Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Humaira Yasmin; Aljawhara H. Almuqrin; Humaira Yasmin; Aljawhara H. Almuqrin

doi:10.3934/math.2024811

AIMS Mathematics

2024, Volume 9, Issue 6: 16721-16752. doi: 10.3934/math.2024811

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Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Humaira Yasmin ^{1,2
,
,},
Aljawhara H. Almuqrin ³

1.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 24 March 2024 Revised: 02 May 2024 Accepted: 07 May 2024 Published: 14 May 2024
MSC : 34G20, 35A20, 35A22, 35R11

Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.
- fractional heat equations,
- diffusion equations,
- Burger's equations,
- Aboodh residual power series method,
- Aboodh transform iterative method,
- Caputo operator
Citation: Humaira Yasmin, Aljawhara H. Almuqrin. Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies[J]. AIMS Mathematics, 2024, 9(6): 16721-16752. doi: 10.3934/math.2024811

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Abstract

Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.

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