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

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.

Analytical Approximate Solutions of Caputo Fractional KdV-Burgers Equations Using Laplace Residual Power Series Technique

“…A variety of techniques is employed to solve several F-PDEs. The L-RPSM was created in 2020 by Eriqat et al [5] to obtain the analytical approximate series solutions of the linear and nonlinear neutral fractional pantograph equations, and this method was subsequently used to investigate the exact and approximate (solitary, vector) solutions for various linear and nonlinear time-F-PDEs [8,9]. The L-RPSM is constructed based on the LT and RPSM by transforming the differential equations to the Laplace space and then using an appropriate expansion to solve the new equation.…”

“…In the past two years, many works have employed the L-RPSM in providing accurate and approximate solutions to many F-PDEs, for example, fractional Fisher's equation and logistic system model [10], nonlinear fractional reaction-diffusion for bacteria growth model [11], fractional Lane-Emden equations [12], fuzzy quadratic Riccati FDE [13], a hyperbolic system of Caputo time-F-PDEs with variable coefficients [9], time-fractional Navier-Stokes equations [6], and time-fractional nonlinear water wave partial differential [7].…”

A modern analytic method to solve singular and non-singular linear and non-linear differential equations

“…In addition, the LT was employed in dealing with non-linear problems because it is known that the LT deals only with some categories of linear equations. In 2021, El-Ajou [38] adapted the LRPS method to establish solitary solutions of non-linear time-fractional dispersive partial differential equations and to introduce a vector series solution of some types of hyperbolic system of Caputo timefractional partial differential equations with variable coefficients [39]. Recently, the LRPS method was used for solving timefractional Navier-Stokes equations [40], fuzzy quadratic Riccati DEs [41], Lane-Emden equations of fractional order [42], a system of fractional initial value problems (IVPs) [43], autonomous n-dimensional fractional non-linear systems [44], and others [45][46][47][48][49].…”

A solution for the neutron diffusion equation in the spherical and hemispherical reactors using the residual power series

“…This modification is a regeneration of RPSM and is presented for the first time in this work. This modification relies on using a fractional derivative operator, such as the Caputo fractional derivative (CFD) [30,31] or the conformable fractional derivative [21,32], in differential equations of the integer order. Since the equations to be solved are singular, we impose the solution in a Frobenius series form.…”