Solution to the Dirichlet Problem of the Wave Equation on a Star Graph

Gaukhar Arepova (), Ludmila Alexeyeva and Dana Arepova
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Gaukhar Arepova: Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
Ludmila Alexeyeva: Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan
Dana Arepova: Institute of Mathematics and Mathematical Modeling, Almaty 050010, Kazakhstan

Mathematics, 2023, vol. 11, issue 20, 1-14

Abstract: In this paper, the solution to the Dirichlet problem for the wave equation on the star graph is constructed. To begin, we solve the boundary value problem on the interval (on one edge of the graph). We use the generalized functions method to obtain the wave equation with a singular right-hand side. The solution to the Dirichlet problem is determined through the convolution of the fundamental solution with the singular right-hand side of the wave equation. Thus, the solution found on the interval is determined by the initial functions, boundary functions, and their derivatives (the unknown boundary functions). A resolving system of two linear algebraic equations in the space of the Fourier transform in time is constructed to determine the unknown boundary functions. Following inverse Fourier transforms, the solution to the Dirichlet problem of the wave equation on the interval is constructed. After determining all the solutions on all edges and taking the continuity condition and Kirchhoff joint condition into account, we obtain the solution to the wave equation on the star graph.

Keywords: star graph; boundary vertices; internal vertex; Dirichlet problem; Kirchhoff joint condition; wave equation; Fourier transform; generalized functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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