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We prove that the mixed problem for the Klein–Gordon–Fock equation utt(x, t) − uxx(x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle QT = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ Lp[0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class Lp(QT) for p ≥ 1. We construct the solution in explicit analytic form.
We consider mixed initial-boundary value problems for longitudinal vibrations described by the telegraph equation in the case of a system consisting of several parts with different densities and elasticities but with equal impedances. We consider the cases of control by displacements at both endpoints of the rod, by elastic forces at both endpoints, and by an elastic force at one endpoint and a displacement...
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