Two modified shifted Chebyshev–Galerkin operational matrix methods for even-order partial boundary value problems

Abstract This paper presents two operational matrices. The first one represents integer-order derivatives of the modified shifted Chebyshev polynomials of the second kind. These polynomials serve as basis functions in two spectral methods, Galerkin and Petrov–Galerkin. These techniques are then applied to solve even-order initial boundary value problems (IBVPs). Additionally, convergence and error analysis is provided and demonstrated. On the other hand, the second operational matrix represents an integer integration of the shifted Chebyshev polynomials of the second kind. The proposed polynomials are used in the expansions of the spectral approach to solve partial boundary value problems (PBVPs). As an example, we consider the one-dimensional linear telegraph equation and solve it using the Galerkin and Petrov–Galerkin methods. The demand for more accurate and efficient spectral approaches to solving BVPs and IBVPs encouraged this study. By using modified shifted Chebyshev polynomials this work aims to improve the accuracy and convergence of even-order IBVPs and PBVPs.