Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method

Khalid K. Ali; Mohamed S. Mohamed; Weam G. Alharbi; M. Maneea; Khalid K. Ali; Mohamed S. Mohamed; Weam G. Alharbi; M. Maneea

doi:10.3934/math.20241201

AIMS Mathematics

2024, Volume 9, Issue 9: 24654-24676. doi: 10.3934/math.20241201

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Research article

Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method

1.
Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt
2.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
4.
Faculty of Engineering, MTI University, Cairo, Egypt

Received: 20 May 2024 Revised: 15 July 2024 Accepted: 12 August 2024 Published: 22 August 2024
MSC : 65Qxx, 97Mxx

This paper presented the formulation and solution of the time fractional q-deformed tanh-Gordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution's behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields.
- time-fractional q-deformed tanh-Gordon equation,
- controlled Picard technique,
- Laplace transform method,
- existence,
- fixed point theorem,
- fractional calculus
Citation: Khalid K. Ali, Mohamed S. Mohamed, Weam G. Alharbi, M. Maneea. Solving the time fractional q-deformed tanh-Gordon equation: A theoretical analysis using controlled Picard's transform method[J]. AIMS Mathematics, 2024, 9(9): 24654-24676. doi: 10.3934/math.20241201

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Abstract

This paper presented the formulation and solution of the time fractional q-deformed tanh-Gordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution's behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields.

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