Showing papers on "Integro-differential equation published in 1989"
TL;DR: In this article, an optimization strategy is presented for optimizing the structure of empirical thermodynamic correlation equations based on a comprehensive functional expression for the physical dependence considered, which is called a "bank of terms," the new procedure optimizes the structure and the length of the equation as well.
Abstract: An optimization strategy is presented for optimizing the structure of empirical thermodynamic correlation equations. Based on a comprehensive functional expression for the physical dependence considered, which is called a “bank of terms,” the new procedure optimizes the structure and the length of the equation as well. The application of this method results in an equation which meets the quality wanted for representing the experimental data with the lowest number of fitted coefficients. The procedure can be used for the determination of the structure of any equation where the method of the linear least squares is applicable. A detailed description of the algorithm is given which includes values for the control parameters for different applications in the field of thermodynamics (vapor pressure equations, equations of state, etc.) and also for applications in other fields. The optimization steps are described using an equation which represents a relationship between variables in a general form. It is demonstrated how even the complex problem of the optimization of a fundamental equation for the Heimholtz energy can be written in terms of this general equation.
111 citations
TL;DR: In this paper, the heat equation in a noncylindrical domain is studied and the unknown source function f is recovered under different assumptions on the smoothness of input data and f itself.
49 citations
01 Nov 1989
47 citations
TL;DR: In this paper, a new spectral transform r(k, l) of the potential u in the time dependent Schrodinger equation (associated to the KPI equation) is defined.
43 citations
TL;DR: The tetrahedron equation and the four-simplex equation are multidimensional generalizations of the Yang-Baxter or triangle equations and common features of these members of the family of "simplex equations" are discussed in this article.
37 citations
TL;DR: Some special types of solutions in the high dimensional sine-Gordon equation can be deformed to that of the double sineGordon equation by solving an ordinary differential equation or using a pure algebraic deformation relation as discussed by the authors.
33 citations
TL;DR: In this article, the authors considered a linear ordinary differential equation in 3×3 matrices and parametrized the asymptotics of all solutions as τ→0, as well as the regular solutions of the nonlinear equation studied as τ →±∞.
Abstract: To study the solutions of the equation (τuτ)τ=eu−eu which is a version of the “degenerate” third Painleve equation we consider a linear ordinary differential equation in 3×3 matrices. By means of the monodromy data of this linear equation we parametrize the asymptotics of all solutions as τ→0, as well as the asymptotics of regular solutions of the nonlinear equation studied as τ→±∞.
21 citations
TL;DR: In this article, the infiltration equation with a time-dependent surface flux is solved numerically for nonlinear conductivities, using first-integral and shooting techniques, using diffusivity and conductivity are both taken as power laws.
Abstract: The infiltration equation with a time-dependent surface flux is solved numerically for nonlinear conductivities, using first-integral and shooting techniques. The diffusivity and conductivity are both taken as power laws. The relative accuracy of the numerical technique is obtained by comparison with the case of linear conductivity, where an exact analytical solution exists. An approximate analytical result is given and compared with the solution for nonlinear conductivities from the shooting technique. It is found that the approximate analytical result is an excellent approximation to the exact numerical solution. It is also suggested that the solution obtained here can be used to validate the numerical solution of Richard's equation for arbitrary soil properties and arbitrary boundary conditions.
19 citations
TL;DR: In this article, a two-variable integro-differential equation, which includes all two-body correlations, produces results for three and four-body bound systems in good agreement with those obtained with the most accurate methods and also for sixteen fermions, interacting by means of local Wigner-type potentials.
Abstract: We find that a single two-variable integro-differential equation, which includes all two-body correlations, produces results for three- and four-body bound systems in good agreement with those obtained with the most accurate methods and also that for sixteen fermions, interacting by means of local Wigner-type potentials our results agree with those obtained with the Variational Monte-Carlo and the Fermi-Hypernetted chain methods. This equation includes a hypercentral potential. When it is set equal to zero it reduces to theS-state projected potential version of the formalism, which for three nucleons is identical to the exact Faddeev equation forS-state projected potentials. We show that the inclusion of the hypercentral component of the two-body local potential, which operates on all orbitals, takes the effect of the higher partial waves largely into account without the need of solving a system of coupled integro-differential equations. We, furthermore, show that by using the Weight-Function Approximation our integro-differential equation is transformed into a simple inhomogeneous differential equation, which becomes accurate forA≲16. The major advantages of our approximate method are, firstly, that unlike the exact Faddeev-Yakubovsky and Alt-Grassberger-Sandhas equations or the Monte-Carlo methods, it does not become rapidly unmanageable, but even simplifies with increasingA and, secondly, the speed and simplicity of our numerical calculations.
18 citations
TL;DR: In this paper, an interface evolution equation has been formulated to describe bombardment-induced etching by an axisymmetric angular distribution of energetic particles where the yield per incident particle is assumed to be a function of its energy and its angle relative to the surface normal.
Abstract: An interface evolution equation has been formulated to describe bombardment‐induced etching by an axisymmetric angular distribution of energetic particles where the yield per incident particle is assumed to be a function of its energy and its angle relative to the surface normal. These assumptions result in a nonlinear integro differential equation, but this equation reduces to a partial differential equation in several important special cases. At points that are not shadowed by a remote part of the surface, the interface evolution equation reduces to a nonlinear hyperbolic conservation law. Such equations have been applied to bombardment‐induced etching by a monodirectional beam with angle‐dependent yields; however, this form of equation applies more generally to raised isolated convex regions (e.g., etching masks) regardless of the angular distribution of the incident particles or the angle dependence of the yield. The essential qualitative feature of the solution in these cases is the spontaneous evolution of facet edges (slope discontinuities) from smooth initial conditions. Shadowing by remote parts of the surface may occur in concave regions (e.g., trenches) where it results in proximity effects.
TL;DR: In this article, the authors present analytical expressions for all the integrals involved in the application of the direct boundary integral equation method for such problems and show that analytic integration has a considerable advantage.
Abstract: In the solution of three-dimensional problems by the boundary integral equation method, linear discretization by triangular elements is simple and flexible. Although it is common to use numerical integration for the resolution of the boundary integrals, analytic integration has a considerable advantage. Important is the case of the «nearly singular» integral ― those integrals which do not contain the observation point but for which the observation point is nearby. Since these integrals are not singular, special numerical integration for singular integrals cannot be used, but ordinary (non-singular) quadrature is inaccurate. This paper contains analytical expressions for all the integrals involved in the application of the direct boundary integral equation method for such problems
Book•
01 Aug 1989
TL;DR: In this paper, the relationship between the light reflected from a surface and its shape was investigated, and it was shown that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point.
Abstract: In this paper we investigate certain first order partial differential equations which formulate the relationship between the light reflected from a surface and its shape. Particular emphasis is given to eikonal equations. Two results are presented. First, we prove that a special type of eikonal equation has only one convex and positive C2 solution in some neighborhood of a singular point. Using this result, we show that a restricted form of this equation has exactly two solutions. These results have application in scanning electron microscopy.
TL;DR: In this paper, an alternative Integral Equation Formulation to deal with thin plates on elastic foundations is developed, where the field equation is decomposed into two equations in partial derivatives of second order, which are formulated in an integral form by application of a reciprocity theorem.
Abstract: An alternative Integral Equation Formulation to deal with thin plates on elastic foundations is developed. The field equation is decomposed into two equations in partial derivatives of second order, which are formulated in an integral form by application of a reciprocity theorem. In this way, no divergent integrals appear in the formulation and the auxiliary function is the fundamental solution of Laplace equation. The domain integrals are transformed into equivalent boundary integrals, although in general it is necessary to introduce some internal points. Two examples will be studied to prove the efficiency of the formulation proposed.
TL;DR: In this article, the authors illustrate the rapid construction of phase plane portraits by a method that has no propagated error and is called the "Newton transform" (NCT).
01 Jan 1989
TL;DR: In this paper, the majorant frequency (MFF) method is proposed for the simulation of rarefied gas flows. But the main difference is that it requires a large number of sampling particles.
Abstract: In this work the direct statistical simulation method for spatially uniform relaxation in rarefied gas is developed on the basis of a probabilistic interpretation for the integral representation of the master kinetic equation (Kats equation). It is shown that under certain conditions the conventional schemes for the relaxation simulation follow directly from this equation. A new accurate simulation scheme is proposed, the majorant frequency scheme, which requires a computing capacity that scales directly with the number of sampling particles. The relation between the solutions of the master kinetic equation of rarefied gas and the Boltzmann equation is studied for the uniform case. It is shown that the correlations occurring in the finite-number particle system affect significantly the statistical simulation results. The criterion for estimating the influence of such correlations during the computation process is suggested. Introduction At present, the direct statistical simulation method, based on splitting up the evolution of a gas system in two stages, is widely used in the dynamics of rarefied gas. The method is realized in the following way. The flowfield computed is divided into cells of a finite size Ax and, according to the initial distribution function, N sampling particles are placed into each cell. Then the spatially-uniform relaxation stage and the stage of a free-molecular transition are successively carried out in all the cells. The free-molecular transition simulation can be performed without difficulties. In this case, the realization of the spatially-uniform relaxation stage is of primary importance. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. * Institute of Theoretical and Applied Mechanics. t Computing Center, 171 172 M. S. IVANOV ET AL For the simulation of collisional relaxation, the numerical schemes suggested in Refs. 1-4 are used. All these schemes are derived from heuristic considerations on the basis of physical understanding of the relaxation process in an actual gas; as a result there is no direct relation with the Boltzmann kinetic equation. The heuristic character of these schemes allow comparative analysis only qualitatively, with the use of the "Boltzmannian" collision frequency as the main criterion." The stochastic process for an approximate solution of the Boltzmann equation is constructed in Ref. 8 using the Euler scheme for the Boltzmann spatiallyuniform equation with its further randomization. In such an approach, for colliding particles the conservation laws are not valid, and this is the basic difference from the schemes given in Refs. 1-4. It seems reasonable to consider the known numerical schemes for the statistical simulation of rarefied gas flows in light of a general theory of Monte Carlo techniques. Such a unified consideration enables one to carry out a comparative analysis to show the inner relation between these schemes and also justifies the use of various Monte Carlo weight techniques.^ Derivation of Simulation Technique from Master Kinetic Equation In the construction of the Monte Carlo numerical technique we shall directly proceed from the master kinetic equation for the N-particle distribution function, which in the spatially-uniform case has the following r°° {fN(t,cjj) fN(t,c)} | vi-vj | by dbij dey l C) K2(t'->t I c) cp (t,c) dc dt + cp0(t,c) /2) Jo J where 9 (t,c) = t) (c) f^ (t,c) is the collision density. DIRECT SIMULATION AND MASTER KINETIC EQUATION 173 9o(t,c)=fN(o,c)\)(c)exp{-\)(c)t} _ . _ • _ _ N _ _• '-* C) = -^Y D'^WCv^V;' I Vi,Vj) II 5(vm-Vm)
TL;DR: In this paper, a new Backlund transformation for the Caudrey-Dodd-Gibbon-Sawada-Kotera equation was derived from the invariance property of the scattering problem.
Abstract: A λ‐modified equation for the Caudrey–Dodd–Gibbon–Sawada–Kotera equation is introduced. A new Backlund transformation for this equation is derived from the invariance property of the scattering problem for the Caudrey–Dodd–Gibbon–Sawada–Kotera equation under a Crum transformation [Q. J. Math. 6, 121 (1955)]. This, in turn, gives rise to a new Backlund transformation for the Caudrey–Dodd–Gibbon–Sawada–Kotera equation.
TL;DR: In this article, it was shown that every solution to the Yang-Baxter equation corresponds with the O R-type algorithm for solving the Riccati equation, and the structure of solutions of the YB equation is described and examples relevant to the algebraic Riemann equation are considered.
Journal Article•
TL;DR: In this article, the authors considered scattering for the Schrodinger equation and for the wave equation with variable speed and derived a new integral equation, which they called the unifying equation.
TL;DR: In this paper, the authors studied the low-frequency asymptotics of the solutions of the exterior Robin problem for the reduced wave equation in two dimensions and of the time independent Schrodinger equation in one and two dimensions with a nonnegative potential with compact support, by employing integral equation methods.
Abstract: We study the low-frequency asymptotics of the solutions of the exterior Robin problem for the reduced wave equation in two dimensions and of the time-independent Schrodinger equation in one and two dimensions with a nonnegative potential with compact support, by employing integral equation methods. Applications to time-dependent problems are indicated. .
TL;DR: In this paper, a transformation of the integro-differential transport equation in terms of radiation intensity to integral equations, where the moments of the radiation intensity depend on position and direction, is presented.
Abstract: Transformation of the integro-differential transport equation in terms of radiation intensity to integral equations in terms of moments of the radiation intensity reduces the computational labor because the former depends on position and direction and the latter depends on position only. Our analysis deals with two cases for which the scattering is linearly anisotropic. One involves radiative transfer in an arbitrary three-dimensional medium with a given inward boundary intensity and the other involves radiative transfer in a three-dimensional rectangular medium with Fresnel boundaries and the top surface exposed to normal incidence. Because the inward boundary intensity is unknown in the second case, an image technique is used to generate integral equations similar to those obtained in the first case. Numerical results for specific examples are given. Comparing the results for a slab with nonreflecting boundaries with existing exact solutions shows that our analysis works quite well.
TL;DR: In this article, the transonic integro-differential equation for two-dimensional flows is solved by boundary element methods, in addition to constant and quadrilateral elements, they develop hybrid elements based on constant elements in the streamwise direction and variable elements in transverse direction.
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: In this article, the authors derived an equation equivalent to the equation which follows from the condition of the vanishing Poisson bracket, which is very suitable for an easier and more efficient handling of terms of the same degree in the velocity components appearing in second integrals of motion.
Abstract: For autonomous dynamical systems with two degrees of freedom the author derives an equation equivalent to the equation which follows from the condition of the vanishing Poisson bracket. The new version of the equation is very suitable for an easier and more efficient handling of terms of the same degree in the velocity components appearing in second integrals of motion. The new variables used are suggested in the light of the inverse problem of dynamics. As an application the author treats the problem of integrals of motion which are homogeneous polynomials in the velocity components x, y.
Dissertation•
01 Jan 1989