Affiliations: [a] Department of Mathematics, Khorramabad Branch, Islamic Azad University, Khorramabad, Iran
| [b] Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
| [c] Department of Mathematics, Ardebil Branch, Islamic Azad University, Ardebil, Iran
Correspondence:
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Corresponding author. Reza Ezzati, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran. E-mail: ezati@kiau.ac.ir.
Abstract: This research paper presents an innovative approach for modeling and analyzing complex systems with uncertain data. Our strategy leverages fuzzy calculus and time-fractional differential equations to achieve this goal. Specifically, we propose the utilization of the fuzzy Atangana-Baleanu time-fractional derivative, which incorporates non-singular kernels for fuzzy functions. This derivative type is particularly suitable for qualitative analysis of fractional differential equations in fuzzy space. We establish the existence and uniqueness of solutions for fuzzy linear time-fractional problems based on this differentiability concept. Additionally, we introduce a numerical solution method, namely the fuzzy homotopy perturbation transform method (FHPTM), to solve these problems. To demonstrate the effectiveness and practical applicability of our approach, we provide concrete examples such as the fuzzy time-fractional Advection-Dispersion equation, the fuzzy time-fractional Diffusion equation, and the fuzzy time-fractional Black-Scholes European option pricing problem. These examples not only illustrate the solution steps involved but also showcase the potential of our method in addressing real-world problems. The outcomes of our research underscore the significance of considering fuzzy calculus and time-fractional differential equations when modeling and analyzing intricate systems with uncertain data.
