New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense

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

doi:10.3934/math.20221016

AIMS Mathematics

2022, Volume 7, Issue 10: 18467-18496. doi: 10.3934/math.20221016

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Research article

New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense

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: 06 July 2022 Revised: 03 August 2022 Accepted: 11 August 2022 Published: 18 August 2022
MSC : 34A08, 34K37, 03E72

The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.
Citation: Muhammad Akram, Ghulam Muhammad, Tofigh Allahviranloo, Ghada Ali. New analysis of fuzzy fractional Langevin differential equations in Caputo's derivative sense[J]. AIMS Mathematics, 2022, 7(10): 18467-18496. doi: 10.3934/math.20221016

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Abstract

The extraction of analytical solution of uncertain fractional Langevin differential equations involving two independent fractional-order is frequently complex and difficult. As a result, developing a proper and comprehensive technique for the solution of this problem is very essential. In this article, we determine the explicit and analytical fuzzy solution for various classes of the fuzzy fractional Langevin differential equations (FFLDEs) with two independent fractional-orders both in homogeneous and non-homogeneous cases. The potential solution of FFLDEs is also extracted using the fuzzy Laplace transformation technique. Furthermore, the solution of FFLDEs is defined in terms of bivariate and trivariate Mittag-Leffler functions both in the general and special forms. FFLDEs are a new topic having many applications in science and engineering then to grasp the novelty of this work, we connect FFLDEs with RLC electrical circuit to visualize and support the theoretical results.

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