Direct Boundary Integral Equation Method for Helmholtz Equations with Dirichlet Boundary Conditions

Abstract

In this thesis, we analyze the Boundary Integral Equation(BIE) methods for the Helmholtz equation with Dirichlet Boundary Conditions in three-dimensions where the boundary is smooth. We present a formulation of solving the BIEs reformulated from the Helmholtz boundary value problems by a direct method. BIE is particularly attractive in developing integral equation methods, it reduces the dimensions of the problem and often transforms a problem in an infinite domain to integrals on the finite boundary in which the far field radiation condition is satisfy automatically. A Boundary value problem for Partial Differential Equation with a constant coefficient can be reduced to a BIE by using the fundamental solution and Green’s representation formula, we reduced the boundary value problem into Direct Boundary Integral Equations. The integral equation formulation has a unique solution at all wave numbers by proving equivalence of the BVP and the integral equation formulation and proving uniqueness of solution for the BVP. The equivalence of the boundary value problem with the reduced integral equation in appropriate Sobolev space are proved.