Geodesic
Classical solution of the mixed problem for the wave equation on a graph with two edges and a cycle
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 5, Tome 194 (2021), pp. 78-91

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In this paper, using the Fourier method, we obtain a classical solution of the mixed problem for the wave equation on the simplest geometric graph consisting of two edges, one of which forms a cycle. We apply an approach based on the method of contour integration of the resolvent of an operator, which allows one to obtain a classical solution to the problem under minimal conditions on the initial data and, at the same time, to avoid a laborious study of the refined asymptotics of the eigenvalues and eigenfunctions of the corresponding operator. The cases of continuous and summable potentials are considered.
Keywords: mixed problem, wave equation, graph, summable potential, Fourier method. 
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     title = {Classical solution of the mixed problem for the wave equation on a graph with two edges and a cycle},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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M. Sh. Burlutskaya; A. V. Kiseleva; Ya. P. Korzhova. Classical solution of the mixed problem for the wave equation on a graph with two edges and a cycle. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 5, Tome 194 (2021), pp. 78-91. http://geodesic.mathdoc.fr/item/INTO_2021_194_a7/