In this present article, a model of the fractional diffusion equation in a fuzzy environment is studied with both singular and nonsingular kernels with a Mittag-Leffler kernel. In this model, initial boundary conditions and coefficients are fuzzy numbers. First ofall, we derive the Legendre operational matrix of fractional differentiation concerning the power kernel and Mittag-Leffler kernel. We used the spectral method in addition to these derived operational matrices to find out the numerical solution of the taken model. This method is easily applicable to fuzzy partial differential equation (PDE) with different fractional operators. It reduced the given model into algebraic equations, which with further solving gives the solution of the model. The feasibility and accuracy of the method on a fractional fuzzy PDE can be seen through the numerical examples in which we incorporated the error table calculated between exact and numerical solution. The dynamics of the model concerning different parameters present in the model are presented in thorough figures. The application of this model in porous media is presented.

中文翻译：

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多孔介质中应用的平动-反应-扩散方程分数模糊模型

在本文中，研究了模糊环境中分数扩散方程的模型，该模型使用奇异核和非奇异核以及 Mittag-Leffler 核。在这个模型中，初始边界条件和系数是模糊数。首先，我们推导了关于幂核和Mittag-Leffler核的分数微分Legendre运算矩阵。除了这些导出的运算矩阵，我们还使用了谱法来找出所采用模型的数值解。该方法很容易适用于具有不同分数算子的模糊偏微分方程（PDE）。它将给定的模型简化为代数方程，通过进一步求解得到模型的解。通过数值例子可以看出该方法在分数模糊偏微分方程上的可行性和准确性，其中我们结合了精确解和数值解之间计算的误差表。关于模型中存在的不同参数的模型动力学以详尽的数字呈现。介绍了该模型在多孔介质中的应用。