A comparison study of Chebyshev spectral collocation based methods for solving nonlinear second order evolution equations.

Abstract

In this study Spectral Quasilinearisation Method (SQLM) coupled with finite differ- ence and Bivariate Spectral Quasilinearisation Method (BSQLM) in solving second order nonlinear evolution partial differential equations are compared. Both meth- ods use Newton-Raphson quasilinearisation method (QLM) and Chebyshev spectral collocation based on Lagrange interpolation to solve the governing equations. The Spectral Quasilinearisation Method coupled with finite difference is obtained by ap- plying the spectral collocation method on space derivatives and finite difference of time derivatives while the BSQLM is a Bivariate Lagrange interpolation based scheme in which the spectral collocation method is applied independently to both time and space derivatives. The applicability of these methods is shown by solving a class of second order nonlinear evolution partial differential equations (NPDEs), namely Burgers equation, Burgers-Fisher, Fisher's equation, Newell-Whitehead-Segel equa- tion and Zeldovich equation that arise in some fields of science and engineering. The numerical approximation results are validated for accuracy by comparing them with exact solutions. Tables for Explicit, Implicit and Crank-Nicolson SQLM and BSQLM with their computational times were generated for comparison; the order of accuracy for each method and error graphs are presented.