Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm

M. Mossa Al-Sawalha; Azzh Saad Alshehry; Kamsing Nonlaopon; Rasool Shah; Osama Y. Ababneh; M. Mossa Al-Sawalha; Azzh Saad Alshehry; Kamsing Nonlaopon; Rasool Shah; Osama Y. Ababneh

doi:10.3934/math.20221082

AIMS Mathematics

2022, Volume 7, Issue 11: 19739-19757. doi: 10.3934/math.20221082

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Research article

Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand
4.
Department of Mathematics, Abdul Wali khan university, Mardan 23200, Pakistan
5.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan

Received: 24 July 2022 Revised: 22 August 2022 Accepted: 26 August 2022 Published: 06 September 2022
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

With effective techniques like the homotopy perturbation approach and the Adomian decomposition method via the Yang transform, the time-fractional vibration equation's solution is found for large membranes. In Caputo's sense, the fractional derivative is taken. Numerical experiments with various initial conditions are carried out through a few test examples. The findings are described using various wave velocity values. The outcomes demonstrate the competence and reliability of this analytical framework. Figures are used to discuss the solution of the fractional vibration equation using the suggested strategies for different orders of memory-dependent derivative. The suggested approaches reduce computation size and time even when the accurate solution of a nonlinear differential equation is unknown. It is helpful for both small and large parameters. The results show that the suggested techniques are trustworthy, accurate, appealing and effective strategies.
- homotopy perturbation method,
- Adomian decomposition method,
- Yang transform,
- fractional vibration equation,
- Caputo operator
Citation: M. Mossa Al-Sawalha, Azzh Saad Alshehry, Kamsing Nonlaopon, Rasool Shah, Osama Y. Ababneh. Approximate analytical solution of time-fractional vibration equation via reliable numerical algorithm[J]. AIMS Mathematics, 2022, 7(11): 19739-19757. doi: 10.3934/math.20221082

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Abstract

With effective techniques like the homotopy perturbation approach and the Adomian decomposition method via the Yang transform, the time-fractional vibration equation's solution is found for large membranes. In Caputo's sense, the fractional derivative is taken. Numerical experiments with various initial conditions are carried out through a few test examples. The findings are described using various wave velocity values. The outcomes demonstrate the competence and reliability of this analytical framework. Figures are used to discuss the solution of the fractional vibration equation using the suggested strategies for different orders of memory-dependent derivative. The suggested approaches reduce computation size and time even when the accurate solution of a nonlinear differential equation is unknown. It is helpful for both small and large parameters. The results show that the suggested techniques are trustworthy, accurate, appealing and effective strategies.

References

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