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.
