Dirichlet and Neumann problems for the heat equation on linear multilink thermal graphs and their solutions
DOI:
https://doi.org/10.70474/3rsyzh34Keywords:
Thermal conductivity, generalized functions, fundamental and generalized solution, Fourier transform, resolving boundary equations, linear graphAbstract
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.
Downloads
References
Dénes K. Theory of finite and infinite graphs, Birkhäuser, 1990.
Lumer G. Connecting of local operators and evolution equations on networks. Lecture Notes in Mathematics, Springer, 1980, 787, 219--234. DOI: https://doi.org/10.1007/BFb0086338
Belov J. Classical solvability of linear parabolic equations on networks, Journal of Differential Equations, 1988, 72, 316--337. 10.1016/0022-0396(88)90158-1 DOI: https://doi.org/10.1016/0022-0396(88)90158-1
Provotorov V. V. Eigen functions of the Sturm-Lowville problem on a star graph, Sb. Mathematics, 2008, 199, 1523--1545. 10.1070/SM2008v199n10ABEH003971 DOI: https://doi.org/10.1070/SM2008v199n10ABEH003971
Yurko V. A. On the reconstruction of Sturm-Liouville operators on graphs, Mathematical Notes, 2006, 79, 4, 619--630. 10.11650/tjm/210302 DOI: https://doi.org/10.1007/s11006-006-0064-0
Yurko V. A. An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs, Analysis on Graphs and Its Applications: Proceedings of Symposia in Pure Mathematics, 2008, 77, 397--408. DOI: https://doi.org/10.1090/pspum/077/2459883
Additional Files
Published
Issue
Section
License
Copyright (c) 2025 Kazakh Mathematical Journal
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
One can find the license terms "CC Attribution-NonCommercial-NoDerivatives 4.0" here.
