A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order

M. S. Alqurashi; Saima Rashid; Bushra Kanwal; Fahd Jarad; S. K. Elagan; M. S. Alqurashi; Saima Rashid; Bushra Kanwal; Fahd Jarad; S. K. Elagan

doi:10.3934/math.2022819

AIMS Mathematics

2022, Volume 7, Issue 8: 14946-14974. doi: 10.3934/math.2022819

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A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order

1.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
2.
Department of Mathematics, Government College University, Faisalabad, Pakistan
3.
Department of Mathematical Sciences, Fatimah jinnah Women university, Rawalpindi, Pakistan
4.
Department of Mathematics, Çankaya University, Ankara, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
6.
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
7.
Department of Mathematics and Computer Sciences, Faculty of Science, Menoufia University, Shebin Elkom 32511, Egypt

Received: 20 March 2022 Revised: 25 May 2022 Accepted: 30 May 2022 Published: 13 June 2022
MSC : 46S40, 47H10, 54H25

The main objective of the investigation is to broaden the description of Caputo fractional derivatives (in short, CFDs) (of order $ 0 < \alpha < r $) considering all relevant permutations of entities involving $ t_{1} $ equal to $ 1 $ and $ t_{2} $ (the others) equal to $ 2 $ via fuzzifications. Under $ {g\mathcal{H}} $-differentiability, we also construct fuzzy Elzaki transforms for CFDs for the generic fractional order $ \alpha\in(r-1, r) $. Furthermore, a novel decomposition method for obtaining the solutions to nonlinear fuzzy fractional partial differential equations (PDEs) via the fuzzy Elzaki transform is constructed. The aforesaid scheme is a novel correlation of the fuzzy Elzaki transform and the Adomian decomposition method. In terms of CFD, several new results for the general fractional order are obtained via $ g\mathcal{H} $-differentiability. By considering the triangular fuzzy numbers of a nonlinear fuzzy fractional PDE, the correctness and capabilities of the proposed algorithm are demonstrated. In the domain of fractional sense, the schematic representation and tabulated outcomes indicate that the algorithm technique is precise and straightforward. Subsequently, future directions and concluding remarks are acted upon with the most focused use of references.
- fuzzy set theory,
- Elzaki transform,
- Adomian decomposition method,
- nonlinear partial differential equation,
- Caputo fractional derivative
Citation: M. S. Alqurashi, Saima Rashid, Bushra Kanwal, Fahd Jarad, S. K. Elagan. A novel formulation of the fuzzy hybrid transform for dealing nonlinear partial differential equations via fuzzy fractional derivative involving general order[J]. AIMS Mathematics, 2022, 7(8): 14946-14974. doi: 10.3934/math.2022819

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Abstract

The main objective of the investigation is to broaden the description of Caputo fractional derivatives (in short, CFDs) (of order $ 0 < \alpha < r $) considering all relevant permutations of entities involving $ t_{1} $ equal to $ 1 $ and $ t_{2} $ (the others) equal to $ 2 $ via fuzzifications. Under $ {g\mathcal{H}} $-differentiability, we also construct fuzzy Elzaki transforms for CFDs for the generic fractional order $ \alpha\in(r-1, r) $. Furthermore, a novel decomposition method for obtaining the solutions to nonlinear fuzzy fractional partial differential equations (PDEs) via the fuzzy Elzaki transform is constructed. The aforesaid scheme is a novel correlation of the fuzzy Elzaki transform and the Adomian decomposition method. In terms of CFD, several new results for the general fractional order are obtained via $ g\mathcal{H} $-differentiability. By considering the triangular fuzzy numbers of a nonlinear fuzzy fractional PDE, the correctness and capabilities of the proposed algorithm are demonstrated. In the domain of fractional sense, the schematic representation and tabulated outcomes indicate that the algorithm technique is precise and straightforward. Subsequently, future directions and concluding remarks are acted upon with the most focused use of references.

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