Numerical Solution of Helmholtz Equation

Abstract

Many natural physical and engineering phenomena appear only in the form of mathematical systems, specifically, as partial differential equations describing the nature of these phenomena. In this thesis, partial differential equations are studied, and the focus is on the Laplace equation and Helmholtz equation with two dimensions. Numerical methods were used to solve these physical problems. We used three numerical methods, the first is the finite difference method with an operator five points, the second is the finite difference method with an operator nine points and the third is the finite elements method to find the approximate solution to those equations, where the equations are converted to another formula Finally, we get a linear system, which can be solved by one of the iterative methods. We found that the finite difference method with operator (nine points) is the best and closest to the exact solution, and with the least possible error in the case of the solution region (domain) of the Laplace equation and Helmholtz's equation with regular geometric shapes (triangle, rectangle,...).