2019
Том 71
№ 11

All Issues

On the periodic solutions of the second-order wave equations. V

Khoma G. P., Mitropolskiy Yu. A.

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Abstract

It is established that the linear problem $u_{tt} - a^2 u_{xx} = g(x, t),\quad u(0, t) = u(\pi, t),\quad u(x, t + T) = u(x, t)$ is always solvable in the space of functions $A = \{g:\; g(x, t) = g(x, t + T) = g(\pi - x, t) = -g(-x, t)\}$ provided that $aTq = (2p - 1)\pi, \quad (2p - 1, q) = 1$, where $p, q$ are integers. To prove this statement, an explicit solution is constructed in the form of an integral operator which is used to prove the existence of a solution to aperiodic boundary value problem for nonlinear second order wave equation. The results obtained can be employed in the study of solutions to nonlinear boundary value problems by asymptotic methods.

English version (Springer): Ukrainian Mathematical Journal 45 (1993), no. 8, pp 1244-1251.

Citation Example: Khoma G. P., Mitropolskiy Yu. A. On the periodic solutions of the second-order wave equations. V // Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1115–1121.

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