A New Scheme for Solving Multiorder Fractional Differential Equations Based on Müntz–Legendre Wavelets

Haifa Bin Jebreen,Fairouz Tchier,Átila Madureira Bueno

doi:10.1155/2021/9915551

Abstract

In this study, we apply the pseudospectral method based on Müntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. Using the operational matrix for the Caputo derivative operator and applying the Chebyshev and Legendre zeros, the problem is reduced to a system of linear algebraic equations. We illustrate the reliability, efficiency, and accuracy of the method by some numerical examples. We also compare the proposed method with others and show that the proposed method gives better results.

Highlights

We apply the pseudospectral method based on Muntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative
E existence and uniqueness of L1 solution of the multiorder fractional differential equation with the Riemann–Liouville and Caputo fractional derivatives under the assumption that f(x, t) ∈ L1[0, 1] satisfies the Lipschitz condition with respect to the second variable are investigated by Kilbas et al [1]. e previous investigation is based on converting equation (2) into the equivalent Volterra integral equation and solving it
We represent the Caputo fractional derivative operator in the Muntz–Legendre wavelets. e results illustrate that by selecting the proper value for μ, the proposed method gives better results than others. e most important advantage of this method over other methods is its flexibility and ease of use

Summary

Introduction

We apply the pseudospectral method based on Muntz–Legendre wavelets to solve the multiorder fractional differential equations with Caputo fractional derivative. The operational matrix of the Caputo fractional derivative has been constructed directly, and using the collocation method, the problem is solved. Dehestani et al [4] applied the fractional-Lucas optimization method to solve the multidimensional and multiorder fractional differential equation with Caputo fractional derivative. To this end, they used the operational matrix of fractional derivative for Lucas functions and reduced the problem into a linear or nonlinear system. A special type of equation (2) is considered by Bhrawy et al [5] as cDαy(t) + cy(t) f(t)

Objectives

Methods

Conclusion