Computational Advancements In Spectral Methods For Solving Fractional-Order Differential Equations

In this paper, we broaden the applicability of the operational matrices associated with Vieta–Lucas polynomials by introducing a modified, computationally efficient method. This method is designed within the spectrum of spectral methods to address Bagley–Torvik types of fractional derivative differential equations (FDDE). The Bagley–Torvik equation involves both integer-order and fractional-order derivatives, along with diverse source terms and non-homogeneous initial conditions, making it challenging to find exact solutions. In response to these complexities, we propose a modified spectral method capable of handling Bagley–Torvik types of FDDE with both homogeneous and non-homogeneous initial conditions as well as diverse source terms. The source terms are approximated using a basis set comprising Vieta–Lucas polynomials, while the fractional-order derivative terms in the problems are approximated using the newly proposed fractional-order integral operational matrix. Through the implementation of these approximations, the FDDE is transformed into a system of matrix equations, without the need for implementing the spectral Tau method. The results derived from this proposed technique are compared with other methods, including existing exact solutions. This comparison shows that the results are computed accurately and within a significantly shorter duration by our proposed method. Additionally, we broaden the domain of the proposed technique by solving more generalized FDDE with initial conditions.