Mixed Convection in an Inclined Channel Filled with Porous Material Having Time-Periodic Boundary Conditions: Steady-Periodic Regime

This paper is concerned with the existence of time-periodic solutions to the nonlinear wave equation with x -dependent coefficients u ( x ) y tt − ( u ( x ) y x ) x + au ( x ) y +| y | p −2 y = f ( x , t ) on (0, π )× under the periodic or anti-periodic boundary conditions y (0, t )=± y ( π , t ), y x (0, t )=± y x ( π , t ) and the time-periodic conditions y ( x , t + T )= y ( x , t ), y t ( x , t + T )= y t ( x , t ). Such a model arises from the forced vibrations of a non-homogeneous string and the propagation of seismic waves in non-isotropic media. A main concept is the notion ‘weak solution’ to be given in §2. For T =2 π / k ( k ∈ ), we establish the existence of time-periodic solutions in the weak sense by investigating some important properties of the wave operator with x -dependent coefficients.

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Numerical simulation of natural convection in an inclined porous cavity under time-periodic boundary conditions with a partially active thermal side wall

RSC Advances ◽

10.1039/c6ra28333k ◽

2017 ◽

Vol 7 (28) ◽

pp. 17519-17530 ◽

Cited By ~ 6

Author(s):

Feng Wu ◽

Gang Wang

Keyword(s):

Numerical Simulation ◽

Natural Convection ◽

Boundary Conditions ◽

Periodic Boundary ◽

Side Wall ◽

Periodic Boundary Conditions ◽

Porous Cavity ◽

Thermal Boundary Conditions ◽

The Right ◽

Time Periodic

Natural convection in an inclined porous cavity with positively or negatively inclined angles is studied numerically for time-periodic boundary conditions on the left side wall and partially active thermal boundary conditions on the right wall.

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Time-periodic solutions to the one-dimensional wave equation with periodic or anti-periodic boundary conditions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210505001174 ◽

2007 ◽

Vol 137 (2) ◽

pp. 349-371 ◽

Cited By ~ 17

Author(s):

Shuguan Ji ◽

Yong Li

Keyword(s):

Wave Equation ◽

Weak Solution ◽

Boundary Conditions ◽

Periodic Solutions ◽

Periodic Boundary ◽

Periodic Boundary Conditions ◽

One Dimensional ◽

Time Periodic Solutions ◽

Time Periodic ◽

Dimensional Wave

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.

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