This article presents a comprehensive study on the convergence behavior of the Dirichlet-Neumann Waveform Relaxation algorithm applied to solve the time fractional sub-diffusion and diffusion-wave equations in multiple subdomains, considering the presence of some heterogeneous media. Our analysis focuses on estimating the convergence rate of the algorithm and investigates how this estimate varies with different fractional orders. Furthermore, we extend our analysis to encompass the 2D sub-diffusion case. To validate our findings, we conduct numerical experiments to verify the estimated convergence rate. The results confirm the theoretical estimates and provide empirical evidence for the algorithm’s efficiency and reliability. Moreover, we push the boundaries of the algorithm’s applicability by extending it to solve the time fractional Allen-Chan equation, a problem that exceeds our initial theoretical estimates. Remarkably, we observe that the algorithm performs exceptionally well in this extended scenario for both short and long-time windows.

本文对狄利克雷-诺依曼波形松弛算法的收敛行为进行了全面的研究，该算法应用于求解多个子域中的时间分数子扩散和扩散波方程，考虑到某些异质介质的存在。我们的分析重点是估计算法的收敛速度，并研究该估计如何随不同分数阶而变化。此外，我们将分析扩展到包括二维子扩散情况。为了验证我们的发现，我们进行了数值实验来验证估计的收敛速度。结果证实了理论估计，并为算法的效率和可靠性提供了经验证据。此外，我们通过扩展算法来解决时间分数 Allen-Chan 方程，从而突破了算法的适用范围，这个问题超出了我们最初的理论估计。值得注意的是，我们观察到该算法在这种扩展场景中无论是短时间窗口还是长时间窗口都表现得非常好。