Approximate Analytical Solution for a Coupled System of Fractional Nonlinear Integrodifferential Equations by the RPS Method

Ayed Al E’Damat

doi:10.1155/2020/1010382

Abstract

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

Highlights

V2(0) v2,0, where 0 < β ≤ 1. e residual power series method (RPSM) has a wide range of applications, especially in simulating nonlinear issues in a fractional meaning, which has been developed and modified over recent years as a powerful mathematical treatment indispensable in dealing with the emerging realistic system in physics, engineering, and natural sciences [25,26,27,28]. It is a modern analytical and approximation technique that relies on the expansion of the fractional power series and residual error functions, which was first proposed in 2013 to provide analytical series solutions to fuzzy differential equations of the first and second orders and minimize the residual errors. is method has many advantages and properties as follows: it is an accurate alternative instrument, it requires less effort to achieve results, it provides a rapid convergence rate to the exact solution, it deals directly with different types of nonlinear terms and complex functions, and it has the ability to choose any point in the integration domain, making the approximate solution applicable
Freihet et al [29] have used the fractional power series for solving the fractional stiff system and introduced some basic theorems related to RPS generalization in the sense of Caputo fractional derivative. e (2 + 1)-dimensional timefractional Burgers–Kadomtsev–Petviashvili equation has been solved by the RPS method [30]
A class of a coupled system of nonlinear fractional integrodifferential equations of fractional order has been discussed by using the RPS method under the Caputo fractional derivative. e RPS algorithm has been given to optimize the approximate solution by minimizing a residual error function with the help of generalized Taylor formula

Summary

Introduction

V2(0) v2,0, where 0 < β ≤ 1. e residual power series method (RPSM) has a wide range of applications, especially in simulating nonlinear issues in a fractional meaning, which has been developed and modified over recent years as a powerful mathematical treatment indispensable in dealing with the emerging realistic system in physics, engineering, and natural sciences [25,26,27,28]. it is a modern analytical and approximation technique that relies on the expansion of the fractional power series and residual error functions, which was first proposed in 2013 to provide analytical series solutions to fuzzy differential equations of the first and second orders and minimize the residual errors. is method has many advantages and properties as follows: it is an accurate alternative instrument, it requires less effort to achieve results, it provides a rapid convergence rate to the exact solution, it deals directly with different types of nonlinear terms and complex functions, and it has the ability to choose any point in the integration domain, making the approximate solution applicable. Is paper aims to introduce a recent analytical as well as numerical method based on the use of fractional residual power series (RPS) technique for obtaining the approximate solution for a class of coupled system of fractional integrodifferential equations in the following form: b 3. Fractional RPS Method for the Coupled System of IDEs e purpose of this section is to construct FPS solution for the coupled system of nonlinear fractional integrodifferential equations (1) and (2) by substituting the FPS expansion among the truncated residual functions.

Results

Conclusion