Author(s)

M. Y. Melnikov & Y. A. Melnikov

Abstract

Employment of Green’s functions is recommended in designing test examples for computer algorithms developed for solution of boundary value problems for elliptic partial differential equations in two dimensions. A number of particular settings is examined for Laplace, Klein-Gordon and biharmonic equations. The focus is on problems posed on multiply connected regions. Illustrative examples have been considered revealing computational potential of the approach and identifying a problem class to which the approach is most effectively applicable. Keywords: test examples, Green’s function modification of the functional equations method, Laplace, Klein-Gordon and biharmonic equations. 1 Introduction Evaluation of the accuracy level represents a crucial stage in the development of numerical procedures for boundary value problems emerging in engineering sciences. This stage implies that some benchmark settings ought to be at hand, for which either exact or well justified and reliable approximate solutions can readily be obtained. But the design of such test settings is not always a trivial exercise. The objective in this study is to show how Green’s functions might serve in checking out results for boundary element method-based procedures. An attractive feature of these functions is that they exactly satisfy not only the governing differential equation but also some of the boundary conditions in the setting. This might make Green’s functions an excellent background for the design of test examples when computer procedures are developed. But, a challenging point is that one of the defining components of Green’s functions (singularity) does not collaborate with this venture.We will show how it can be taken care of.

Keywords

test examples, Green’s function modification of the functional equations method, Laplace, Klein-Gordon and biharmonic equations