Spectral methods for solution of differentiatial and functional equations

Abstract

An operational approach developed earlier for the spectral method that uses Legendre polynomials is generalized here for arbitrary systems of basis functions (not necessarily orthogonal) that satisfy only two conditions: the result of multiplication by x or of differentiation with respect to x is expressed in the same functions. All systems of classical orthogonal polynomials satisfy these conditions. In particular, we construct a spectral method that uses Chebyshev polynomials, which is most effective for numerical computations. This method is applied for numerical solution of the linear functional equations that appear in problems of generalized summation of series as well as in the problems of analytical continuation of discrete maps. We also demonstrate how these methods are used for solution of nonstandard and nonlinear boundary value problems for which ordinary algorithms are not applicable.