On using third and fourth kinds Chebyshev polynomials for solving the integrated forms of high odd-order linear boundary value problems

This article presents some spectral Petrov–Galerkin numerical algorithms based on using Chebyshev polynomials of third and fourth kinds for solving the integrated forms of high odd-order two point boundary value problems governed by homogeneous and nonhomogeneous boundary conditions. The principle idea behind obtaining the proposed numerical algorithms is based on constructing trial and test functions as compact combinations of shifted Chebyshev polynomials of third and fourth kinds. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. Some numerical examples are illustrated for the sake of demonstrating the validity and the applicability of the proposed algorithms. The presented numerical results indicate that the proposed algorithms are reliable and very efficient.