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A time-periodic solution of the equation of forced vibrations of a string with homogeneous boundary conditions. (English. Russian original) Zbl 1068.35073

Differ. Equ. 39, No. 11, 1633-1638 (2003); translation from Differ. Uravn. 39, No. 11, 1550-1555 (2003).
Summary: We consider the problem \[ \begin{aligned} u_{tt}-u_{xx}+g(u)= f(x,t),\;0<x<\pi, \quad & t\in\mathbb R,\tag{1}\\ u(0,t)-h_1u_x'(0,t)=0,\;u(\pi,t)+h_2u_x'(\pi,t)=0, \quad & t\in\mathbb R,\tag{2}\\ u(x,t+T)=u(x,t),\;0<x<\pi,\quad & t\in\mathbb R.\tag{3} \end{aligned} \] Here \(h_1,h_2>0\), \(T=2 \pi a/b\), where \(a\) and \(b\) are coprime positive integers, and \(f(x,t)\) is a given \(T\)-periodic function of \(t\). In the case of Dirichlet boundary conditions \((h_1=h_2=0)\) the inverse of the linear part of (1) is a compact operator on the complement of the kernel. This is not true for the problem with general boundary conditions (2), which is the main difficulty.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B10 Periodic solutions to PDEs
74K05 Strings

Keywords:

Fourier series
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