Showing papers on "Integro-differential equation published in 1976"
TL;DR: In this article, a model kinetic equation for a gas with rotational degrees of freedom is obtained by averaging of the distribution function over quantities corresponding to the rotational degree of freedom, which is then reduced to a closed system of two kinetic equations.
Abstract: A model kinetic equation for a gas with rotational degrees of freedom is obtained. By averaging of the distribution function over quantities corresponding to the rotational degrees of freedom this equation is reduced to a closed system of two kinetic equations, each of which is analogous to the kinetic equation of a monatomic gas.
130 citations
TL;DR: In this article, the eigenvalues of the Helmholtz equation Δ2o + k2o = 0 in a two-dimensional area when o vanishes on the boundary are calculated.
Abstract: A method is presented to calculate the eigenvalues of the Helmholtz equation Δ2o + k2o = 0 in a two-dimensional area when o vanishes on the boundary. The method is based on an integral equation, which can be easily solved numerically. Results obtained for circular and rectangular geometries are also given and compared to the exact values.
85 citations
TL;DR: In this article, the path integral solution of a stochastic process described by a generalized Langevin equation with coordinate-dependent fluctuating forces and white spectrum was determined, and the exponent of the exponential function occurring in that solution was interpreted as generalization of the Onsager-Machlup function.
Abstract: We determine the path integral solution of a stochastic process described by a generalized Langevin equation with coordinate-dependent fluctuating forces and white spectrum. Since such equations do not permit a unique determination of the distribution function but require the Ito or Stratonovich prescription, we first pass over to the corresponding Fokker-Planck equation adopting such a prescription. By means of the one-dimensional case we show that the path integral solutions are not uniquely determined in form but allow for a class of equivalent representations. Adopting an especially suitable representation we then present the path integral solution of the multi-dimensional Fokker-Planck equation. The exponent of the exponential function occurring in that solution can be interpreted as generalization of the Onsager-Machlup function. Finally we give a path-integral solution of the master equation. In the “Fokker-Planck limit” again a generalized Onsager-Machlup function is obtained.
82 citations
TL;DR: In this paper, it was shown that testing Pocklington's equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallen's equation when a common set of basis functions is used with each.
Abstract: It is shown that testing Pocklington's equation with piecewise sinusoidal functions yields an integro-difference equation whose numerical solution is identical to that of the point-matched Hallen's equation when a common set of basis functions is used with each. For any choice of basis functions, the integro-difference equation has the simple kernel, the fast convergence, the simplicity of point-matching, and the adequate treatment of rapidly varying incident fields, but none of the additional unknowns normally associated with Hallen's equation. Furthermore, for the special choice of piecewise sinusoids as the basis functions, the method reduces to Richmond's piecewise sinusoidal reaction matching technique, or Galerkin's method. It is also shown that testing with piecewise linear (triangle) functions yields an integro-difference equation whose solution converges asymptotically at the same rate as that of Hallen's equation. The resulting equation is essentially that obtained by approximating the second derivative in Pocklington's equation by its finite difference equivalent. The authors suggest a simple and highly efficient method for solving Pocklington's equation. This approach is contrasted to the point-matched solution of Pocklington's equation and the reasons for the poor convergence of the latter are examined.
64 citations
39 citations
TL;DR: In this article, reflection of transverse electromagnetic waves is used to determine the electron density in a cold, collisionless, unmagnetized plasma, and the applicable Gel'fand-Levitan integral equation is extended so that it is valid for all times Laplace transformation of the extended equation gives a linear functional equation containing the complex reflection coefficient.
Abstract: In the one‐dimensional form of the plasma inverse problem, reflection of transverse electromagnetic waves is used to determine the electron density in a cold, collisionless, unmagnetized plasma We extend the applicable Gel’fand–Levitan integral equation so that it is valid for all times Laplace transformation of the extended equation gives a linear functional equation containing the complex reflection coefficient We solve the functional equation analytically in special cases, and classify reflection coefficients by their analytic properties
19 citations
TL;DR: In this paper, a method is presented which transforms certain nonlinear differential equations of dynamics into linear equations by introducing an independent variable and utilizing the integrals of motion, and the linearization of unperturbed and perturbed Keplerian motions are discussed.
Abstract: A method is presented which transforms certain nonlinear differential equations of dynamics into linear equations by introducing an independent variable and utilizing the integrals of motion. As examples of special interest, the linearizations of unperturbed and perturbed Keplerian motions are discussed.
14 citations
TL;DR: In this paper, an integral equation for the current induced in a dielectric body by incident electromagnetic radiation is solved for the case of a thin flat circular disc of homogeneous lossy Dielectric material, which is reduced by using the method of moments to a linear algebraic matrix equation for coefficients in an expansion of the current density as a linear combination of basis functions.
12 citations
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TL;DR: In this article, the equivalence of the Alt-Grassberger-Sandhas equation to the Yakubovskiy equation is discussed and an equation is derived which is a natural extention of the three-body Faddeev equation.
Abstract: Publisher Summary This chapter focuses on four-body equation. It discusses the equivalence of Yakubovski equation and six coupled equations. It gives an expression for the wave function of the four-body system referring to the final state interactions. The equivalence of the Alt–Grassberger–Sandhas equation to the Yakubovski equation is discussed. An equation is derived which is a natural extention of the three-body Faddeev equation. Seven coupled equations are also described.
11 citations
TL;DR: In this article, a new approach for the general study of 1.1 equations of type (1) is developed, consisting in the determination of the periodical steady solutions in a suitable parametric form with which the equation is transformed into a non linear integral equation.
Abstract: A new approach for the general study of the equations of type (1.1) is developed, consisting in the determination of the periodical steady solutions in a suitable parametric form with which the equation is transformed into a non linear integral equation. The procedure is developed in details in the case of the Duffing equation, for which it is obtained an approximate solution valid even if the non-linearity of the system is very high, based on the study of a Volterra integral equation.
TL;DR: In this article, it was shown that the Whittaker-shannon interpolation formula provides representations for particular classes of entire functions of exponential type, where every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions.
Abstract: The Whittaker-Shannon interpolation formula, or "cardinal series", is a special case of the more general linear integro-differential equation with constant complex coefficients En=0a f(n)(z) = ff(Z-t)d,(t) where the integral is taken over the whole real line with respect to the measure A. In this study, I show that many of these equations provide representations for particular classes of entire functions of exponential type. That is, every function in the class satisfies the equation and conversely every solution of the equation is a member of the class of functions. When the measure in the convolution integral above is chosen to be discrete, a particular form of the above type of equation is an equation of periodicity f(z) = f(z + T). Following an extensive treatment of the general equation written above, the study concludes by offering a generalization in terms of these convolution equations of a classical theorem in complex analysis concerning periodic entire functions.
TL;DR: In this article, a method is described of transforming the one-dimensional electromagnetic wave propagation problem into an integral equation of the Volterra type, which can be solved easily and within a short time.
Abstract: One‐dimensional electromagnetic wave propagation problems can be formulated exactly in terms of an integrodifferential equation of the Fredholm type. The numerical solution of this equation is often complicated and time consuming. In this paper a method is described of transforming this equation into an integral equation of the Volterra type. The solution of the latter equation can be carried out easily and within a short time.
01 Dec 1976
TL;DR: In this paper, the Boundary Integral Equation (BIE) method is applied to boundary value problems governed by Laplace's equation and two other numerical approximations, quadratic shape function and cubic spline function representations, are adopted.
Abstract: : The Boundary Integral Equation (BIE) method is applied to boundary value problems governed by Laplace's equation. In addition to the piecewise constant approximation for boundary functions, two other numerical approximations, quadratic shape function and cubic spline function representations, are adopted. Comparisons are discussed. Topics for further research are indicated. (Author)
TL;DR: In this paper, the use of the Euler-Jacobi equation is considered in the study of a quadratic functional defined on a cone, such functionals occur in the variation of optimal-control problems.
Abstract: The use of the method of the Euler-Jacobi equation is considered in the study of a quadratic functional defined on a cone. Such functionals occur in the variation of optimal-control problems. Several concepts are introduced with the aid of which the Euler-Jacobi equation is extended and the application of this method is justified also in the case that the equation is not a linear differential equation.
TL;DR: In this article, a new method of solving a class of integral equations with difference kernels is given, which is based on establishing a connection between the solution of the given equation and that of the corresponding equation on the half-axis.
Abstract: In this note a new method of solving a class of integral equations with difference kernels is given. It is based on establishing a connection between the solution of the given equation and that of the corresponding equation on the half-axis. This method allows us to reduce the given equation to a new integral equation with the kernel of a simple structure.
TL;DR: In this paper, a quantum-mechanical formula for the atomic inversion of the Glauber distribution function of the field and a closed equation for the field distribution function is deduced.
Abstract: The method given in part I of the present paper is applied to the single-mode laser model. The treatment is based on two coupled equations for the Glauber distribution function of the field and for another distribution, which is linked in a simple way to the atomic inversion. It is shown that such equations are the operator analogue, in the Schrodinger picture, of the semi-classical equations of the laser. The analysis leads to a quantum-mechanical formula for the atomic inversion, which describes the effect of fluctuations. A closed equation is deduced for the field distribution function. In the markoff approximation such an equation reduces to the Fokker-Planck equation recently deduced by Mandel, which generalizes the Risken equation to the case of high-intensity lasers. The present method completely avoids the resummation of infinite terms of the kernel of the generalized master equation, performed by Mandel.
TL;DR: In this paper, a potential problem is solved in a geometry with a slotted boundary, where the normal component of the electric field should vanish at a slot, and special difficulties arise if a line charge density is put on the slot, in order to solve the field problem by an ordinary integral equation.
Abstract: A potential problem is solved in a geometry with a slotted boundary. If the normal component of the electric field should vanish at a slot, special difficulties arise if a line charge density is put on the slot, in order to solve the field problem by an ordinary integral equation. Two modified integral equation methods are then presented to eliminate these difficulties.
TL;DR: In this article, an exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation.
Abstract: An exact non-equilibrium Ornstein-Zernike (OZ) equation is derived for lattice fluid systems whose time development is given by a generalized master equation. The derivation is based on a generalization of the Montroll-Weiss continuous-time random walk on a lattice, and on their relationship with master equation solutions. Time dependent direct and total correlation functions are defined in terms of the generating functions for the probability densities of the random walker, such that, in the infinite time limit the equilibrium OZ equation is recovered. A perturbative analysis of the time dependent OZ equation is shown to be formally analogous to the perturbation of the Bloch equation in quantum field theory. Analytic results are obtained, under the mean spherical approximation, for the time dependent total correlation function for a one-dimensional lattice fluid with exponential attraction.
TL;DR: In this article, definite integrals which occur in transport problems through a scattering medium are studied in terms of such functions as the exponential integral of the first and second order, the dilogarithm, and a newly introduced and tabulated function.
Abstract: Some definite integrals which occur in transport problems through a scattering medium are studied. They are expressed in terms of such functions as the exponential integral of the first and second order, the dilogarithm, and a newly introduced and tabulated function.
TL;DR: In this paper, a stochastic integral equation of the Ito-Doob type is considered and the form given by it is analyzed. But it is not a nonlinear integral equation.
Abstract: There are two basic classes of random or stochastic integral equations cur ren t ly under s tudy by probabilists and mathemat ical statisticians. Those integral equations involving Ito or Ito-Doob type stochastic integrals and those which can be considered as probabilistic analogues of classical determinist ic integral equations whose formulation involves only the Lebesque integral. I t has been shown [1]-[4], [6]-[11], [13], [14], [16], [17], among others tha t the theory of random integral equations of the l a t t e r ca tegory are ex t remely important in stochastically character izing many physical situations in life sciences and engineering. In this paper we shall be concerned with a stochastic integral equation of the Ito-Doob type. We shall be concerned with a nonlinear stochastic integral equation of the form given by
TL;DR: In this article, it is demonstrated that the kinetic equation of Davis's effective potential theory follows directly from the application of well-defined approximations to the three-body correlations involved in the second equation of the BBGKY hierarchy.
Abstract: It is demonstrated that the kinetic equation of Davis's effective potential theory follows directly from the application of well-defined approximations to the three-body correlations involved in the second equation of the BBGKY hierarchy. The same, simple mathematical techniques involved in this demonstration are used to derive two other kinetic equations, one of which is a generalization to high densities of the Boltzmann equation. In order to facilitate its application to the calculation of the van Hove and other correlation functions, the kinetic equation of the effective potential theory is Fourier-Laplace transformed: explicit formulae are given for the matrix elements of all operators that occur in this equation.
TL;DR: In this article, the multi-dimensional Fokker-planck equation is solved by Feynman path integrals and the solution may be interpreted as generalized Onsager-Machlup function.
TL;DR: In this paper, the authors investigated the problem of (2) by making use of the theory of Nevanlinna, and they obtained results concerning the growth of f(z) in some special cases, and obtained some other results.
Abstract: and has shown that this has a meromorphic solution f ( z ) if there is a value a such that q ( a ) = a with q ' (a)= a and [a I > 1. Myrberg [6] investigated the solutions of (3) when q(z) is a polynomial which takes certain special forms and f ( z ) was an entire function. The purpose of this note is to investigate the solution of (2) by making use of the theory of Nevanlinna. We obtain results concerning the growth of f(z) , give the solution of (2) in some special cases, and obtain some other results.
20 Jul 1976