Green's Functions for Mixed Boundary Value Problems in Regions of Irregular Shape

Abstract

A semi-analytic approach is applied to the construction of Green's functions and matrices of Green's type for Laplace and Klein-Gordon equation in two dimensions. Mixed boundary value problems posed on multiply connected regions are considered. Statements of the problems are complicated by intricate geometry of the regions and different types of boundary conditions imposed on different fragments of the boundary. The approach is based on a combination of the Green's function method and the method of functional equations. Kernels of resolving potentials representing the regular components of the Green's functions to be constructed are built with Green's functions obtained for regions of standard shape.

Equivalence between the Trefftz method and the method of fundamental solution for the annular Green's function using the addition theorem and image concept

“…Melnikov [19][20][21] utilized the method of modified potentials (MMP) to solve BVPs from various areas of computational mechanics. Later, Melnikov and Melnikov [22] studied in computing Green's functions and matrices of Green's type for mixed BVPs stated on 2-D regions of irregular configuration. For the image method, Thomson [23] proposed the concept of reciprocal radii to find the image source to satisfy the homogeneous Dirichlet boundary condition.…”

An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains

“…where 2 ( , ) ln , U s x r r = (4) in which r is the distance between the source point s and the field point x(r ≡ | x − s |). After exchanging with the variables x and s, we have four boundary integral equations as shown in the next section.…”

“…Timoshenko and Woinowsky-Krieger [3] also treated the circular plate problems in their books. In the Melnikov's paper [4], a semi-analytical approach was applied to construct Green's functions and matrices of Green's type for the Laplace and Klein-Gordon equations in two dimensions. Mixed boundary value problems posed in multiplyconnected regions were also concerned by Melnikov [5].…”

Derivation of Green’s function using addition theorem

“…Melnikov (1982Melnikov ( , 1995; Melnikov and Melnikov (2001) utilized the method of modified potentials (MMP) to solve boundary value problems from various areas of computational mechanics. Later, Melnikov and Melnikov (2006) studied computing Green's functions and matrices of Green's type for mixed boundary value problems (BVP) posed on 2-D regions of irregular configuration. For the image method, Thomson (Thomson, 1848) proposed the concept of reciprocal radii to find the image source to satisfy the homogeneous Dirichlet boundary condition.…”