Adapted Shifted ChebyshevU Operational Matrix of Derivatives: Two Algorithms for Solving Even-Order BVPs

Herein, modified orthogonal polynomials are introduced. These polynomials are generated from the second kind of shifted Chebyshev polynomials on the interval [α, β]. The operational matrix of its derivative is constructed. The Tau and Galerkin method with the proposed orthogonal polynomials is used to solve the boundary value problems (BVPs) with even order. The effectiveness of these methods is proved through their application to several BVPs