The fractional derivative with nonsingular kernel has been widely used to many physical fields which was shown to offer a new insight into the mathematical modeling of natural phenomena. In this paper, we first study the well-posedness and regularity of the multidimensional variable-order time-fractional reaction–diffusion equation with Caputo–Fabrizio fractional derivative and then present and analyze a Galerkin finite element approximation to the proposed model based on the proved smoothing properties of the solutions. To improve the computational efficiency of the variable-order Caputo–Fabrizio derivative, we approximate the kernel of the fractional derivative by the K-term truncation of its Taylor expansion at a fixed fractional order to develop a fast evaluation scheme which significantly reduces the memory requirement from $\mathcal{O}(N)$ to $\mathcal{O}(K)$ and the computational complexity from $\mathcal{O}(N^2)$ to $\mathcal{O}(K^{2}N)$ where N refers to the number of time steps. We accordingly develop a fast Galerkin finite element method to the proposed model. Numerical experiments are carried out to substantiate the theoretical findings and to show the performance of the fast method.

中文翻译：

---

非奇异核变阶时间分数阶反应扩散方程的数学分析与高效有限元逼近

具有非奇异核的分数阶导数已被广泛应用于许多物理领域，它被证明为自然现象的数学建模提供了新的见解。在本文中，我们首先研究了具有 Caputo-Fabrizio 分数阶导数的多维变阶时间分数阶反应扩散方程的适定性和正则性，然后基于已证明的模型提出并分析了所提出模型的 Galerkin 有限元近似解决方案的平滑特性。为了提高变阶 Caputo–Fabrizio 导数的计算效率，我们通过 K 来近似分数阶导数的核- 以固定分数阶截断其泰勒展开式以开发快速评估方案，显着降低内存需求$\mathcal{欧}(否)$到$\mathcal{欧}(钾)$以及来自的计算复杂度$\mathcal{欧}({否}^{\mathrm{2个}})$到$\mathcal{欧}({钾}^{\mathrm{2个}}否)$其中N指的是时间步数。因此，我们为所提出的模型开发了一种快速的 Galerkin 有限元方法。进行了数值实验以证实理论发现并显示快速方法的性能。