New spectral algorithm for fractional delay pantograph equation using certain orthogonal generalized Chebyshev polynomials

This article presents a novel computational algorithm for solving the fractional pantograph differential equation (FPDE). The algorithm is based on introducing a new family of orthogonal polynomials, generalizing the second-kind Chebyshev polynomials family. Specifically, we use the shifted generalized Chebyshev polynomials of the second kind (SGCPs) as basis functions, approximating the solutions as linear combinations of these new polynomials. First, we establish new theoretical results related to these polynomials, which form the foundation of our proposed method. Then, we apply the spectral Galerkin method to convert the FPDE with its initial conditions into an algebraic system that can be numerically solved. Additionally, we analyze the convergence of the proposed expansion. Finally, numerical examples are provided to validate the theoretical results.