Adaptive-Coefficient Finite Difference Frequency Domain Method for Solving Time-Fractional Cattaneo Equation with Absorbing Boundary Condition

Fractal and Fractional Pub Date : 2024-02-29 DOI:10.3390/fractalfract8030146

Wenhao Xu, Jing Ba, Jianxiong Cao, Cong Luo

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Abstract

The time-fractional Cattaneo (TFC) equation is a practical tool for simulating anomalous dynamics in physical diffusive processes. The existing numerical solutions to the TFC equation generally deal with the Dirichlet boundary conditions. In this paper, we incorporate the absorbing boundary condition as a complex-frequency-shifted (CFS) perfectly matched layer (PML) into the TFC equation. Then, we develop an adaptive-coefficient (AC) finite-difference frequency-domain (FDFD) method for solving the TFC with CFS PML. The corresponding analytical solution for homogeneous TFC equation with a point source is proposed for validation. The effectiveness of the developed AC FDFD method is verified by the numerical examples of four typical TFC models, including the different orders of time-fractional derivatives for both the homogeneous model and the layered model. The numerical examples show that the developed AC FDFD method is more accurate than the traditional second-order FDFD method for solving the TFC equation with the CFS PML absorbing boundary condition, while requiring similar computational costs.

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自适应系数有限差分频域法求解带吸收边界条件的时分数卡塔尼奥方程

时间分数卡塔尼奥（TFC）方程是模拟物理扩散过程中异常动态的实用工具。现有的 TFC 方程数值解一般采用 Dirichlet 边界条件。本文将吸收边界条件作为复频偏移（CFS）完全匹配层（PML）纳入 TFC 方程。然后，我们开发了一种自适应系数（AC）有限差分频域（FDFD）方法，用于求解带有 CFS PML 的 TFC。我们还提出了带有点源的均质 TFC 方程的相应解析解，以进行验证。通过四个典型 TFC 模型的数值示例验证了所开发的交流 FDFD 方法的有效性，其中包括均质模型和分层模型的不同阶次时间分导数。数值示例表明，在求解具有 CFS PML 吸收边界条件的 TFC 方程时，所开发的交流 FDFD 方法比传统的二阶 FDFD 方法更精确，而所需的计算成本却相差无几。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Fractal and Fractional

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