AI 中文总结
该研究针对度量星形图建模的分支准一维区域的时变热方程，结合透明边界条件与网络偏微分方程理论，推导了消除热回流的精确顶点条件，经数值方法验证，为低维结构热扩散控制提供了数学框架。
AI 中文摘要
我们研究以度量星形图建模的分支准一维区域中，时变热方程的透明边界条件（TBCs）。通过结合经典的TBC概念与网络上的偏微分方程（PDE）理论，我们推导了一种精确的顶点条件，确保热流在节点处无阻碍地传输。具体而言，我们推导了扩散系数的求和规则，消除了节点处的热回流。我们使用Crank-Nicolson有限差分法通过数值方法验证了该分析模型的有效性，证实了热通过节点的平滑、无阻碍传播。研究结果为低维结构中热扩散的可调控制与优化提供了实用的数学框架，特别是结合了知名的TBC概念与度量图上PDE的理论，提出了一种用于网络热扩散控制的数学模型工具。
英文摘要
We study transparent boundary conditions (TBCs) for the time-dependent heat equation in a branching, quasi-one-dimensional domain modeled as a metric star graph. By combining the classical concept of TBCs with the theory of partial differential equations (PDEs) on networks, we derive an exact vertex condition that ensures unobstructed thermal flow across junctions. Specifically,we derive a sum rule for the diffusion coefficients that eliminates thermal backflow at the vertex.We demonstrate the validity of this analytical model numerically using a Crank-Nicolson finite-difference method, which confirms smooth, unobstructed heat propagation through the network.These results provide a practical mathematical framework for tunable control and optimization of thermal diffusion in low-dimensional structures. In particular, combining the well-known concept of the TBCs and theory for PDE on metric graphs, we propose a mathematical model providing a control tool for thermal diffusion in networks.
Comments8 pages, 7 figures
