A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations

Shabir Ahmad; Aman Ullah; Ali Akgül; Fahd Jarad; Shabir Ahmad; Aman Ullah; Ali Akgül; Fahd Jarad

doi:10.3934/math.2022521

AIMS Mathematics

2022, Volume 7, Issue 5: 9389-9404. doi: 10.3934/math.2022521

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A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations

1.
Department of Mathematics, University of Malakand, Chakdara, Dir Lower, Khyber Pakhtunkhwa, Pakistan
2.
Art and Science Faculty, Department of Mathematics, Siirt University, TR-56100 Siirt, Turkey
3.
Department of Mathematics, Cankaya University, Etimesgut 06790, Ankara, Turkey
4.
King Abdulaziz University Jeddah, Saudi Arabia
5.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan

Received: 08 December 2021 Revised: 21 February 2022 Accepted: 06 March 2022 Published: 11 March 2022
MSC : 35R11

It is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. To obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.
- Yang transform,
- homotopy perturbation method,
- power law kernel
Citation: Shabir Ahmad, Aman Ullah, Ali Akgül, Fahd Jarad. A hybrid analytical technique for solving nonlinear fractional order PDEs of power law kernel: Application to KdV and Fornberg-Witham equations[J]. AIMS Mathematics, 2022, 7(5): 9389-9404. doi: 10.3934/math.2022521

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Abstract

It is important to deal with the exact solution of nonlinear PDEs of non-integer orders. Integral transforms play a vital role in solving differential equations of integer and fractional orders. To obtain analytical solutions to integer and fractional-order DEs, a few transforms, such as Laplace transforms, Sumudu transforms, and Elzaki transforms, have been widely used by researchers. We propose the Yang transform homotopy perturbation (YTHP) technique in this paper. We present the relation of Yang transform (YT) with the Laplace transform. We find a formula for the YT of fractional derivative in Caputo sense. We deduce a procedure for computing the solution of fractional-order nonlinear PDEs involving the power-law kernel. We show the convergence and error estimate of the suggested method. We give some examples to illustrate the novel method. We provide a comparison between the approximate solution and exact solution through tables and graphs.

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