Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions
2007 ◽
Vol 137
(2)
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pp. 349-371
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Keyword(s):
This paper is devoted to the study of time-periodic solutions to the nonlinear one-dimensional wave equation with x-dependent coefficients u(x)ytt – (u(x)yx)x + g(x,t,y) = f(x,t) on (0,π) × ℝ under the periodic boundary conditions y(0,t) = y(π,t), yx(0,t) = yx(π,t) or anti-periodic boundary conditions y(0, t) = –y(π,t), yx[0,t) = – yx(π,t). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. Our main concept is that of the ‘weak solution’. For T, the rational multiple of π, we prove some important properties of the weak solution operator. Based on these properties, the existence and regularity of weak solutions are obtained.
2008 ◽
Vol 465
(2103)
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pp. 895-913
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2020 ◽
Vol 139
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pp. 356-382
2019 ◽
Vol 39
(2)
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pp. 403-412
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Keyword(s):
2019 ◽
Vol 32
(2)
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pp. 1065-1084
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2006 ◽
Vol 229
(2)
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pp. 466-493
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2003 ◽
Vol 46
(15)
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pp. 2911-2916
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2018 ◽
Vol 124
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pp. 533-542
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Keyword(s):