Spectral Solutions for Differential and Integral Equations with Varying Coefficients Using Classical Orthogonal Polynomials

Spectral methods for solving differential/integral equations are characterized by the representation of the solution by a truncated series of smooth functions. The unknowns to be determined are the expansion coefficients in such a representation. The goal of this article is to give an overview of numerical problems encountered when determining these coefficients and the rich variety of techniques proposed to solve these problems. Therefore, a series of explicit formulae expressing the derivatives, integrals and moments of a class of orthogonal polynomials of any degree and for any order in terms of the same polynomials are addressed. We restrict the current study to the orthogonal polynomials of the Hermite, generalized Laguerre, Bessel, and Jacobi (including Legendre, Chebyshev, and ultraspherical) families. Moreover, formulae expressing the coefficients of an expansion of these polynomials which have been differentiated or integrated an arbitrary number of times in terms of the coefficients of the original expansion are given. In addition, formulae for the polynomial coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients are also presented. A simple approach to build and solve recursively for the connection coefficients between different orthogonal polynomials is established. The essential results are summarized in tables which could serve as a useful reference to numerical analysts and practitioners. Finally, applications of these results in solving differential and integral equations with varying polynomial coefficients, by reducing them to recurrence relations in the expansion coefficients of the solution, are implemented.