Transparent Boundary Conditions for the Heat Equation on Metric Graphs

Abstract

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.