NUMERICAL METHODS FOR SOLVING SOME PARABOLIC EQUATIONS

Abstract

In this thesis we are devoted to applying the spectral methods with

• different basis functions, for studying some linear (Heat equation) and nonlinear (Burgers' equation) Parabolic equations. Theoretical background as well as, detailed informations for the numerical schemes used in solving these equations are given. Most of the prevwus work on these equationwas devoted to finite difference and finite element methods. Spectral methods are also used but widely with Chebyshev polynomials as basis for the space of solution. In present work we extend our applications to the use of Legendre polynomials as basis. The necessary formulas for the method were concluded and shown in details throughout this thesis. Also the resulting systems of algebraic equations were handled in two different ways as to obtain better numerical results. We have concluded that spectral methods are good and convenient for such problems, and better results are obtained in case of using Legendre polynomials rather than the widely used Chebyshev polynomials. We feel that these results can give rise to apply the scheme to different linear and nonlinear problems.