Spatially One-Dimensional Boundary Value Problems of Coupled Thermoelasticity: Generalized Functions Method

Alexeyeva, L. A.; Akhmetzhanova, M. M.

doi:10.3103/s0025654422080325

Cited by 1 publication

(2 citation statements)

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“…To solve this Dirichlet problem, we use the method of generalized functions [32][33][34]. For a regular function u(x, t) in the space of generalized functions of slow growth S (R 2 ), we obtain a wave equation with a singular right-hand side:…”

Section: Statement Of the Dirichlet Problem On One Edgementioning

confidence: 99%

“…The basic element for a wave equation on a graph is a finite-length edge; therefore, Dirichlet problems are considered on each edge. The novelty of this work is that a method of generalized functions has been developed to solve these problems [32][33][34]. This method transforms the Dirichlet boundary value problems on each edge into wave equations with a singular right-hand side in the space of generalized functions.…”

Section: Introductionmentioning

confidence: 99%

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

Arepova,

Alexeyeva,

Arepova

2023

Mathematics

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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.

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Section: Statement Of the Dirichlet Problem On One Edgementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%