On the solution of nonlinear fractional-order shock wave equation via analytical method

Azzh Saad Alshehry; Naila Amir; Naveed Iqbal; Rasool Shah; Kamsing Nonlaopon; Azzh Saad Alshehry; Naila Amir; Naveed Iqbal; Rasool Shah; Kamsing Nonlaopon

doi:10.3934/math.20221061

AIMS Mathematics

2022, Volume 7, Issue 10: 19325-19343. doi: 10.3934/math.20221061

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Research article

On the solution of nonlinear fractional-order shock wave equation via analytical method

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Humanities and Sciences, School of Electrical Engineering and Computer Science (SEECS), National University of Sciences and Technology (NUST), Islamabad, Pakistan
3.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
4.
Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan
5.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received: 17 July 2022 Revised: 20 August 2022 Accepted: 23 August 2022 Published: 31 August 2022
MSC : 34A34, 35A20, 35A22, 44A10, 33B15

In this study, we propose a method to study fractional-order shock wave equations and wave equations arising from the motion of gases. The fractional derivative is taken in Caputo manner. The approaches we used are the combined form of the Yang transform (YT) together with the homotopy perturbation method (HPM) called homotopy perturbation Yang transform method (HPYTM) and also Yang transform (YT) with the Adomian decomposition method called Yang transform decomposition method (YTDM). The HPYTM is a combination of the Yang transform, the homotopy perturbation method and He's polynomials, whereas the YTDM is a combination of the Yang transform, the decomposition method and the Adomian polynomials. Adomian and He's polynomials are excellent tools for handling nonlinear terms. The manipulation of the recurrence relation, which generates the series solutions in a limited number of iterations, is the essential innovation we describe in this study. We give several graphical behaviors of the exact and analytical results, absolute error graphs, and tables that highly agree with one another to demonstrate the reliability of the suggested methodologies. The results we obtained by implementing the proposed approaches indicate that it is easy to implement and computationally very attractive.
- homotopy perturbation method,
- Adomian decomposition method,
- Yang transform,
- fractional-order nonlinear shock wave equation,
- Caputo operator
Citation: Azzh Saad Alshehry, Naila Amir, Naveed Iqbal, Rasool Shah, Kamsing Nonlaopon. On the solution of nonlinear fractional-order shock wave equation via analytical method[J]. AIMS Mathematics, 2022, 7(10): 19325-19343. doi: 10.3934/math.20221061

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Abstract

In this study, we propose a method to study fractional-order shock wave equations and wave equations arising from the motion of gases. The fractional derivative is taken in Caputo manner. The approaches we used are the combined form of the Yang transform (YT) together with the homotopy perturbation method (HPM) called homotopy perturbation Yang transform method (HPYTM) and also Yang transform (YT) with the Adomian decomposition method called Yang transform decomposition method (YTDM). The HPYTM is a combination of the Yang transform, the homotopy perturbation method and He's polynomials, whereas the YTDM is a combination of the Yang transform, the decomposition method and the Adomian polynomials. Adomian and He's polynomials are excellent tools for handling nonlinear terms. The manipulation of the recurrence relation, which generates the series solutions in a limited number of iterations, is the essential innovation we describe in this study. We give several graphical behaviors of the exact and analytical results, absolute error graphs, and tables that highly agree with one another to demonstrate the reliability of the suggested methodologies. The results we obtained by implementing the proposed approaches indicate that it is easy to implement and computationally very attractive.

References

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