Enhanced shifted Tchebyshev operational matrix of derivatives: two spectral algorithms for solving even-order BVPs

Abstract

Herein, new orthogonal polynomials have been generated from shifted Chebyshev polynomials that fulfill a given set of homogeneous boundary conditions and the necessary formulae have been established. Moreover, an integer order derivative operational matrix has been introduced. Then, the presented novel polynomials are used together with the two spectral methods, namely, the Galerkin and Tau methods, as the basis functions. The convergence and error analyses were introduced and proved. Finally, some even-order boundary value problems (BVPs) have been approximated using the presented method.