Abstract
On the basis of the exact solution of the linear Dirichlet problem \(u_{tt} - u_{xx} = f\left( {x,t} \right)\), \(u\left( {0,t} \right) = u\left( {\pi ,t} \right) = 0,{\text{ }}u\left( {x,0} \right) = u\left( {x,2\pi } \right) = 0,\) \(0 \leqslant x \leqslant \pi ,{\text{ }}0 \leqslant t \leqslant 2\pi ,\) we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u tt − u xx = f(x, t, u, u t).
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REFERENCES
Yu. A. Mitropol'skii and N. G. Khoma, “Periodic solutions of second-order quasilinear hyperbolic equations,” Ukr. Mat. Zh., 47, No.10, 1370–1373 (1995).
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Mitropol'skii, Y.A., Khoma, N.G. & Khoma, S.G. Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order. Ukrainian Mathematical Journal 52, 1068–1074 (2000). https://doi.org/10.1023/A:1005277616733
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DOI: https://doi.org/10.1023/A:1005277616733