Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations

M. Mossa Al-Sawalha; Osama Y. Ababneh; Rasool Shah; Nehad Ali Shah; Kamsing Nonlaopon; M. Mossa Al-Sawalha; Osama Y. Ababneh; Rasool Shah; Nehad Ali Shah; Kamsing Nonlaopon

doi:10.3934/math.2023264

AIMS Mathematics

2023, Volume 8, Issue 3: 5266-5280. doi: 10.3934/math.2023264

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Research article

Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13110, Jordan
3.
Department of Mathematics, Abdul Wali khan university Mardan, 23200, Pakistan
4.
Department of Mechanical Engineering, Sejong University, Seoul 05006, Korea
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: 21 September 2022 Revised: 05 November 2022 Accepted: 17 November 2022 Published: 14 December 2022
MSC : 33B15, 34A34, 35A20, 35A22, 44A10

This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.
- residual power series,
- fractional-order partial differential equations,
- Laplace transform,
- Caputo operator
Citation: M. Mossa Al-Sawalha, Osama Y. Ababneh, Rasool Shah, Nehad Ali Shah, Kamsing Nonlaopon. Combination of Laplace transform and residual power series techniques of special fractional-order non-linear partial differential equations[J]. AIMS Mathematics, 2023, 8(3): 5266-5280. doi: 10.3934/math.2023264

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Abstract

This paper investigates fractional-order partial differential equations analytically by applying a modified technique called the Laplace residual power series method. The analytical solution was utilized to test the accuracy and precision of the proposed methodologies and shown by tables and graphs. The solution is a convergent series established on Taylor's new form. When determining the series coefficients like RPSM, the fractional derivatives must be calculated every time. We only need to perform a few computations to obtain the coefficients because LRPSM only requires the concept of an infinite limit. The advantage of this method is that it does not require Adomian polynomials or he's polynomials to solve nonlinear problems. As a result, the method's reduced computation size is a strength. The outcome we got supports the idea that the suggested method is the best one for handling any non-linear models that appear in technology and science.

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