Boundary value problem with Dirichlet, Neumann and Robin conditions on separate parts of the boundary for an inhomogeneous equation of string vibration
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Otar Jokhadze
and
Sergo Kharibegashvili
Abstract
In this paper, the well-posedness of a boundary value problem for an inhomogeneous equation of string vibration with Dirichlet, Neumann and Robin conditions on separate parts of the boundary is proved. In this case, the solution is given in quadratures. If the Robin condition turns into the Neumann condition, then the problem becomes ill-posed. In particular, it is shown that the corresponding homogeneous problem has an infinite number of linearly independent solutions that can be written explicitly.
Funding statement: The work was supported by the Shota Rustaveli National Science Foundation, Grant No. FR-21-7307.
References
[1] A. V. Bitsadze, Equations of Mathematical Physics. Translated from the Russian by V. M. Volosov and I. G. Volosova, “Mir”, Moscow, 1980. Search in Google Scholar
[2] D. G. Bourgin and R. Duffin, The Dirichlet problem for the virbrating string equation, Bull. Amer. Math. Soc. 45 (1939), 851–858. 10.1090/S0002-9904-1939-07103-6Search in Google Scholar
[3] J. Hadamard, On the Dirichlet problem for the hyperbolic case, Proc. Natl. Acad. Sci. USA 28 (1942), 258–263. 10.1073/pnas.28.6.258Search in Google Scholar PubMed PubMed Central
[4]
A. Huber,
Die erste Randwertaufgabe für geschlossene Bereiche bei der Gleichung
[5] F. John, The Dirichlet problem for a hyperbolic equation, Amer. J. Math. 63 (1941), 141–154. 10.2307/2371285Search in Google Scholar
[6] O. M. Jokhadze and S. S. Kharibegashvili, On the Dirichlet type problem for the inhomogeneous equation of string oscillation, Reports of Qualitde 3 (2024), 107–111. Search in Google Scholar
[7] O. M. Jokhadze and S. S. Kharibegashvili, Solution of some problems for the string vibration equation in a half-strip by quadratures, Differ. Equ. 60 (2024), no. 2, 169–179. 10.1134/S0012266124020034Search in Google Scholar
[8] V. I. Korzyuk, The characteristic parallelogram method in an example of the first mixed problem for the one-dimensional wave equation, Dokl. Nats. Akad. Nauk Belarusi 61 (2017), no. 3, 7–13. Search in Google Scholar
[9] A. Sommerfeld, Randwertaufgabe in der Theorie der partielen Differentialgleichungen, Encykl. d. math. Wiss. 2 (1900), 504–560. Search in Google Scholar
[10] N. N. Vakhania, On boundary value problems for the hyperbolic case, J. Complexity 10 (1994), no. 3, 341–355. 10.1006/jcom.1994.1017Search in Google Scholar
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