In this research an analytic method is used to solve fuzzy heat equation with fuzzy initial values, which is based on GENERALIZED HUKUHARA differentiability. At first, we define fuzzy laplace transform in t considering x as a parameter on fuzzy functions u(x,t) and their DERIVATIVEs by using GENERALIZED HUKUHARA differentiability. The fuzzy laplace transform is used in an analytic method for solving the fuzzy partial differential equation with GENERALIZED HUKUHARA differantiation. Finally, the method is explained by presenting examples.

In this study, an economic production quantity (EPQ) model with deterioration is developed where the production rate is stock dependent and the demand rate is unit selling price and stock dependent. The low unit selling price and more stocks correspond high demand but more stock corresponds to slow production because of the avoidance of unnecessary stocks. First of all, we develop the production model by solving some ordinary differential equations having deterministic profit function under some specific assumptions. Later, we develop the fuzzy model by solving the fuzzy differential equations using GENERALIZED HUKUHARA DERIVATIVE. In fact, the differential equation of the model has been split into two parts namely gH(L-R) and gH(R-L) on the basis of left(L) and right(R) α- cuts of fuzzy numbers for which the problem itself is transformed into multi-objective EPQ problem. A new formula of aggregation of several objective values obtained at different aspiration levels has been discussed to defuzzify the fuzzy multi-objective problems. We solve the crisp and fuzzy models using LINGO software. Numerical and graphical illustrations confirm that the model under GENERALIZED HUKUHARA DERIVATIVE of (R-L) type contributes more profit which is one of the basic novelties of the proposed approach.

In this present study, the tumor growth model using the Gompertz equation with the Allee effect is developed under a fuzzy environment using the GENERALIZED HUKUHARA DERIVATIVE  (GHD) approach. To capture the tumor growth patterns with the Allee threshold, the parameters present in the model vary from time to time, and in real life, it is very difficult to estimate the exact cell count. In this vague situation, the initial condition, coefficient, and both together are taken as the fuzzy number. In this paper, the  GHD  approach is used to solve the fuzzy tumor growth model in which four different cases are considered with respect to (i)-gH differentiability and (ii)-gH differentiability concept. The main objective of this study is to present a significant reduction in uncertainty while modeling the tumor growth in a fuzzy environment with the Allee effect. Finally, the proposed model and technique are illustrated by numerical simulation and analysis of tumor growth is conducted.

An analytical fuzzy solution is achieved by means of the fuzzy d’, Alembert formula for the fuzzy one-dimensional homogeneous wave equation in a half-line considering the GENERALIZED HUKUHARA partial differentiability of the solution. In the current article, the exclusive solution and the stability of the homogeneous fuzzy wave equations are brought into existence. Eventually, given the various instances represented, the efficacy and accuracy of the method are scrutinized.

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.