Showing papers on "Integro-differential equation published in 1973"
TL;DR: In this article, the inverse scattering method was used to solve the initial value problem for a broad class of nonlinear evolution equations, including sine-Gordon, sinh-Gordon and Benney-Newell.
Abstract: We present the inverse scattering method which provides a means of solution of the initial-value problem for a broad class of nonlinear evolution equations. Special cases include the sine-Gordon equation, the sinh-Gordon equation, the Benney-Newell equation, the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, and generalizations.
925 citations
TL;DR: In this article, the numerical results from a computer solution of the time-dependent thin-wire electric-field integral equation described in Part I of this paper are validated by their Fourier transform of the frequency domain, where they are compared with independently computed data.
147 citations
TL;DR: In this article, it was shown that the existence of bounded solutions under weaker assumptions on the smoothness condition of a(t) and absolute continuity of f(i) was not required.
37 citations
TL;DR: In this paper, a semi-infinite strip held rigidly on its short end is considered, and the integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation.
Abstract: A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. The integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. The stress singularity at the strip corner is obtained from the singular integral equation, which is then solved numerically. Stresses along the rigid end are determined, and the effect of the material properties on the stress-intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.
33 citations
TL;DR: In this paper, it is shown that if the collective wave function occuring in the Griffin-Hill-Wheeler integral equation is slowly varying, the equation can be transformed into a Schrodinger equation.
Abstract: It is shown that if the collective wave function occuring in the Griffin-Hill-Wheeler integral equation is slowly varying, the equation can be transformed into a Schrodinger equation. The conditions under which this can be done, are studied in detail.
31 citations
27 citations
TL;DR: In this paper, the radial factor of a separable solution of the wave equation in Schwarzschild's space-time satisfies a second-order linear differential equation, and the behavior of the solutions near the singular points (the origin, the horizon, and infinity) of the equation is analyzed.
Abstract: The radial factor of a separable solution of the wave equation in Schwarzschild's space‐time satisfies a second‐order linear differential equation. This equation is studied in detail. The behavior of the solutions near the singular points (the origin, the horizon, and infinity) of the equation is analyzed. By an appropriate transformation two simpler differential equations are obtained corresponding to retarded and advanced solutions with characteristic asymptotic expansions. Their properties permit the expression of the general solution of the radial equation in terms of a single contour integral. Finally, through a ``matching'' technique, the behavior of a solution at the singular points is determined from its behavior at a single singular point.
25 citations
22 citations
01 Mar 1973
TL;DR: In this paper, a nonlinear stochastic integral equation of the Hammerstein type in the form x(t, c) = h(t; c) + f k(s, s; co)f (s, x(s; co)) dy (s) is studied where t E S, a v-finite measure space with certain properties, co E Q, the supporting set of a probability measure space (Q, A, P), and the integral is a Bochner integral.
Abstract: A nonlinear stochastic integral equation of the Hammerstein type in the form x(t; c) = h(t; co) + f k(t, s; co)f (s, x(s; co)) dy (s) is studied where t E S, a v-finite measure space with certain properties, co E Q, the supporting set of a probability measure space (Q, A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be a second order vector-valued stochastic process x(t; co) on S which satisfies the equation almost certainly. Using certain spaces of functions, which are spaces of second order vector-valued stochastic processes on S, and fixed point theory, several theorems are proved which give conditions such that a unique random solution exists.
16 citations
TL;DR: In this paper, a master equation for the diagonal part of the density matrix is derived in the basis of the irreducible representations of the symmetry group of the hamiltonian.
15 citations
TL;DR: In this paper, the Percus-Yevick Equation (PYE) was used to solve the problem of self-consistency of the liquid pressure in a system of hard spheres.
Abstract: CONTENTS1. Introduction 592 a) Problem of the Theory of the Liquid State 592 b) The Method of Integral Equations 593 2. The Percus-Yevick Equation 594 a) Functional Definition of the Direct Correlation Function 594 b) The Percus Approximation 595 c) Analytic Solution of the Percus-Yevick Equation for a System of Hard Spheres 597 3. Results of Numerical Calculations by the Integral Equation Method 599 a) Calculation of the RDF and of the Thermodynamic Characteristics 599 b) Determination of the Interparticle Potential 603 c) Problem of Self Consistency of the Liquid Pressure 604 d) Phase Transitions in the Percus-Yevick Theory 605 Bibliography 606
01 Jan 1973
TL;DR: In this article, it was shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition, which is essentially the Galerkin expansion of the solution to a given quasi-linear wave equation.
Abstract: In this paper it is shown that under specified conditions on the initial data a certain infinite coupled system of ordinary differential equations has a solution satisfying an auxiliary convergence condition. The infinite system discussed is essentially the Galerkin expansion of the solution to a given quasi-linear wave equation. The results obtained suffice to prove the existence of a solution to this wave equation.
TL;DR: In this paper, a fixed point theorem is obtained for an equation of the form u = T[p, / + GLuJ(t, /+ GLuJJ) for a hyperbolic equation in a Banach space.
Abstract: A fixed point theorem is obtained for an equation of the form u = T[p, / + GLuJJ. This theorem is then applied to a functionally perturbed ordinary differential equation of the form u'kt) f(t) + A(t, u(t)) + G[uJ(t); u(0)=p, and, as a consequence of this, Fredholm integrodifferential equations of the form z'(t) = /(t) + (s), u(0, t) =>/<«). These last two equations are set in a Banach space so as to allow applications to integrodifferential equations such as d2 / d â \\ iz(s, t, z) = f(s, t, z) + HI z, u(s, t, z), — u(s, t, z),— u(s, t, z)\\ dsdt \\ ds dt ) r\\ ( d d \\ , + I K[z, r, u(s, t, r),— u(s, t, r),— u(s, t, r)\\ dr, JO \\ ds dt / u(s, 0, z) m o-(s, z), u(0, t, z) = T(t, z). Presented in part to the Society, November 19, 1971 under the title Existence on prescribed rectangles for a hyperbolic equation in a Banach space; received by the editors January 18, 1972. AMS (MOS) subject classifications (1970). Primary 34D10, 45D05, 47H10; Secondary 35L60, 45K05, 34G05.
TL;DR: In this paper, an exact, low-density equation of motion for the generating operator G q (t) = exp (iLt) |j??? i | was derived for the case of foreign gas pressure broadening.
Journal Article•
TL;DR: In this paper, a nonlinear perturbed stochastic integral equation of the form where ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ) is studied.
Abstract: The object of this present paper is to study a nonlinear perturbed stochastic integral equation of the formwhere ω ∈ Ω, the supporting set of the complete probability measure space (Ω A, μ). We are concerned with the existence and uniqueness of a random solution to the above equation. A random solution, x(t; ω), of the above equation is defined to be a vector random variable which satisfies the equation μ almost everywhere.
01 Jan 1973
TL;DR: In this paper, the authors discussed contour integral solutions of the Laplace linear differential equation of order n. They showed how these solutions can be expressed in terms of confluent forms of Lauricella's hypergeometric function FD(n−1) of n−1 variables.
Abstract: This paper discusses certain contour integral solutions of the Laplace linear differential equation of order n. It is shown, to quote one of the observations made here, how these solutions can be expressed in terms of confluent forms of Lauricella's hypergeometric function FD(n−1) of n−1 variables.
TL;DR: In this paper, an approach for designing a recursive algorithm for finding the root or roots of the equation ǫ(x) = 0, where ǒ is a continuously differentiable function from Rn → Rn, is presented.
Abstract: An approach is presented for designing a recursive algorithm for finding the root or roots of the equation ƒ(x) = 0, where ƒ is a continuously differentiable function from Rn → Rn . A differential equation, or difference equation, is written with ƒ as the dependent variable and a control vector u as the forcing function. Optimal control theory is then used to drive ƒ from a given initial value to zero.
TL;DR: In this article, a projection operator is defined in terms of three conditions, which are postulated as being necessary and sufficient conditions for the existence of an exact markovian equation.
TL;DR: In this article, a new integral method for solving the Laplace equation is suggested, where the differential problem is transferred into an integral equation with a kernel that defines a positive-definite operator and the solution is given as the limit of a converging sequence of approximations.
Abstract: So far, numerical methods for solving the Laplace equation have consisted mainly of either a finite-difference or a variational finite-element scheme. The variational approach, however, always dealt with a positive-definite differential operator −∇2. In this letter, a new integral method for solution is suggested. The differential problem is transferred into an integral equation with a kernel that defines a positive-definite operator. Then, by the Ritz method,1 the solution is given as the limit of a converging sequence of approximations. Two examples are presented: a boundary-value problem for which there is an exact solution, and the square parallel-plate capacitor fundamental in the study of microstrip-line propagation.
TL;DR: An integral equation with two variables, depending on time and location on the section was introduced for obtaining the source density distributions over the immersed surface of the cylinder by using the time dependent Green's function.
Abstract: When a cylinder floating on the free surface starts an arbitrary oscillation, the flow field around the body should change in complicated manner. This paper deals with a wave making theory of the cylinder at the early stage of the oscillation.An integral equation with two variables, depending on time and location on the section was introduced for obtaining the source density distributions over the immersed surface of the cylinder by using the time dependent Green's function.The kernel function of the integral equation is not suitable to integrate itself numericallly in direct, when the time variables become large. In order to elude the difficalty of the integration for that case, the function was expanded to an appropriate asymptotic series.Then, the numerical solution of the integral equation was obtained. However, it becomes clear that a reasonable solution is hardly obtained, as the integral equation involves an eigen function. Only an approximate solution was obtained in this paper. Further investigation is necessary to introduce an integral equation without eigen function.
TL;DR: First integrals of the nonlinear parabolic equation t_0 are considered, i.e. functionals that are constant on solutions of (1): every first integral satisfies a first-order variational differential equation.
Abstract: First integrals of the nonlinear parabolic equation t_0, \qquad$ SRC=http://ejioporg/images/0025-5734/21/3/A01/tex_sm_2021_img1gif/>(1)are considered, ie functionals that are constant on solutions of (1): Every first integral satisfies a first-order variational differential equation A solution of the Cauchy problem is constructed for this equation The method of constructing these solutions, ie first integrals, affords a number of corollaries concerning statistical characteristics of solutions of (1)Bibliography : 4 items
TL;DR: In this paper, the Krook kinetic equation is used as the governing equation in an investigation of a class of exterior flow problems involving flow past an elliptic cylinder, and several sets of solutions are presented.
Abstract: The Krook kinetic equation is used as the governing equation in an investigation of a class of exterior flow problems involving flow past an elliptic cylinder. A numerical method for solving the integral equation form of the Krook equation for this class of problems is described, and several sets of solutions are presented.
TL;DR: In this paper, the generalized Bethe-Goldstone equation from the es-tlieory is used to perform a calculation in the harmonic oscillator model, taking into account hole-hole diagrams and eliminating the model-dependent particle energies.
Abstract: The generalized Bethe-Goldstone equation from the es-tlieory is used to perform a calculation in the harmonic oscillator model. It takes into account hole-hole diagrams and eliminates the model-dependent particle energies. Both effects tend to increase the binding energy and to decrease the nuclear radius.