Comparison of two modified analytical approaches for the systems of time fractional partial differential equations

Aisha Abdullah Alderremy; Rasool Shah; Nehad Ali Shah; Shaban Aly; Kamsing Nonlaopon; Aisha Abdullah Alderremy; Rasool Shah; Nehad Ali Shah; Shaban Aly; Kamsing Nonlaopon

doi:10.3934/math.2023360

AIMS Mathematics

2023, Volume 8, Issue 3: 7142-7162. doi: 10.3934/math.2023360

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Research article

Comparison of two modified analytical approaches for the systems of time fractional partial differential equations

1.
Department of Mathematics, Faculty of Science, King Khalid University, Abha 61413, Saudi Arabia
2.
Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
3.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea
4.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 11884, Egypt
5.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
† These authors contributed equally to this work and are co-first authors

Received: 12 August 2022 Revised: 24 November 2022 Accepted: 25 November 2022 Published: 11 January 2023
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

The aim of this article is to present a comparison of two analytical approaches toward obtaining the solution of the time-fractional system of partial differential equations. The newly proposed approaches are the new approximate analytical approach (NAAA) and Mohand variational iteration transform approach (MVITA). The NAAA is based on the Caputo-Riemann operator and its basic properties with the decomposition procedure. The NAAA provides step wise series form solutions with fractional order, which quickly converge to the exact solution for integer order. The MVITA is based on a variational iteration procedure and uses the Mohand integral transform. The MVITA also provides a series solution without a stepwise solution. Both approaches provide a series form of solutions to the proposed problems. The analytical procedures and obtained results are compared for the proposed problems. The obtained results were also compared with exact solutions for the problems. The obtained result and plots have shown the validity and applicability of the proposed algorithms. Both approaches can be extended for the analytical solution of other physical phenomena in science and technology.
Citation: Aisha Abdullah Alderremy, Rasool Shah, Nehad Ali Shah, Shaban Aly, Kamsing Nonlaopon. Comparison of two modified analytical approaches for the systems of time fractional partial differential equations[J]. AIMS Mathematics, 2023, 8(3): 7142-7162. doi: 10.3934/math.2023360

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Abstract

The aim of this article is to present a comparison of two analytical approaches toward obtaining the solution of the time-fractional system of partial differential equations. The newly proposed approaches are the new approximate analytical approach (NAAA) and Mohand variational iteration transform approach (MVITA). The NAAA is based on the Caputo-Riemann operator and its basic properties with the decomposition procedure. The NAAA provides step wise series form solutions with fractional order, which quickly converge to the exact solution for integer order. The MVITA is based on a variational iteration procedure and uses the Mohand integral transform. The MVITA also provides a series solution without a stepwise solution. Both approaches provide a series form of solutions to the proposed problems. The analytical procedures and obtained results are compared for the proposed problems. The obtained results were also compared with exact solutions for the problems. The obtained result and plots have shown the validity and applicability of the proposed algorithms. Both approaches can be extended for the analytical solution of other physical phenomena in science and technology.

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