Integral Representations of Solutions to Initial-Boundary Value Problems on Mixed Metric Graphs

Mardonbek Eshimbetov

doi:10.5269/bspm.81722

Mardonbek Eshimbetov National University of Uzbekistan named after Mirzo Ulugbek

Abstract

In this paper, initial-boundary value problems for the heat equation on symmetric metric graphs
are investigated. Based on the Fokas unified transform method, global relations are derived and used to
Establish a correspondence between Dirichlet and Neumann boundary conditions at the vertices of the graph.
The problem is reduced to a system of algebraic equations with respect to the unknown values of the solution
at the branching points of the graph. As a result, an explicit integral representation of the solution in terms
of given initial and boundary data is constructed. The convergence properties of the contour integrals are
analyzed, and the conditions ensuring exponential decay of the integrands are justified. The obtained results
extend the analytical framework for studying diffusion processes on metric graphs and provide a theoretical
basis for modeling heat transfer in complex branched structures.

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