Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations

Hussam Aljarrah,Anuar Ishak,Mohammad Alaroud,Maslina Darus

doi:10.3390/math9222868

Abstract

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering.

Highlights

Fractional partial differential equations (FPDEs) play a substantial role in converting several physical systems into mathematical models
It is normally advised to utilize this formula to keep historical cases of the original systems, in addition to its behavior. This will aid in attaining a superior comprehension of those physical systems, decrease computational complexity, and moderate the controller designing without losing any hereditary behaviors
Various numerical and analytical techniques have been developed to investigate the solutions for the systems of FPDEs, such as the Adomian decomposition technique, homotopy perturbation technique, variational iteration technique, reproducing kernel technique, and homotopy perturbation transform technique [5–9]

Summary

Introduction

Fractional partial differential equations (FPDEs) play a substantial role in converting several physical systems into mathematical models. In view of the residual error functions, the main advantage of this RPS algorithm enables the simplicity of computing coefficients within imposed conditions using only differential operators by using the Mathematica software package, unlike other analytical methods that require integral operators which are difficult in the fractional sense. It is not affected by round arithmetic errors and can be implemented without any restrictions on the nature of the system and the type of classification.

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Concepts and

Exact Solution