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.
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.
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
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.
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.
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.
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.
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.
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.
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 -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
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
A spectral boundary integral equation method for the 2-D Helmholtz equation
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.
The 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.
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.
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.
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.
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
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.
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.
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…
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.
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.
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
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.
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.
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…
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.
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
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.
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
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
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.
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.
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.
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.
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).
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.
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.
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
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.
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.
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
On the solution of integral equations with strong ly singular kernels
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1985-01-01
In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.
On the 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.
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.
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 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.
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.
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
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. PMID:24832290
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
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.
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
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
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
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.
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.
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.