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

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 s...

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Main Authors:	Hussam Aljarrah, Mohammad Alaroud, Anuar Ishak, Maslina Darus
Format:	article
Language:	EN
Published:	MDPI AG 2021
Subjects:	residual function approximate solution Caputo fractional derivative multiple fractional power series Mathematics QA1-939
Online Access:	https://doaj.org/article/a1ed6a12b9e5493b8107e3409e29903d
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spelling	oai:doaj.org-article:a1ed6a12b9e5493b8107e3409e29903d2021-11-25T18:16:44ZAdaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations10.3390/math92228682227-7390https://doaj.org/article/a1ed6a12b9e5493b8107e3409e29903d2021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2868https://doaj.org/toc/2227-7390In 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.Hussam AljarrahMohammad AlaroudAnuar IshakMaslina DarusMDPI AGarticleresidual functionapproximate solutionCaputo fractional derivativemultiple fractional power seriesMathematicsQA1-939ENMathematics, Vol 9, Iss 2868, p 2868 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	residual function approximate solution Caputo fractional derivative multiple fractional power series Mathematics QA1-939
spellingShingle	residual function approximate solution Caputo fractional derivative multiple fractional power series Mathematics QA1-939 Hussam Aljarrah Mohammad Alaroud Anuar Ishak Maslina Darus Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
description	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.
format	article
author	Hussam Aljarrah Mohammad Alaroud Anuar Ishak Maslina Darus
author_facet	Hussam Aljarrah Mohammad Alaroud Anuar Ishak Maslina Darus
author_sort	Hussam Aljarrah
title	Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
title_short	Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
title_full	Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
title_fullStr	Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
title_full_unstemmed	Adaptation of Residual-Error Series Algorithm to Handle Fractional System of Partial Differential Equations
title_sort	adaptation of residual-error series algorithm to handle fractional system of partial differential equations
publisher	MDPI AG
publishDate	2021
url	https://doaj.org/article/a1ed6a12b9e5493b8107e3409e29903d
work_keys_str_mv	AT hussamaljarrah adaptationofresidualerrorseriesalgorithmtohandlefractionalsystemofpartialdifferentialequations AT mohammadalaroud adaptationofresidualerrorseriesalgorithmtohandlefractionalsystemofpartialdifferentialequations AT anuarishak adaptationofresidualerrorseriesalgorithmtohandlefractionalsystemofpartialdifferentialequations AT maslinadarus adaptationofresidualerrorseriesalgorithmtohandlefractionalsystemofpartialdifferentialequations
_version_	1718411400090484736

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

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