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.