Dirichlet and Neumann problems for the heat equation on linear multilink thermal graphs and their solutions

Boundary value problems of thermal conductivity on a line thermal graph are considered, which can be used to study various structures under conditions of thermal heating (cooling). Here, based on the generalized function method, a unified technique has been developed for solving boundary value problems of thermal conductivity, typical for engineering applications. Generalized solutions to nonstationary and stationary boundary value problems of heat conduction on an edge and on a thermal line graph are constructed under various boundary conditions at the ends of the graph and generalized Kirchhoff conditions at its node. Using the properties of the Fourier transformant of the fundamental solution, regular integral representations of solutions to boundary value problems are obtained in analytical form. The solutions obtained make it possible to simulate heat sources of various types, including using singular generalized functions. The method of generalized functions presented here makes it possible to solve a wide class of boundary value problems with local and connected boundary conditions at the ends of the edges of the graph and different transmission conditions at its nodes.