Chebyshev and Legendre Spectral Methods in Algebraic Manipulation Languages

In this tutorial, we show how to combine Galerkin and collocation methods with algebraic manipulation languages. Because polynomial and trigonometric manipulations are a special strength of REDUCE, Maple, and other symbolic languages, it is easy to use Chebyshev and Legendre polynomials and Fourier series as the basis sets. However, because the truncation is small and calculations are performed symbolically in rational arithmetic, many of the standard precepts for numerical use of spectral methods must be modified or reversed. We offer many guidelines and suggestions. Six examples programmed in REDUCE and Maple—eigenproblems, bifurcations, and nonlinear diffusion—illustrate the possibilities.