Integrals of Chebyshev polynomials of third and fourth kinds: An application to solution of boundary value problems with polynomial coefficients

Abstract

Two new formulae expressing explicitly the repeated integrals of Chebyshev polynomials of third and fourth kinds of arbitrary degree in terms of the same polynomials are derived. The method of proof is novel and essentially based on making use of the power series representation of these polynomials and their inversion formulae. Using the Galerkin spectral method, we show that these formulae can be used to solve some high-order boundary value problems with varying coefficients, and propose two Galerkin-type algorithms for solving the integrated forms of some high-order boundary value problems with polynomial coefficients. A numerical example is discussed, which shows that the proposed algorithms are more accurate and efficient compared with the analytical ones.