This paper develops a numerical approach for solving two-dimensional fractal-fractional parabolic partial differential equations. The fractal-fractional derivative is defined in the Atangana-Riemann-Liouville sense with Mittage-Leffler kernel. To solve this equation, we first eliminate the spatial derivatives using peridynamic differential operators. Then, we derive an operational matrix (OM) of fractal-fractional derivative in terms of the Legendre polynomials to simplify the time derivative. The aim of the formulated method is to transform the original problem into an uncomplicated system of linear algebraic equations which can be solved by mathematical software. The applicability of the approach is examined for several examples and numerical results show the computational efficiency of the method.

本文开发了一种求解二维分形分数抛物线偏微分方程的数值方法。分形-分数导数用 Mittage-Leffler 核在 Atangana-Riemann-Liouville 意义下定义。为了求解这个方程，我们首先使用近场动力学微分算子消除空间导数。然后，我们根据勒让德多项式推导出分形分数导数的运算矩阵（OM），以简化时间导数。公式化方法的目的是将原始问题转化为可以通过数学软件求解的简单线性代数方程组。该方法的适用性通过几个例子进行了检验，数值结果显示了该方法的计算效率。