An integral equation formulation of three dimensional anisotropic elastostatic boundary value problems

1973 ◽  
Vol 3 (3) ◽  
pp. 203-216 ◽  
Author(s):  
S. M. Vogel ◽  
F. J. Rizzo
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Equation Formulation ◽  
Integral Equation Formulation

scholarly journals A unified boundary integral equation method for a class of second order elliptic boundary value problems

1984 ◽  
Vol 25 (4) ◽  
pp. 501-517 ◽  
Author(s):  
M. Rezayat ◽  
F. J. Rizzo ◽  
D. J. Shippy
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Second Order ◽  
Elliptic Type ◽  
Integral Equation Formulation

AbstractA generalized integral equation formulation and a systematic numerical solution procedure are presented for a class of boundary value problems governed by a general second-order differential equation of elliptic type. Diverse numerical examples include problems of plane-wave scattering, three-dimensional fluid flow, and plane heat transfer for a body with a moving flame boundary. The last example employs certain representation functions useful to increase solution effectiveness in problems with an isolated integrable singularity.

A well-posed integral equation formulation for three-dimensional rough surface scattering

2006 ◽  
Vol 462 (2076) ◽  
pp. 3683-3705 ◽  
Author(s):  
Simon N Chandler-Wilde ◽  
Eric Heinemeyer ◽  
Roland Potthast
Keyword(s):  
Integral Equation ◽  
Rough Surface ◽  
Double Layer ◽  
Inverse Operator ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Wave Number ◽  
Layer Potential ◽  
Equation Formulation ◽  
Integral Equation Formulation

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage–Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space when the scattering surface does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, κ , for κ >0, if the coupling parameter η is chosen proportional to the wave number. In the case when is a plane, we show that the choice is nearly optimal in terms of minimizing the condition number.

scholarly journals Integral Equation Formulation of some Three Part Boundary Value Problems

1963 ◽  
Vol 13 (4) ◽  
pp. 317-323 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Fredholm Integral Equation ◽  
Simple Form ◽  
Boundary Value ◽  
Outer Radius ◽  
Electrostatic Problem ◽  
Circular Annulus ◽  
Inner Radius ◽  
Integral Equation Formulation

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).

Solution of three‐dimensional boundary value problems for magnetostatics by a constrained Fredholm integral equation

1985 ◽  
Vol 57 (8) ◽  
pp. 3832-3834 ◽  
Author(s):  
I. D. Mayergoyz ◽  
J. D’Angelo ◽  
C. Crowley
Keyword(s):  
Integral Equation ◽  
Boundary Value Problems ◽  
Fredholm Integral Equation ◽  
Boundary Value ◽  
Three Dimensional ◽  
Dimensional Boundary

scholarly journals On the solutions of mixed plane problems that are unbounded where the boundary conditions change

2001 ◽  
Vol 32 (3) ◽  
pp. 173-180
Author(s):  
M. G. El-Sheikh ◽  
V. N. Gavdzinski ◽  
A. E. Radwan
Keyword(s):  
Integral Equation ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Infinite System ◽  
Original Method ◽  
Plane Boundary ◽  
Algebraic Equations ◽  
Typical Problem ◽  
Equation Formulation ◽  
Integral Equation Formulation

In this paper, we apply a modification to the method of the integral equation formulation of mixed plane boundary value problems so that it enables us to obtain the solutions unbounded at the points where the boundary conditions change. Such solutions are of great physical interest. The modification is illustrated by means of a typical problem. As is it the case in the original method proposed by Cherskii [1], the problem is reduced to an infinite system of algebraic equations. The justification of the truncation of such systems has been established.

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