Towards Efficient Solutions of Space-Time Fractional Fuzzy Diffusion Equations: A Methodological Approach

This paper aims to introduce a groundbreaking methodology for deriving analytical solutions to the space-time fractional fuzzy diffusion equation. Our approach uniquely incorporates a Caputo generalized Hukuhara fractional derivative (of order \beta \in (0,2]) for the second-order spatial derivative, alongside a fuzzy Caputo-Katugampola generalized Hukuhara time-fractional derivative (of order \alpha \in (0,1)) for the first-order temporal derivative. The primary objective is to develop explicit and fundamental solutions for both the space-time fractional fuzzy diffusion equation and the time fractional fuzzy diffusion equation, encompassing various forms of fuzzy Caputo-Katugampola generalized Hukuhara time-fractional differentiability. We initiate our study by thoroughly analyzing the fuzzy Fourier and fuzzy \wp-Laplace transforms of the equation. To demonstrate the practical utility and effectiveness of our proposed method, we apply it to two specific models: a fuzzy groundwater flow model for computing pressure head, and a fuzzy model for determining the concentration of tumor cells. The results obtained highlight the method's efficiency and precision in addressing the complexities of both the space-time fractional fuzzy diffusion equation and the time fractional fuzzy diffusion equation.