Boundary Value Problems on a Star Thermal Graph and their Solutions

Abstract

In this study, heat conductivity boundary value problems on a star graph are considered, inspired by engineering applications, e.g., heat conduction phenomena in mesh-like structures. Based on the generalized function method, a unified technique for solving boundary value problems on such graphs is developed. Generalized solutions to transient and stationary boundary value problems are constructed for different conditions at the end edges, with the Kirchhoff conditions at the common node. Regular integral representations of solutions to boundary value problems are obtained using the properties and symmetry of the fundamental solution’s Fourier transform. The derived results allow the action of various heat sources to be simulated, including concentrated ones by using singular generalized functions. The generalized function method enables a wide variety of boundary value problems to be tackled, including those with local boundary conditions at the ends of the graph, and various transmission conditions at the common node. Based on the research, the authors propose an analytical solution method under the action of various heat sources to solve various boundary value problems on a star thermal graph.