A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

Ahmad El-Ajou,Zeyad Al-Zhour

doi:10.3389/fphy.2021.525250

Abstract

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

Highlights

Many natural phenomena have been modeled through partial differential equations (PDEs), especially in physics, engineering, chemistry, and biology, as well as in humanities [1, 2]
The present work aims to apply the L-residual power series method (RPSM) to construct approximate solutions (ASs) of a hyperbolic system of time-fractional partial differential equations (T-FPDEs) with variable coefficients in the sense of Caputo’s fractional derivative (FD), which are given in the form of the following model: Ut(α)(x, t) A(x, t)Ux(β)(x, t) + B(x, t)U(x, t) + F(x, t), 0 < α, β ≤ 1, x ∈ I, t ≥ 0, (11)
We present some definitions and theories regarding matrix analysis and the matrix fractional power series (FPS), which are playing a central role in constructing the L-RPSM solution to a hyperbolic system of T-FPDEs with variable coefficients

Summary

INTRODUCTION

Many natural phenomena have been modeled through partial differential equations (PDEs), especially in physics, engineering, chemistry, and biology, as well as in humanities [1, 2]. In the last five years, the residual power series method (RPSM) has achieved an advanced rank among the methods used to find ASs for many fractional differential and integral equations. The proposed method called the Laplace-RPSM (L-RPSM) was first introduced by the authors in [28] and used for introducing exact and approximate SSs to the linear and nonlinear neutral FDEs. El-Ajou [29] adapted the new method in creating solitary solutions for the nonlinear time-fractional partial differential equations (T-FPDEs). The present work aims to apply the L-RPSM to construct ASs of a hyperbolic system of T-FPDEs with variable coefficients in the sense of Caputo’s FD, which are given in the form of the following model: Ut(α)(x, t) A(x, t)Ux(β)(x, t) + B(x, t)U(x, t) + F(x, t), 0 < α, β ≤ 1, x ∈ I, t ≥ 0, (11).

PRELIMINARIES OF MATRIX FPS

APPLYING THE L-RPSM TO THE HYPERBOLIC SYSTEM OF T-FPDES

APPLICATIONS AND NUMERICAL SIMULATIONS

CONCLUSION