A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

Muhammad Akram; Ghulam Muhammad; Tofigh Allahviranloo; Ghada Ali; Muhammad Akram; Ghulam Muhammad; Tofigh Allahviranloo; Ghada Ali

doi:10.3934/math.2023011

AIMS Mathematics

2023, Volume 8, Issue 1: 228-263. doi: 10.3934/math.2023011

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Research article

A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations

1.
Department of Mathematics, University of the Punjab, New Campus, Lahore-54590, Pakistan
2.
Department of Mathematics, Lahore Garrison University, Lahore 54000, Pakistan
3.
Faculty of Engineering and Natural Sciences, Istinye University, Istanbul, Turkey
4.
Department of Mathematics, King Abdulaziz University Jeddah, Saudi Arabia

Received: 23 August 2022 Revised: 20 September 2022 Accepted: 22 September 2022 Published: 28 September 2022
MSC : 34A08, 03E72, 33E12

The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.
- system of fractional differential equations,
- Mittag-Leffler function,
- fuzzy fractional calculus,
- Caputo fractional derivative,
- diffusion process
Citation: Muhammad Akram, Ghulam Muhammad, Tofigh Allahviranloo, Ghada Ali. A solving method for two-dimensional homogeneous system of fuzzy fractional differential equations[J]. AIMS Mathematics, 2023, 8(1): 228-263. doi: 10.3934/math.2023011

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Abstract

The purpose of this study is to extend and determine the analytical solution of a two-dimensional homogeneous system of fuzzy linear fractional differential equations with the Caputo derivative of two independent fractional orders. We extract two possible solutions to the coupled system under the definition of strongly generalized $ H $-differentiability, uncertain initial conditions and fuzzy constraint coefficients. These potential solutions are determined using the fuzzy Laplace transform. Furthermore, we extend the concept of fuzzy fractional calculus in terms of the Mittag-Leffler function involving triple series. In addition, several important concepts, facts, and relationships are derived and proved as property of boundedness. Finally, to grasp the considered approach, we solve a mathematical model of the diffusion process using proposed techniques to visualize and support theoretical results.

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