# A linear periodic boundary-value problem for a second-order hyperbolic equation

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## Abstract

We study the boundary-value problem*u* _{ tt } -*u* _{ xx } =*g*(*x, t*),*u*(0,*t*) =*u* (π,*t*) = 0,*u*(*x, t* +*T*) =*u*(*x, t*), 0 ≤*x* ≤ π,*t* ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of\(\frac{\pi }{q} - , \frac{{2\pi }}{{2s - 1}} - \), and\(\frac{{4\pi }}{{2s - 1}}\)-periodic functions (*q* and s are natural numbers). We obtain the results only for sets of periods\(T_1 = (2p - 1)\frac{\pi }{q}, T_2 = (2p - 1)\frac{{2\pi }}{{2s - 1}}\), and\(T_3 = (2p - 1)\frac{{4\pi }}{{2s - 1}}\) which characterize the classes of π-, 2π -, and 4π-periodic functions.

## Keywords

Periodic Solution Natural Number Classical Solution Naukova Dumka Unique Function
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## References

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© Kluwer Academic/Plenum Publishers 1999