Showing papers on "Integro-differential equation published in 1974"
TL;DR: In this paper, a method to find N-soliton solutions of the K.d.V.like equation is presented, a method which can be also applicable to the so-called Schrodinger equation, which belongs to another series than the class of equations solvable bv inverse scattering method.
Abstract: To find N-soliton solutions of the K.d.V. equation, a method which can be also applicable to the so-called K.d.V.-like equation is presented. This equation belongs to another series than the class of equations solvable bv inverse scattering method for Schrodinger equation.
483 citations
TL;DR: In this paper, an integral equation method is used to derive the electromagnetic response of a three-dimensional heterogeneity in a 3-layer medium, which consists of replacing the heterogeneity with point dipole scattering currents.
Abstract: Summary
An integral equation method is used to derive the electromagnetic response of a three-dimensional heterogeneity in a three-layer medium The method consists of replacing the heterogeneity with point dipole scattering currents The kernel of the integral equation is a tensor Green's function which is derived for the three-layer case It is shown how the use of iteration methods and Simpson's rule integration might significantly reduce computing requirements The problem of initial estimates is discussed The method is extended to include the case of a simultaneous conductivity and permeability anomaly
176 citations
TL;DR: The Boltzmann equation is formulated in a hydrodynamic representation (modified Eulerian) in terms of velocities relative to an accelerated medium as discussed by the authors, where the target particles are treated as fixed in moving hydrodynamics zones, or cells.
Abstract: The Boltzmann equation is formulated in a hydrodynamic representation (modified Eulerian) in terms of velocities relative to an accelerated medium. Treating the target particles as fixed in moving hydrodynamic zones, or cells, and employing a representation in relative velocities gives a well‐defined cross section and permits use of the standard set of velocities and cross sections for computational purposes. Specific one‐dimensional forms for the transport equation are also given in plane and spherical geometries.
30 citations
TL;DR: In this paper, an integro-differential equation (IDE) on a finite closed interval is studied, and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides.
Abstract: In this paper we study an ordinary second-order integro-differential equation (IDE) on a finite closed interval. We demonstrate the equivalence of this equation to a certain integral equation, and deduce that the homogeneous IDE may have either 2 or 3 linearly independent solutions, depending on the value of a parameter λ. We study a Cauchy problem for the IDE, both by this integral equation approach and by an independent approach, based on the perturbation theory for linear operators. We give necessary and sufficient conditions for the Cauchy problem to be solvable for arbitrary right-hand sides—these conditions again depend on λ—and specify the behaviour of the IDE when these conditions are not satisfied. At the end of the paper some examples are given of the type of behaviour described.
13 citations
TL;DR: In this article, a new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed, where the original nonlinear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order.
Abstract: A new method of analytic solution of the Percus-Yevick equation for the radial distribution functiong(r) of hard-sphere fluid is proposed. The original non-linear integral equation is reduced to non-homogeneous linear integral equation of Volterra's type of the second order. The kernel of this new equation has a polynomial form which allows to find analytic expression forg(r) itself without using the Laplace transformation. In addition, the first three moments of the total correlation function can be found.
13 citations
01 Jan 1974
TL;DR: In this article, an integral equation method is presented for solving the standard boundary value problem for generalized analytic functions, which combines a Fredholm equation of the second kind in the domain with Fredholm equations of the first and second kind on the boundary for single boundary and surface layers.
Abstract: An integral equation method is presented for solving the standard boundary value problem for generalized analytic functions. The method combines a Fredholm equation of the second kind in the domain with Fredholm equations of the first and the second kind on the boundary for single boundary and surface layers. An approximation method for solving these equations is prescribed. The method provides an application for solving semilinear problems by an imbedding method combined with Newton's approximation. For the standard problem a few numerical results are given in the appendix.
12 citations
10 citations
9 citations
TL;DR: In this paper, it was proved that the Cauchy problem for the Navier-Stokes system has a unique solution that is analytic in and defined in a neighbourhood of zero of the corresponding function space.
Abstract: First integrals are constructed for non-linear parabolic systems (in the sense of Petrovskii) of differential equations with periodic boundary conditions; these are functionals taking a constant value with respect to on any solution of the original system: . First integrals are looked for as solutions of a certain first order partial differential equation in infinitely many variables. It is proved that the Cauchy problem for this equation in the case of analytic initial values has a unique solution that is analytic in and defined in a neighbourhood of zero of the corresponding function space. The result is used for the construction of moment functions and the characteristic functional of a statistical solution of the original parabolic system. All the results of this article are valid also for the Navier-Stokes system.
8 citations
TL;DR: In this article, a new way of deducing the Boltzmann transport equation from the Liouville equation is presented, which provides for generalization by means of less restrictive assumptions.
Abstract: A new way of deducing the Boltzmann transport equation from the Liouville equation is presented. The major advantage of the new method is that it provides for generalization by means of less restrictive assumptions.
7 citations
TL;DR: In this paper, a new method for evaluating the path integral corresponding to the harmonic oscillator with time-dependent frequency Ω(t) and acted on by a timedependent perturbative force is given.
TL;DR: In this paper, it was shown that the BBGKY hierarchy equation can rigorously be transformed into a closed non-markoffian equation for f. The latter equation contains infinite sets of collision and initial correlation terms which are most conveniently represented by connected diagrams and which can be expressed in terms of f and initial (arbitrarily given) correlation functions.
TL;DR: In this paper, Liouville equation and projection operator techniques were used to obtain a compact equation for the rate of change of then-particle momentum distribution function to any order in the density.
Abstract: Starting from the Liouville equation and making use of projection operator techniques we obtain a compact equation for the rate of change of then-particle momentum distribution function to any order in the density. This equation is exact in the thermodynamic limit. The terms up to second order in the density are studied and expressions are given for the errors committed when one makes the usual hypothesis to derive generalized Boltzmann equations. Finally the Choh-Uhlenbeck operator is obtained under additional assumptions.
TL;DR: In this article, it was pointed out that integrals over arbitrary ranges and indefinite integrals may often be obtained very simply by the methods of contour integration, and it was shown that contour integrals can be obtained easily by the method of contours integration.
Abstract: It is pointed out that integrals over arbitrary ranges and indefinite integrals may often be obtained very simply by the methods of contour integration.
TL;DR: In this article, the exact N-particle kinetic equation for a classical gas with central-force interactions, previously obtained from the Liouville equation by projection-operator methods, was obtained.
TL;DR: DigiZeitschriften e.V. as mentioned in this paper gewährt ein nicht exklusives, nicht übertragbares, persönliches and beschränktes Recht auf Nutzung dieses Dokuments.
Abstract: DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht kommerziellen Gebrauch bestimmt. Das Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhabern. Als Nutzer sind Sie sind nicht dazu berechtigt, eine Lizenz zu übertragen, zu transferieren oder an Dritte weiter zu geben. Die Nutzung stellt keine Übertragung des Eigentumsrechts an diesem Dokument dar und gilt vorbehaltlich der folgenden Einschränkungen: Sie müssen auf sämtlichen Kopien dieses Dokuments alle Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen Schutz beibehalten; und Sie dürfen dieses Dokument nicht in irgend einer Weise abändern, noch dürfen Sie dieses Dokument für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder anderweitig nutzen; es sei denn, es liegt Ihnen eine schriftliche Genehmigung von DigiZeitschriften e.V. und vom Herausgeber oder sonstigen Rechteinhaber vor. Mit dem Gebrauch von DigiZeitschriften e.V. und der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an.
TL;DR: In this article, a hyperbolic equation whose discontinuity waves are all exceptional and propagate with velocity λ was obtained, and it was shown that λ is related to the dispersion relationE(p).
Abstract: We obtain a hyperbolic equation whose discontinuity waves are all exceptional and propagate with velocity λ. When λ → ∞ or λ=c, this equation becomes identical to the Schrodinger equation and to the Klein-Gordon equation respectively. We also show that λ is related to the dispersion relationE(p).
TL;DR: In this article, the cooperative problem for a lattice gas on a plane, square lattice and on a simple cubic lattice is solved by a system of two coupled, transcendental equations, derived by a combinatorial method, which describes a homogeneous or periodical particle density on the lattice as a function of the temperature and the chemical potential of the gas.
Abstract: The cooperative problem for a lattice gas on a plane, square lattice and on a simple cubic lattice is solved by a system of two coupled, transcendental equations, derived by a combinatorial method, which describes a homogeneous or periodical particle density on the lattice as a function of the temperature and the chemical potential of the lattice-gas. For the particle interaction a Hard-Core potential (nearest neighbour exclusion) with a soft long-range tail is assumed. The zero-component of the Fourier-transform of this long-range interaction part can be positive or negative. The system of transcendental equations is solved by a graphic method. As a result, the complete pressure-density state diagram and the pressure-temperature phase diagram can be drawn. The lattice-gas exists in three stable phases: gas, liquid and solid. Three phase changes are possible: condensation, crystallization and sublimation. Critical points of condensation and freezing are examined. The number of possible phases and phase changes at a fixed temperature depends on the geometric structure of the particle interaction.
TL;DR: DigiZeitschriften e.V. as discussed by the authors gewährt ein nicht exklusives, nicht übertragbares, persönliches and beschränktes Recht auf Nutzung dieses Dokuments.
Abstract: DigiZeitschriften e.V. gewährt ein nicht exklusives, nicht übertragbares, persönliches und beschränktes Recht auf Nutzung dieses Dokuments. Dieses Dokument ist ausschließlich für den persönlichen, nicht kommerziellen Gebrauch bestimmt. Das Copyright bleibt bei den Herausgebern oder sonstigen Rechteinhabern. Als Nutzer sind Sie sind nicht dazu berechtigt, eine Lizenz zu übertragen, zu transferieren oder an Dritte weiter zu geben. Die Nutzung stellt keine Übertragung des Eigentumsrechts an diesem Dokument dar und gilt vorbehaltlich der folgenden Einschränkungen: Sie müssen auf sämtlichen Kopien dieses Dokuments alle Urheberrechtshinweise und sonstigen Hinweise auf gesetzlichen Schutz beibehalten; und Sie dürfen dieses Dokument nicht in irgend einer Weise abändern, noch dürfen Sie dieses Dokument für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, aufführen, vertreiben oder anderweitig nutzen; es sei denn, es liegt Ihnen eine schriftliche Genehmigung von DigiZeitschriften e.V. und vom Herausgeber oder sonstigen Rechteinhaber vor. Mit dem Gebrauch von DigiZeitschriften e.V. und der Verwendung dieses Dokuments erkennen Sie die Nutzungsbedingungen an.
TL;DR: In this article, it was shown that the Riccati equation can be integrated by parts so that, for any local potential, the integrand decreases as the cyclic folding procedure is applied.
Abstract: The radial one-electron Schrodinger equation can be written as a nonlinear first-order differential equation by making a suitable logarithmic transformation. The resulting Riccati equation has the equivalent Hammerstein integral representation [1],
where the kernel, N(r, r′) is
and H(r, r′) is the Heaviside unit step function. This kernel is a more general one than that developed in ref. [1]. Both kernels apply in cases where the Riccati equation corresponds to a Sturm–Liouville problem.
It is shown that this integral equation can be integrated by parts so that, for any local potential, the integrand decreases as the cyclic folding procedure is applied. During this cyclic folding, the kernel generates an equation that contains only coefficients of β(r)0 and β(r)1. Consequently, after truncating at the end of the nth cycle, it is possible to write down a Pade-type approximation to the logarithmic derivative as a known function of the independent variable. All coefficients in this approximation can be evaluated as simple algebraic formulations of P(r), R(r), and integrals over P(r).
TL;DR: In this article, the authors discussed the consequences of the modification of Kubo's stochastic Liouville equation, which the authors proposed in a recent publication, and showed that these calculations are actually based on the assumption of the modified equation and that the extension to saturated spectra can now readily be established.
01 Jan 1974
TL;DR: The chapter describes an integral that is found by multiplying together Sommerfeld's contour integrals for the Hankel functions that is uniformly valid in a closed α interval containing α = 0.
Abstract: This chapter provides an overview of integrals. It has been observed that by admitting sectionally continuous behavior in f ( v ), the integrals of the same form can be treated over any finite interval (0, k ). The motivating idea of Laplace is applied to integrals in which the large parameter x enters in a more general way. The chapter describes an integral that is found by multiplying together Sommerfeld's contour integrals for the Hankel functions. The chapter presents a method that yields a generalized asymptotic expansion for I (α, x ) that is uniformly valid in a closed α interval containing α = 0.
TL;DR: In this article, the problem of determining a function from a knowledge of integrals of the function along families of curves with a known weight function is reduced to the solution of an integrodifferential equation.
Abstract: We consider the problem of determining a function from a knowledge of integrals of the function along families of curves with a known weight function. For sufficiently general assumptions on the family of curves and on the weight function the problem is reduced to the solution of an integrodifferential equation. We establish the uniqueness of the solution of this equation in certain classes of functions.