SPECTRAL BASED COMPUTATIONAL METHODS FOR THE SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS

Abstract

In this work, a class of spectral based computational method for the solution of partial differential equation is developed. Theoretical solution to a given general second order partial differential equation is partly differentiated at constant time t but varying on the space (x) and substituted to the aforementioned equation; leading to a general parabolic partial differential equation and; by adopting the above spectral based theoretical solution to this general parabolic equation and further simplifications were performed resulting into development of a class of spectral based computational methods for solution of partial differential equations at step numbers j = 2, 3, and 4, respectively. Nine distinct spectral based new computational schemes were obtained. The accuracy, consistency, stability and convergence properties of these methods were determined. The methods are implemented on some sampled problems that involve both constant and, variable coefficients parabolic partial differential equations; and evaluated by comparing them with some existing difference methods. The results obtained are found to be more rapidly converging as the step lengths h and k approaches zeros. This work provided accurate numerical solutions to a class of dynamical problems having time dependent boundary conditions. Higher ordered parabolic partial differential equations with defined theoretical solutions and given boundary conditions can be solved directly using any of these methods which amount to no time wastage, reduction of antiqueness and expenses.