Chebyshev polynomials in the solution of ordinary and partial differential equations

Chebyshev polynomials are used to obtain accurate numerical solutions of ordinary and partial differential equations. For functions of one or two variables, expressed in terms of Chebyshev polynomials, generalisations are obtained of formulae for finding function and derivative values. Then, for ordinary differential equations a standard method of solution is extended for use at cirbitrary points, and is applied to a number of differential equations associated with functions of mathematical physics. Elliptic partial differential equations, similar to Laplace equations, are examined and it is shown how Chebyshev series solutions cam be found, the coefficients in the solution being obtained in a manner related to that for calculating function values at grid points, when a finite difference method is used.