Numerical Solution Of 2-D Helmholtz Equation With(Variable Or Constant Coefficients ) By Boundary Domain Integral Equation Method

Abstract

In this thesis the Boundary Domain Integral Method (BDIEM) is applied to the solution of the nonhomogeneous or homogeneous Helmholtz equation with variable or constant coefficients. This thesis deals with the extension of the BDIE formulations to the treatment of the two-dimensional Helmholtz equation with variable coefficients. The concept of linear partial differential equation should be examined.In this study PDE, BVP, BEM, green’s function, Laplace equation, Poisson’s equation and reduction of Helmholtz equation to an integral equation are studied.Boundary domain integral equation method is aimed to obtain exact solution and approximate solution. Four possible cases are investigated. First of all when both material parameters and wave number are constant and f(x)=0. Secondly when both the parametric a(x) and wave number k(x) are constants and f (x) ̸= 0. Thirdly when the parametric a(x) is constant, wave number k(x) is variable and f (x) ̸= 0, the standard fundamental solution for the Laplace equation is used in the formulation.Lastly when the parametric is constant and wave number are variables and f(x)=0. Moreover, when the material parameters are variable (with constant or variable wave number), a parametric is adopted to reduce the Helmholtz equation to a BDIE. To illustrate the efficiency the proposed approaches,numerical examples for several basic problems and exact solutions are presented.In general i applied test examples with two homogeneous Helmholtz equation and two test examples with non-homogeneous Helmholtz equations.The accuracy and convergence of the MFS numerical technique used in this thesis is investigated using certain test examples for various geometry domains. So it is important to solve Helmholtz equation with variable coefficient for researchers on applied mathematics and physics.