Showing papers on "Integro-differential equation published in 1987"
TL;DR: In this article, the first-passage-time p.d. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions.
Abstract: The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein-Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.
217 citations
TL;DR: In this article, an extension of the dual reciprocity boundary element method (DRBEM) to nonlinear diffusion problems is presented, in which thermal conductivity, specific heat, and density coefficients are all functions of temperature.
Abstract: This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.
182 citations
TL;DR: In this article, an averaging for the nonlinear Schroedinger equation using the technique of finite-gap averaging is investigated, and the results for the single-gap case are given explicitly.
Abstract: An averaging is investigated for the nonlinear Schroedinger equation using the technique of finite-gap averaging. For the single-gap case, the results are given explicitly. Some characteristics of the original equation needed for applied calculations are averaged. Finally, recursion and functional formulas connecting the densities of the integrals of the motion of the Schroedinger equation, the fluxes, and the variational derivatives are given.
90 citations
TL;DR: In this article, the Miura transformation between KdV and MKdV solutions is generalized to the two-dimensional case and an integrable equation associated with the twodimensional Dirac operator -the modified Veselov-Novikov equation -is introduced.
Abstract: The Miura transformation between KdV and MKdV solutions is generalized to the two-dimensional case. An integrable equation associated with the two-dimensional Dirac operator - the modified Veselov-Novikov equation - is introduced.
80 citations
TL;DR: In this article, the authors established the uniform convergence and obtained convergence rates for several algorithms for solving a class of Hadamard singular integral equations, and improved on the mean square convergence shown in [2].
73 citations
TL;DR: In this paper, the authors proposed a singular equation approach for crack problems in plane elasticity, where the right-hand term of the integral equation is changed from tractions to resultant forces.
52 citations
TL;DR: In this paper, the asymptotic expansion of the Korteweg-de Vries-Burgers equation is presented, and the expansion is shown to be linear in the number of Ks.
Abstract: The asymptotic expansion of the Korteweg-de Vries-Burgers equation is presented in this paper.
33 citations
27 citations
TL;DR: In this paper, a group-theoretical scheme for constructing higher stationary Landau-Lifshitz equations in the form of Euler equations on the orbits of the algebra of su(2)-valued functions on an elliptic curve was developed.
Abstract: A group-theoretical scheme is developed for constructing higher stationary Landau-Lifshitz equations in the form of Euler equations on the orbits of the algebra of su(2)-valued functions on an elliptic curve. Duality between the Landau-Lifshitz equation and the equation for the currents of an asymmetric chiral O(3) field is established.
26 citations
TL;DR: In this paper, a simple product integration scheme is proposed to solve a nonlinear Volterra integro-differential equation with a weakly singular kernel and a non-smooth solution.
25 citations
TL;DR: In this article, the large-time behavior of solutions of the one-dimensional discrete Boltzmann equation is studied under suitable assumptions, and it is proved that as time tends to infinity, the solution approaches a function which is constructed explicitly in terms of the self-similar solution of the Burgers equation and the linear heat equation.
Abstract: Large-time behavior of solutions of the one-dimensional discrete Boltzmann equation is studied. Under suitable assumptions it is proved that as time tends to infinity, the solution approaches a function which is constructed explicitly in terms of the self-similar solutions of the Burgers equation and the linear heat equation.
IBM1
TL;DR: In this article, the two-electron integrals over Cartesian Gaussian functions occurring in the Hylleraas-CI method were extended to three and four-electrons.
Abstract: In a previous paper, formulas for two-electron integrals over Cartesian Gaussian functions occurring in the Hylleraas-CI method have been given. This paper reports on the extension of those formula to three- and four-electron integrals and presents test results obtained by using the two-electron integrals for H2 With only 2 p-type polarization functions on each atom, the ground state energy calculated for H2 is only about 20 cm−1 higher than the exact value.
TL;DR: In this paper, the problem of diffusion and reaction on a porous nonisothermal finite cylindrical catalyst pellet in the presence of external heat and mass-transfer resistances is solved by the integral equation method.
TL;DR: In this article, the classical non-linear Boltzmann equation is solved using random quadrature formulas, and the solution of the set of finite-difference equations is obtained by one of the iteration methods.
Abstract: Methods of solving the classical non-linear Boltzmann equation, based on the discrete approximation of the distribution function in a phase space, the finite-difference approximation of a transfer operator and the calculation of collision integrals using random quadrature formulas, are considered. The solution of the set of finite-difference equations is obtained by one of the iteration methods.
TL;DR: In recent years, there has been a revival of interest in the method of integral equations for the solution of boundary value problems as mentioned in this paper, and the boundary element method has become an important tool for the computational solution of partial differential equations.
Abstract: In recent years there has been a revival of interest in the method of integral equations for the solution of boundary value problems. On the theoretical side, classical results have been extended by relaxing assumptions on the smoothness of the boundary, and in applications the boundary element method has become an important tool for the computational solution of partial differential equations. Both these developments are obviously of great interest to numerical analysts, especially since the integral equation formulations lead to numerical schemes which are completely independent of finite difference and finite element methods.
TL;DR: In this article, the authors considered the equation of motion for a complex classical field in a (3+1)-dimensional 6 model, and they used the results of the symmetry reduction method to exactly solve both the carrier equations and the envelope equation.
Abstract: The authors consider the equation of motion for a complex classical field in a (3+1)-dimensional 6 model. The resultant complex non-linear Klein-Gordon equation is solved using an ansatz in which the envelope satisfies a scalar non-linear Klein-Gordon equation and the carrier satisfies either a wave equation or Laplace's equation with additional constraints. They use the results of the symmetry reduction method to exactly solve both the carrier equations and the envelope equation. The latter has recently been analysed and here they only briefly discuss the types of the appropriate solutions. However, they present a detailed discussion on one particular solution which is physically important. It bifurcates both in real space and in phase space. Possible physical applications have been outlined in the last section and they include superfluidity, superconductivity, liquid crystals and helicoidal metamagnets.
01 Jan 1987
TL;DR: In this article, the Schrodinger equation satisfied by the square root of the electron density is derived without approximation from the theory of marginal and conditional amplitudes, which is an exact dynamical model for computing effective one-electron potentials from known N −electron wavefunctions or from experimental electron densities.
Abstract: The Schrodinger equation satisfied by the square root of the electron density is derived without approximation from the theory of marginal and conditional amplitudes. The equation arises from a factorization of the total N—electron wavefunction defined by the normalisation appropriate to a conditional amplitude. The effective potential is, in contrast to that in the one—electron Dyson equation, a scalar, local potential that only depends upon one stationary state of the N—electron system. The equation is easily transformed into an exact differential equation for the electron density itself; the transformation leaves the potential energy and the energy eigenvalue unchanged. In this exact equation the kinetic energy is identical with the kinetic energy in the Thomas-Fermi-Weizsacker differential equation. The exact equation applies to excited states as well as to the ground state, thus extending the Hohenberg-Kohn density-functional theorem to excited states. The exact equation provides a basis for self-consistent-density calculations. This Schrodinger equation is an exact dynamical model for computing effective one-electron potentials from known N—electron wavefunctions, or from experimental electron densities. Two forms of the theory are presented: 1) the static nuclei model pertinent to theoretical calculations within the Born-Oppenheimer separation, and 2) the non-Born-Oppenheimer (vibrationally averaged) model appropriate to computation of the effective one-electron potential from experimental electron densities.
TL;DR: In this article, a Lienard-type model equation describing stationary nonlinear wave motion in an active medium is investigated, which is able to describe both a threshold effect and the possibility of amplification or attenuation.
Abstract: A Lienard-type model equation describing stationary nonlinear wave motion in an active medium is investigated. This model equation has been derived from the evolution equation governing pulse transmission in a nerve fibre. Two models (the classical third-order FitzHugh-Nagumo equation and the model second-order equation) are compared and analysed. The possibility of calculating a single pulse in the whole `time' range, without any convergence problems, offers a certain advantage to a Lienard-type model equation. In physical terms, this equation is able to describe both a threshold effect and the possibility of amplification or attenuation. A qualitative analysis of the phase portrait of this model is presented. It is proved that limit cycles in the phase plane cannot exist under the conditions arising from the physics of the nerve fibre. The threshold problem is analysed in detail and an algorithm is presented to find the threshold distinguishing the processes of amplification and attenuation. The results obtained in this work permit a full map of the solutions to be given for a general Lienard-type equation. The model equation under consideration describes a single pulse, but the full map of the solutions also contains periodic solutions corresponding to the well-known Van der Pol equation.
TL;DR: The moment equation with different wavenumbers and different transverse coordinates for wave propagation in a random medium is a linear differential equation as mentioned in this paper, and the solution of the moment equation is a Dyson expansion.
Abstract: The moment equation with different wavenumbers and different transverse coordinates for wave propagation in a random medium is a linear differential equation. It often appears in the study of problems related to wave propagation in a random medium. The differential equation can be converted into an integral equation by using Green's functions and the integral equation can be solved by iteration. The moment equation is solved by the method of successive scatters, too. The solution of the moment equation is a Dyson expansion. The physical implication of the successive solution of the moment equation with different wavenumbers is explained.
TL;DR: In this article, an exact three-point integration rule is derived from the integral equation equivalent to the one-dimensional Schrodinger equation with the boundary conditions appropriate for a bound state problem, and the approximate evaluation of this rule by means of the Euler-McLaurin sum rule gives rise to various O(h2) and O (h4) integration methods associated with the different ways of splitting the starting equation into a free part and an interaction term.
TL;DR: DeSanto et al. as discussed by the authors proposed a method to transform numerical solutions of the standard parabolic equation of ocean acoustics into the solution of the Helmholtz equation.
Abstract: A method is presented for transforming numerical solutions of the standard parabolic equation of ocean acoustics into solutions of the Helmholtz equation. The method is based on an established integral transform [J. A. DeSanto, J. Acoust. Soc. Am. 62, 295–297 (1977)] that is exact for range‐independent media. The Fourier–Bessel transform (with respect to range) of the Helmholtz equation is shown to be related to the Fourier transform of the parabolic equation by means of a nonlinear mapping between horizontal wavenumbers. As a result, fast field program (FFP) techniques can be applied to the Fourier transform of the solution to the standard parabolic equation to obtain an approximate solution to the Helmholtz equation. Several numerical examples are presented to illustrate this postprocessing approach to correcting the phase errors inherent in standard parabolic equation predictions.
TL;DR: An integro-differential equation model of tree height growth is developed, together with a biological interpretation of its coefficients, and the model fit the data well and was montonically increasing with an upper asymptote, although several other curve forms are possible.
Abstract: An integro-differential equation model of tree height growth is developed, together with a biological interpretation of its coefficients. The integro-differential equation is reduced to a second order linear differential equation with constraints on its initial conditions. Because of the constraints, fitting of the differential equation is best accomplished using a multipoint boundary value approach. An example using stem analysis data is presented. The model fit the data well and was montonically increasing with an upper asymptote, although several other curve forms are possible.
TL;DR: In this paper, it was shown that the resolvent kernel of a Volterra integrodifferential equation in Hilbert space is integrable on (0, oo).
Abstract: It is proved that the resolvent kernel of a certain Volterra integrodifferential equation in Hilbert space is absolutely integrable on (0, oo).Weaker assumptions on the convolution kernel appearing in the integral term are used than in existing results. The equation arises in the linear theory of isotropic viscoelastic rods and plates.
TL;DR: In this article, a quadrature formula for the evaluation of singular integrals is obtained, which has the same step length h as the formula for regular integrals derived by F. Stenger.
Abstract: Using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained. The formula has the same step length h as the formula for regular integrals derived by F. Stenger. These quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity.
Using these formulae the solution of singular integral equations does not depend on the order of the end point singularities. Furthermore the collocation points are given by a very simple equation and, in the case of constant coefficients, by a closed-form formula.
TL;DR: In this paper, strong and classical solutions of the Hopf equation, a statistical version of the Navier-Stokes equation on a compact Riemannian manifold with or without boundary, were constructed.
Abstract: In this note, we construct, strong and classical solutions of the Hopf equation, a statistical version of the Navier-Stokes equation on a compact Riemannian manifold with or without boundary. Our points are to regard the Hopf equation as a given Functional Derivative Equation (F.D.E. for short) of second order, to derive the Navier-Stokes equation as the characteristic equation of it and to give an exact meaning to the 'trace* of the second order functional derivatives which appear in the Hopf equation. To construct a solution of the Hopf-Foias equation with the energy in- equality of strong form, we apply Foia§'s argument with slight modifica- tions instead of using Prokhorov's compactness argument.
01 Jan 1987
TL;DR: In this paper, it was shown that a new two-variable integro-differential equation for an A-boson bound system which takes all two-body correlations exactly into account and is exact for A = 3 and S-state projected potentials, produces 3 and 4-body binding energies in excellent agreement with the best results in the literature.
Abstract: It is shown that a new two-variable integro-differential equation for an A-boson bound system which takes all two-body correlations exactly into account and is exact for A = 3 and S-state projected potentials, produces 3- and 4-body binding energies in excellent agreement with the best results in the literature. The adiabatic approximation to this equation provides close upper and lower bounds and describes three-body scattering.
01 Jan 1987
TL;DR: In this paper, the transonic integro-differential equation for two-dimensional flows is solved by boundary element methods, in addition to constant and quadrilateral elements, and hybrid elements based on constant elements in the streamwise direction and variable elements in transverse direction.
Abstract: Abstract The transonic integro-differential equation for two-dimensional flows is solved by boundary element methods. In addition to constant and quadrilateral elements we develop hybrid elements based on constant elements in the streamwise direction and variable elements in the transverse direction. Computation is carried out for parabolic-arc and NACA0012 airfoils and the results, which converge fast, compare favourably with finite-difference solutions. The hybrid elements are to be preferred because they yield results which are more accurate than constant elements without the computational complexity associated with quadrilateral elements. Moreover, they can be applied with a small number of nodes by using only one strip of rectangular elements.
TL;DR: Evaluated surface integrals in the same one-dimensional three-body model employed recently to demonstrate the nonuniqueness explicitly confirm the conclusion that the many-particle Lippmann-Schwinger equation has nonunique solutions in actual three-dimensional collisions, and support the belief that the derivation of the real energy Lippman-SchWinger equation is mathematically sound.
Abstract: At real energies E, the derivation of the Lippmann-Schwinger integral equation from the Schroedinger equation involves various surface integrals at infinity in configuration space. Plausible assumptions about the values of these surface integrals made originally by Gerjuoy imply that the many-particle (n>2) Lippmann-Schwinger equation generally has nonunique solutions. This paper evaluates these surface integrals in the same one-dimensional three-body model (of McGuire) employed recently to demonstrate the nonuniqueness explicitly. The computed values of the surface integrals agree precisely with Gerjuoy's hypotheses. These results further confirm the conclusion that the many-particle Lippmann-Schwinger equation has nonunique solutions in actual three-dimensional collisions, and support the belief that the aforesaid derivation of the real energy Lippmann-Schwinger equation is mathematically sound.