Numerical Methods Based on the Hybrid Shifted Orthonormal Polynomials and Block-Pulse Functions for Solving a System of Fractional Differential Equations

Abstract

This paper develops two numerical methods for solving a system of fractional differential equations based on hybrid shifted orthonormal Bernstein polynomials with generalized block-pulse functions (HSOBBPFs) and hybrid shifted orthonormal Legendre polynomials with generalized block-pulse functions (HSOLBPFs). Using these hybrid bases and the operational matrices method, the system of fractional differential equations is reduced to a system of algebraic equations. Error analysis is performed and some simulation examples are provided to demonstrate the efficacy of the proposed techniques. The numerical results of the proposed methods are compared to those of the existing numerical methods. These approaches are distinguished by their ability to work on the wide interval [0, a], as well as their high accuracy and rapid convergence, demonstrating the utility of the proposed approaches over other numerical methods.