SPECTRAL METHODS FOR SOLVING DIFFERENTIAL AND FUNCTIONAL EQUATIONS

The operator approach previously developed for the spectral method using Legendre polynomials is generalized here to any systems of basis functions (not necessarily orthogonal) that satisfy two conditions: the result of the operation of multiplication by x or differentiation with respect to x is expressed in the same functions. All systems of classical orthogonal polynomials meet these conditions. In particular, a spectral method utilizing Chebyshev polynomials is constructed, which is most efficient for numerical calculations. This method is applied for the numerical solution of linear functional equations that arise in generalized series summation problems, aswell as in problems of analytic continuation of discrete mappings. It is also shown how these methods solve nonstandard and nonlinear boundary value problems for which conventional algorithms are not applicable.

spectral methods, Chebyshev polynomials, boundary value problems, functional equations, high-precision computations