Compatibility of the Paraskevopoulos’s algorithm with operational matrices of Vieta–Lucas polynomials and applications

Abstract

In this study, the numerically stable operational matrices are proposed to approximate the Caputo fractional-order derivatives by introducing an algorithm. The proposed operational matrices are named fractional Vieta–Lucas differentiation matrices that are constructed by using the basis of shifted Vieta–Lucas polynomials (VLPs) and Caputo-fractional derivatives. In addition, we numerically solve the multi-order linear and nonlinear fractional-order differential equations by introducing a new numerical algorithm that is based on Paraskevopoulos’s algorithm together with the newly proposed operational matrices of shifted VLPs. The applicability of Paraskevopoulos’s algorithm was previously studied with the Adomian decomposition technique to solve fractional-order ordinary differential equations. We extend its applicability to the operational matrices technique. However, to the best of our knowledge, no previous study has reported that discusses the applicability of Paraskevopoulos’s algorithm with the operational matrices of shifted VLPs. To demonstrate the advantages of the newly proposed numerical algorithm, the multi-order linear and nonlinear Caputo fractional-order differential equations are solved numerically. The solutions of the first, third, fourth, and fifth examples obtained by using the proposed algorithm are compared with the solutions obtained otherwise by using various numerical approaches including stochastic approach, Taylor matrix method, Bessel collocation method, shifted Jacobi collocation method, spectral Tau method, and Chelyshkov collocation method. It is shown that the proposed numerical algorithm and the fractional Vieta–Lucas differentiation matrices are highly efficient in solving all the aforementioned examples.