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<p>We implement an approach to description of the solution of the initial-boundary value problem for the wave equation on a finite and bounded geometrical graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. We consider the linear transmission conditions of a more general form than in previous works. The approach is based on interpreting the behavior of the solution at the vertices of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> as boundary regimes with respect to adjacent edges. The set of these boundary regimes turns out to be a solution to the initial-value problem for a system of differential equations with delayed arguments on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\([0;+\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>;</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with infinitely increasing number of delayed arguments as the argument infinitely grows.</p>

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THE METHOD OF BOUNDARY REGIMES FOR SOLUTION OF THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE WAVE EQUATION ON GEOMETRICAL GRAPH

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • V. L. Pryadiev

摘要

We implement an approach to description of the solution of the initial-boundary value problem for the wave equation on a finite and bounded geometrical graph \(\Gamma \) Γ . We consider the linear transmission conditions of a more general form than in previous works. The approach is based on interpreting the behavior of the solution at the vertices of \(\Gamma \) Γ as boundary regimes with respect to adjacent edges. The set of these boundary regimes turns out to be a solution to the initial-value problem for a system of differential equations with delayed arguments on \([0;+\infty )\) [ 0 ; + ∞ ) with infinitely increasing number of delayed arguments as the argument infinitely grows.