Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

Hegagi Mohamed Ali; Kottakkaran Sooppy Nisar; Wedad R. Alharbi; Mohammed Zakarya; Hegagi Mohamed Ali; Kottakkaran Sooppy Nisar; Wedad R. Alharbi; Mohammed Zakarya

doi:10.3934/math.2024274

AIMS Mathematics

2024, Volume 9, Issue 3: 5671-5685. doi: 10.3934/math.2024274

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Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

1.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia
2.
Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia
3.
Physics Department, College of Science, University of Jeddah, Jeddah 23890, Saudi Arabia
4.
Department of Mathematics, College of Science, King Khalid University, P.O. Box 9004, Abha 61413, Saudi Arabia

Received: 20 November 2023 Revised: 12 January 2024 Accepted: 15 January 2024 Published: 30 January 2024
MSC : 33E12, 35R11, 74H10

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.
- nonlinear coupled Jaulent–Miodek equation,
- fractional partial differential equations,
- Mittag-Leffler function,
- approximate solutions,
- nonlinear problems
Citation: Hegagi Mohamed Ali, Kottakkaran Sooppy Nisar, Wedad R. Alharbi, Mohammed Zakarya. Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order[J]. AIMS Mathematics, 2024, 9(3): 5671-5685. doi: 10.3934/math.2024274

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Abstract

In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.

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