Spectral Tau Algorithm for Certain Coupled System of Fractional Differential Equations via Generalized Fibonacci Polynomial Sequence

Abstract

This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.