Abstract
ABSTRACT NUMERICAL SOLUTION OF DIFFERANTLAL EQUATIONS WITH FINITE DIFFERENCE METHODS Canan AKTAŞ Balıkesir University, Institute of Science, Department of Mathematics Education (M.Sc Thesis / Supervisor : Prof. Dr. Aydın OKÇU) Bahkesir-Turkey, 1995 Parabolic equations are solved by means of finite difference methods. Finite difference approximations are studied in two groups, namely explicit and implicit. The explicit method is the direct computation of U(x, t+k) unknown values by using U(x, t) known values in a step-by-step manner. Thus, this method operates noniteratively in the direction oft. When applied to a non-linear equation, it does yield an equation system, but when it comes to stability, it is not efficient as it brings about certain restrictions. The implicit method is the utilization of an iterative operation in the solution. Although this is efficient as far as stability and convergence are concerned, it yields a nonlinear equation when applied to another nonlinear equation. Detailed explanations concerning these methods have been given in section 2 and 3 of this thesis. Equation âV â2U ât âx2, 0<x<X,t>0 can be reduced to the ordinary differential equation - = AV+b dt by using finite difference methods. Here, A and B are independent of t; and V(t) satisfies V(0)=g initial condition. A is a matrix in the form of 111/ having/an (N-l) order. The solution of ordinary differential equation in the form of ÜX = AV+b dt has been shown in section 3. The subject of stability has been given in section 4. In section 5 numerical methods for diffusion and reaction-diffusion have been explained. dV Calculation method of eigenvalues of differential equation - = AV + b and dt Basic-Language computer program for classical explicit approximation have been given in the appendix section. KEY WORDS : partial differential equations / finite difference methods / classical explicit approximation / classical implicit approximation / Crank-Nicolson method / method of lines IV
