Dissertation (MA Mathematics)This dissertation is a study of finite difference numerical methods that are based on the principle of discretization. In chapter one presents the general overview of partial differential equations, Initial boundary value problems and derivations of Finite difference methods for one dimensional heat equation. A brief discussion of the truncation error is also presented. The second chapter is concerned with consistency, stability and convergence of the finite difference schemes for one dimensional and two dimensional parabolic partial differential equations. Two method of stability analysis, Fourier series method and Matrix Analysis method are used to analyze the stability of Crank-Nicolson and Alternating Direction Implicit (ADI) methods. Chapter three is devoted to numerical simulation of two dimensional parabolic equations. Two finite difference schemes, namely, Crank-Nicolson and Alternating Direction Implicit (ADI) methods are applied to the numerical solution of a two dimensional parabolic partial differential equation. Two Matlab routines one for Crank-Nicolson Scheme and another for ADI Scheme and dynamic pictures are included.