Geodesic
A problem with nonlocal condition for one-dimensional hyperbolic equation
Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 7-14

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In this paper, we study the problem with a dynamic nonlocal condition for the one-dimensional hyperbolic equation, which occurs in the study of rod vibrations. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. Conditions have been obtained for input data, providing unambiguous resolution of the task, proof of the existence and singularity of the problem in the space of Sobolev. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.
Keywords: hyperbolic equation, nonlocal problem, integral conditions, singularity of the solution, solveability of the problem. 
@article{VSGU_2021_27_1_a0,
     author = {A. V. Bogatov},
     title = {A problem with nonlocal condition for one-dimensional hyperbolic equation},
     journal = {Vestnik Samarskogo universiteta. Estestvennonau\v{c}na\^a seri\^a},
     pages = {7--14},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a0/}
}
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A. V. Bogatov. A problem with nonlocal condition for one-dimensional hyperbolic equation. Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 27 (2021) no. 1, pp. 7-14. http://geodesic.mathdoc.fr/item/VSGU_2021_27_1_a0/