Dissertation (MSc Mathematics)This dissertation provides general review of Partial Differential Equations. The Finite Difference Method which uses Taylor’s Theorem for solving Partial Differential Equations and Solution of elliptic partial differential equations, it also introduces the concepts of convergence, consistency and stability of finite difference scheme, Finite Difference Method (FDM) involving direct method and Iterative method like Gauss Seidel or Gauss Seidel with Successive Over Relaxation are applied on specific chosen example then simulation of Two Dimensional Laplace’s Partial Differential Equations of each method is to be done by MATLAB. Finally analysis of the results based on the three methods is made to indicate which method performs better than others. With this dissertation those wishing to apply Partial Differential Equations especially elliptic will be interested to use Gauss Seidel with Successive over Relaxation method since analysis shows that this method gives better results when solving itteratively.