DRM solutions to two-dimensional linear wave equations

Abstract

In this thesis, two numerical methods called the Finite Difference Dual Reciprocity Method (FDDRM) and the Laplace Transform Dual Reciprocity Method (LTDRM) are developed for solving Linear Wave Equations (LWEs) in R2. Both proposed methods are based on the Dual Reciprocity Method (DRM) which is the efficient method for solving Poisson equations. According to FDDRM, an LWE is transformed into the Poisson equation in the time space using some finite difference techniques. On the other hand, LTDRM uses the Laplace transform to transform an LWE into the Poisson equation in the Laplace space. After transformation, the DRM technique is then to solve the transformed equation. With these methods, boundary-only integral equations can be derived and the dimension of the problem is reduced by two. Since FDDRM uses some finite difference techniques, a solution at any specific time can be attained with a step-by-step calculation in time, while LTDRM needs a numerical inversion of the Laplace transform to convert a solution obtained in the Laplace space into a solution in the time space. In this research, a numerical Laplace transform inversion called "Stehfest's algorithm" is chosen. The numerical solutions obtained from FDDRM and LTDRM for several test examples are presented herein. It will be seen that LTDRM is more efficient than FDDRM when a solution at a large time is required.