illustrated at atmospheric temperatures TA = 10∘C and 20∘C.
Mrugalla, Florian; Kast, Stefan M
2016-09-01
Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute-solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute-solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems. PMID:27366935
NASA Astrophysics Data System (ADS)
Mrugalla, Florian; Kast, Stefan M.
2016-09-01
Complex formation between molecules in solution is the key process by which molecular interactions are translated into functional systems. These processes are governed by the binding or free energy of association which depends on both direct molecular interactions and the solvation contribution. A design goal frequently addressed in pharmaceutical sciences is the optimization of chemical properties of the complex partners in the sense of minimizing their binding free energy with respect to a change in chemical structure. Here, we demonstrate that liquid-state theory in the form of the solute–solute equation of the reference interaction site model provides all necessary information for such a task with high efficiency. In particular, computing derivatives of the potential of mean force (PMF), which defines the free-energy surface of complex formation, with respect to potential parameters can be viewed as a means to define a direction in chemical space toward better binders. We illustrate the methodology in the benchmark case of alkali ion binding to the crown ether 18-crown-6 in aqueous solution. In order to examine the validity of the underlying solute–solute theory, we first compare PMFs computed by different approaches, including explicit free-energy molecular dynamics simulations as a reference. Predictions of an optimally binding ion radius based on free-energy derivatives are then shown to yield consistent results for different ion parameter sets and to compare well with earlier, orders-of-magnitude more costly explicit simulation results. This proof-of-principle study, therefore, demonstrates the potential of liquid-state theory for molecular design problems.
Mann, Sarah L; Selby, Edward A; Bates, Marsha E; Contrada, Richard J
2015-10-01
High frequency heart rate variability (HRV) is a measure of neurocardiac communication thought to reflect predominantly parasympathetic cardiac regulation. Low HRV has been associated empirically with clinical and subclinical levels of anxiety and depression and, more recently, high levels of HRV have been associated with better performance on some measures of executive functioning (EF). These findings have offered support for theories proposing HRV as an index measure of a broad, self-regulatory capacity underlying aspects of emotion regulation and executive control. This study sought to test that proposition by using a structural equation modeling approach to examine the relationships of HRV to negative affect (NA) and EF in a large sample of U.S. adults ages 30s-80s. HRV was modeled as a predictor of an NA factor (self-reported trait anxiety and depression symptoms) and an EF factor (performance on three neuropsychological tests tapping facets of executive abilities). Alternative models also were tested to determine the utility of HRV for predicting NA and EF, with and without statistical control of demographic and health-related covariates. In the initial structural model, HRV showed a significant positive relationship to EF and a nonsignificant relationship to NA. In a covariate-adjusted model, HRV's associations with both constructs were nonsignificant. Age emerged as the only significant predictor of NA and EF in the final model, showing inverse relationships to both. Findings may reflect population and methodological differences from prior research; they also suggest refinements to the interpretations of earlier findings and theoretical claims regarding HRV. PMID:26168884
NASA Astrophysics Data System (ADS)
Tai, Ta-Sheng
2013-09-01
Integrable models in two dimensions are well-studied. Their appearance proved to be so universal in various kinds of topics including 2D conformal field theory, 3D Chern-Simons theory, to name a few. We present how 4D supersymmetric gauge theory also gets related to spin-chain models. More precisely, we study Baxter's T-Q equation of XXX spin-chain models under the semiclassical limit where an intriguing SU(N)/SU(2)N-3 correspondence emerges. That is, two kinds of 4D {N} = 2 superconformal field theories having the above different gauge groups are encoded simultaneously in one Baxter's T-Q equation which captures their spectral curves. For example, while one is SU(Nc) with Nf = 2Nc flavors the other turns out to be SU(2)Nc-3 with Nc hyper-multiplets (Nc > 3). It is seen that the corresponding Seiberg-Witten differential supports our proposal.
Nebraska Statewide Wind Integration Study: Executive Summary
EnerNex Corporation, Knoxville, Tennessee; Ventyx, Atlanta, Georgia; Nebraska Power Association, Lincoln, Nebraska
2010-03-01
Wind generation resources in Nebraska will play an increasingly important role in the environmental and energy security solutions for the state and the nation. In this context, the Nebraska Power Association conducted a state-wide wind integration study.
Solution and Study of the Two-Dimensional Nodal Neutron Transport Equation
Panta Pazos, Ruben; Biasotto Hauser, Eliete; Tullio de Vilhena, Marco
2002-07-01
In the last decade Vilhena and coworkers reported an analytical solution to the two-dimensional nodal discrete-ordinates approximations of the neutron transport equation in a convex domain. The key feature of these works was the application of the combined collocation method of the angular variable and nodal approach in the spatial variables. By nodal approach we mean the transverse integration of the SN equations. This procedure leads to a set of one-dimensional S{sub N} equations for the average angular fluxes in the variables x and y. These equations were solved by the old version of the LTS{sub N} method, which consists in the application of the Laplace transform to the set of nodal S{sub N} equations and solution of the resulting linear system by symbolic computation. It is important to recall that this procedure allow us to increase N the order of S{sub N} up to 16. To overcome this drawback we step forward performing a spectral painstaking analysis of the nodal S{sub N} equations for N up to 16 and we begin the convergence of the S{sub N} nodal equations defining an error for the angular flux and estimating the error in terms of the truncation error of the quadrature approximations of the integral term. Furthermore, we compare numerical results of this approach with those of other techniques used to solve the two-dimensional discrete approximations of the neutron transport equation. (authors)
Modeling for System Integration Studies (Presentation)
Orwig, K. D.
2012-05-01
This presentation describes some the data requirements needed for grid integration modeling and provides real-world examples of such data and its format. Renewable energy integration studies evaluate the operational impacts of variable generation. Transmission planning studies investigate where new transmission is needed to transfer energy from generation sources to load centers. Both use time-synchronized wind and solar energy production and load as inputs. Both examine high renewable energy penetration scenarios in the future.
Leading with integrity: a qualitative research study.
Storr, Loma
2004-01-01
This research paper gives an account of a study into the relationship between leadership and integrity. There is a critical analysis of the current literature for effective, successful and ethical leadership particularly, integrity. The purpose and aim of this paper is to build on the current notions of leadership within the literature, debate contemporary approaches, focussing specifically on practices within the UK National Health Service in the early 21st century. This leads to a discussion of the literature on ethical leadership theory, which includes public service values, ethical relationships and leading with integrity. A small study was undertaken consisting of 18 interviews with leaders and managers within a District General HospitaL Using the Repertory Grid technique and analysis 15 themes emerged from the constructs elicited, which were compared to the literature for leadership and integrity and other studies. As well as finding areas of overlap, a number of additional constructs were elicited which suggested that effective leadership correlates with integrity and the presence of integrity will improve organisational effectiveness. The study identified that perceptions of leadership character and behaviour are used to judge the effectiveness and integrity of a leader. However, the ethical implications and consequences of leaders' scope of power and influence such as policy and strategy are somewhat neglected and lacking in debate. The findings suggest that leaders are not judged according to the ethical nature of decision making, and leading and managing complex change but that the importance of integrity and ethical leadership correlated with higher levels of hierarchical status and that it is assumed by virtue of status and success that leaders lead with integrity. Finally, the findings of this study seem to suggest that nurse leadership capability is developing as a consequence of recent national investment. PMID:15588012
NASA Astrophysics Data System (ADS)
Plakhov, Iu. V.; Mytsenko, A. V.; Shel'Pov, V. A.
A numerical integration method is developed that is more accurate than Everhart's (1974) implicit single-sequence approach for integrating orbits. This method can be used to solve problems of space geodesy based on the use of highly precise laser observations.
Two-component integrable generalizations of Burgers equations with nondiagonal linearity
NASA Astrophysics Data System (ADS)
Talati, Daryoush; Turhan, Refi˙k.
2016-04-01
Two-component second- and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New integrable systems are obtained. Master symmetries of the obtained symmetry integrable systems, and bi-Poisson structures of those that also possess conservation laws, are given.
An efficient step-size control method in numerical integration for astrodynamical equations
NASA Astrophysics Data System (ADS)
Liu, C. Z.; Cui, D. X.
2002-11-01
Using the curvature of the integral curve, a step-size control method is introduced in this paper. This method will prove to be the efficient scheme in the sense that it saves computation time and improve accuracy of numerical integration.
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…
Teaching Linear Equations: Case Studies from Finland, Flanders and Hungary
ERIC Educational Resources Information Center
Andrews, Paul; Sayers, Judy
2012-01-01
In this paper we compare how three teachers, one from each of Finland, Flanders and Hungary, introduce linear equations to grade 8 students. Five successive lessons were videotaped and analysed qualitatively to determine how teachers, each of whom was defined against local criteria as effective, addressed various literature-derived…
Numerical Study of Fractional Ensemble Average Transport Equations
NASA Astrophysics Data System (ADS)
Kim, S.; Park, Y.; Gyeong, C. B.; Lee, O.
2014-12-01
In this presentation, a newly developed theory is applied to the case of stationary and non-stationary stochastic advective flow field, and a numerical solution method is presented for the resulting fractional Fokker-Planck equation (fFPE), which describes the evolution of the probability density function (PDF) of contaminant concentration. The derived fFPE is evaluated for three different form: 1) purely advective form, 2) second-order moment form and 3) second-order cumulant form. The Monte Carlo analysis of the fractional governing equation is then performed in a stochastic flow field, generated by a fractional Brownian motion for the stationary and non-stationary stochastic advection, in order to provide a benchmark for the results obtained from the fFPEs. When compared to the Monte Carlo simulation based PDFs and their ensemble average, the second-order cumulant form gives a good fit in terms of the shape and mode of the PDF of the contaminant concentration. Therefore, it is quite promising that the non-Fickian transport behavior can be modeled by the derived fractional ensemble average transport equations either by means of the long memory in the underlying stochastic flow, or by means of the time-space non-stationarity of the underlying stochastic flow, or by means of the time and space fractional derivatives of the transport equations.
Airframe Integration Trade Studies for a Reusable Launch Vehicle
NASA Technical Reports Server (NTRS)
Dorsey, John T.; Wu, Chauncey; Rivers, Kevin; Martin, Carl; Smith, Russell
1999-01-01
Future launch vehicles must be lightweight, fully reusable and easily maintained if low-cost access to space is to be achieved. The goal of achieving an economically viable Single-Stage-to-Orbit (SSTO) Reusable Launch Vehicle (RLV) is not easily achieved and success will depend to a large extent on having an integrated and optimized total system. A series of trade studies were performed to meet three objectives. First, to provide structural weights and parametric weight equations as inputs to configuration-level trade studies. Second, to identify, assess and quantify major weight drivers for the RLV airframe. Third, using information on major weight drivers, and considering the RLV as an integrated thermal structure (composed of thrust structures, tanks, thermal protection system, insulation and control surfaces), identify and assess new and innovative approaches or concepts that have the potential for either reducing airframe weight, improving operability, and/or reducing cost.
Giacometti, Achille; Gögelein, Christoph; Lado, Fred; Sciortino, Francesco; Ferrari, Silvano
2014-03-07
Building upon past work on the phase diagram of Janus fluids [F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev. Lett. 103, 237801 (2009)], we perform a detailed study of integral equation theory of the Kern-Frenkel potential with coverage that is tuned from the isotropic square-well fluid to the Janus limit. An improved algorithm for the reference hypernetted-chain (RHNC) equation for this problem is implemented that significantly extends the range of applicability of RHNC. Results for both structure and thermodynamics are presented and compared with numerical simulations. Unlike previous attempts, this algorithm is shown to be stable down to the Janus limit, thus paving the way for analyzing the frustration mechanism characteristic of the gas-liquid transition in the Janus system. The results are also compared with Barker-Henderson thermodynamic perturbation theory on the same model. We then discuss the pros and cons of both approaches within a unified treatment. On balance, RHNC integral equation theory, even with an isotropic hard-sphere reference system, is found to be a good compromise between accuracy of the results, computational effort, and uniform quality to tackle self-assembly processes in patchy colloids of complex nature. Further improvement in RHNC however clearly requires an anisotropic reference bridge function.
Giacometti, Achille; Gögelein, Christoph; Lado, Fred; Sciortino, Francesco; Ferrari, Silvano; Pastore, Giorgio
2014-03-01
Building upon past work on the phase diagram of Janus fluids [F. Sciortino, A. Giacometti, and G. Pastore, Phys. Rev. Lett. 103, 237801 (2009)], we perform a detailed study of integral equation theory of the Kern-Frenkel potential with coverage that is tuned from the isotropic square-well fluid to the Janus limit. An improved algorithm for the reference hypernetted-chain (RHNC) equation for this problem is implemented that significantly extends the range of applicability of RHNC. Results for both structure and thermodynamics are presented and compared with numerical simulations. Unlike previous attempts, this algorithm is shown to be stable down to the Janus limit, thus paving the way for analyzing the frustration mechanism characteristic of the gas-liquid transition in the Janus system. The results are also compared with Barker-Henderson thermodynamic perturbation theory on the same model. We then discuss the pros and cons of both approaches within a unified treatment. On balance, RHNC integral equation theory, even with an isotropic hard-sphere reference system, is found to be a good compromise between accuracy of the results, computational effort, and uniform quality to tackle self-assembly processes in patchy colloids of complex nature. Further improvement in RHNC however clearly requires an anisotropic reference bridge function. PMID:24606350
Integrated inversion using combined wave-equation tomography and full waveform inversion
NASA Astrophysics Data System (ADS)
Wang, Haiyang; Singh, Satish C.; Calandra, Henri
2014-07-01
Wave-equation tomography (WT) and full waveform inversion (FWI) are combined through a hybrid misfit function to estimate high-resolution subsurface structures starting from a poorly constrained initial velocity model. Both methods share the same wavefield forward modelling and inversion schemes, while they differ only on the ways to calculate misfit functions and hence the ways to sample in the model space. Aiming at minimizing the cross-correlation phase delay between synthetic and real data, WT can be used to retrieve the long- and middle-wavelength model components, which are essential to FWI. Compared to ray-based traveltime tomography that is based on asymptotic high-frequency approximation, WT provides a better resolution by exploring the band-limited feature of seismic wavefield. On the other hand, FWI is capable of resolving the short-wavelength model component, complementing the WT. In this study, we apply WT to surface first-arrival refraction data, and apply FWI to both refraction and reflection data. We assign adaptive weights to the two different misfit measurements and build a progressive inversion strategy. To illustrate the advantage of our strategy over conventional `ray tomography + FWI' approach, we show in a synthetic lens test that WT can provide extra subsurface information that is critical for a successful FWI application. To further show the efficiency, we test our strategy on the 2-D Marmousi model where satisfactory inversion results are achieved without much manual intervention. Finally, we apply the inversion strategy to a deep-water seismic data set acquired offshore Sumatra with a 12-km-long streamer. In order to alleviate several practical problems posed by the deep-water setting, we apply downward continuation (DC) to generate a virtual ocean bottom experiment data set prior to inversion. The new geometry after DC boosts up the shallow refractions, as well as avoiding cumbersome modelling through the thick water column, thus
Haghtalab, Mohammad; Faraji-Dana, Reza
2012-05-01
Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green's functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly. PMID:22561933
A new treatment of boltzmann-like collision integrals in nuclear kinetic equations
Lang, A.; Babovsky, H.; Cassing, W.; Mosel, U.; Reusch, H.G.; Weber, K. )
1993-06-01
The authors present a new method to solve the collision-integrals in BUU- type simulations of heavy-ion reactions and compare it to the two currently used full- and parallel-ensemble schemes. 25 refs., 8 figs.
Integrability and supersymmetry of Schrödinger-Pauli equations for neutral particles
NASA Astrophysics Data System (ADS)
Nikitin, A. G.
2012-12-01
Integrable quantum mechanical systems for neutral particles with spin 1/2 and nontrivial dipole momentum are classified. It is demonstrated that such systems give rise to new exactly solvable problems of quantum mechanics with clear physical content. Solutions for three of them are given in explicit form. The related symmetry algebras and superalgebras are discussed. The presented classification is restricted to two-dimensional systems, which admit matrix integrals of motion linear in momenta.
Integrability and supersymmetry of Schroedinger-Pauli equations for neutral particles
Nikitin, A. G.
2012-12-15
Integrable quantum mechanical systems for neutral particles with spin (1/2) and nontrivial dipole momentum are classified. It is demonstrated that such systems give rise to new exactly solvable problems of quantum mechanics with clear physical content. Solutions for three of them are given in explicit form. The related symmetry algebras and superalgebras are discussed. The presented classification is restricted to two-dimensional systems, which admit matrix integrals of motion linear in momenta.
The solution of the relaxation problem for the Boltzmann equation by the integral iteration method
NASA Technical Reports Server (NTRS)
Limar, Y. F.
1972-01-01
The Boltzmann equation is considered in terms of the problem of relaxation of some initial distribution function which depends only on velocities, to Maxwell's distribution function. The Boltzmann equation is given for the relaxation problem in which the distribution function f(t, u, v) is time dependent and is also dependent on two other variables u and v (the velocities of rigid spherical molecules). An iteration process is discussed in which the velocity space u, v is subdivided into squares, the distribution function in each square being approximated by the second-order surface from the values of the distribution function at nine points. The set of all of these points forms a network of u, v values at the nodes of which the distribution function can be found.
Fast methods to numerically integrate the Reynolds equation for gas fluid films
NASA Technical Reports Server (NTRS)
Dimofte, Florin
1992-01-01
The alternating direction implicit (ADI) method is adopted, modified, and applied to the Reynolds equation for thin, gas fluid films. An efficient code is developed to predict both the steady-state and dynamic performance of an aerodynamic journal bearing. An alternative approach is shown for hybrid journal gas bearings by using Liebmann's iterative solution (LIS) for elliptic partial differential equations. The results are compared with known design criteria from experimental data. The developed methods show good accuracy and very short computer running time in comparison with methods based on an inverting of a matrix. The computer codes need a small amount of memory and can be run on either personal computers or on mainframe systems.
New types of multisoliton solutions of some integrable equations via direct methods
NASA Astrophysics Data System (ADS)
Burde, Georgy I.
2016-06-01
Exact explicit solutions, which describe new multisoliton dynamics, have been identified for some KdV type equations using direct methods devised for this purpose. It is found that the equations, having multi-soliton solutions in terms of the KdV-type solitons, possess also an alternative set of multi-soliton solutions which include localized static structures that behave like (static) solitons when they collide with moving solitons. The alternative sets of solutions include the steady-state solution describing the static soliton itself and unsteady solutions describing mutual interactions in a system consisting of a static soliton and several moving solitons. As distinct from common multisoliton solutions those solutions represent combinations of algebraic and hyperbolic functions and cannot be obtained using the traditional methods of soliton theory.
NASA Astrophysics Data System (ADS)
Kovalchuk, Valery I.
2014-11-01
In this paper, a method has been developed to solve three-particle Faddeev equations in the configuration space making use of a series expansion in hyperspherical harmonics. The following parameters of the bound state of triton and helium-3 nuclei have been calculated: the binding energies, the weights of symmetric and mixed-symmetry components of the wave function, the magnetic moments, and the charge radii.
Monte Carlo studies of multiwavelength pyrometry using linearized equations
NASA Astrophysics Data System (ADS)
Gathers, G. R.
1992-03-01
Multiwavelength pyrometry has been advertised as giving significant improvement in precision by overdetermining the solution with extra wavelengths and using least squares methods. Hiernaut et al. [1] have described a six-wavelength pyrometer for measurements in the range 2000 to 5000 K. They use the Wien approximation and model the logarithm of the emissivity as a linear function of wavelength in order to produce linear equations. The present work examines the measurement errors associated with their technique.
Bottomonium in a Bethe-Salpeter-equation study
Blank, M.; Krassnigg, A.
2011-11-01
Using a well-established effective interaction in a rainbow-ladder truncation model of QCD, we fix the remaining model parameter to the bottomonium ground-state spectrum in a covariant Bethe-Salpeter equation approach and find surprisingly good agreement with the available experimental data including the 2{sup --} {Upsilon}(1D) state. Furthermore, we investigate the consequences of such a fit for charmonium and light-quark ground states.
Geodynamical studies using integrated gravity studies
NASA Astrophysics Data System (ADS)
Fernandez, J.; Tiampo, K. F.; Rundle, J. B.
2014-12-01
The expansion and proliferation of new data, at regional and global scales, over the past 30 years has allowed us to measure different geophysical signals (displacement, gravity, seismicity, etc.) with unprecedented spatial resolution and precision. Here we consider observations with both terrestrial and space origin, as well as new data obtained from the fusion of both types of observations. Advances in statistical geodynamics requires improved tools for data processing, fusion, modelling and interpretation in order to obtain the maximum value from these new, large data sets, in conjunction with the development of new applications. A clear example is the space gravimetry carried out using different satellites (e.g., GRACE and GOCE) which has allowed, using the available data, the development of combined gravity models such as GGMPlus (http://geodesy.curtin.edu.au/research/models/GGMplus/), with a spatial resolution of 200 m within ±60º geographic latitude. An additional example is the use of the gravity gradients determined by the GOCE satellite to estimate the stress field and its temporal variations at global scales. Here we will present a new research project aimed at providing estimates of gravity, strain and stress at varying spatial scales, integrated using advanced techniques for statistical data assimilation.