Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

Authors:
E. H. Doha;W. M. Abd-Elhameed
Affiliations:
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt;Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt
Venue:
Journal of Computational and Applied Mathematics
Year:
2005

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Abstract

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.