Numerical Approximation of the One-Dimensional Vlasov–Poisson System with Periodic Boundary Conditions

1996 ◽  
Vol 33 (4) ◽  
pp. 1377-1409 ◽  
Author(s):  
Stephen Wollman ◽  
Ercument Ozizmir
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Numerical Approximation ◽  
Periodic Boundary Conditions ◽  
Poisson System ◽  
One Dimensional ◽  
The One

scholarly journals PERIODIC SOLUTIONS OF THE VLASOV–POISSON SYSTEM WITH BOUNDARY CONDITIONS

2000 ◽  
Vol 10 (05) ◽  
pp. 651-672 ◽  
Author(s):  
M. BOSTAN ◽  
F. POUPAUD
Keyword(s):  
Boundary Conditions ◽  
Periodic Solutions ◽  
Periodic Boundary ◽  
Space Dimension ◽  
Periodic Boundary Conditions ◽  
Small Data ◽  
Poisson System ◽  
One Dimensional ◽  
The One ◽  
Time Periodic

We study the Vlasov–Poisson system with time periodic boundary conditions. For small data we prove existence of weak periodic solutions in any space dimension. In the one-dimensional case the result is stronger: we obtain existence of mild solution and uniqueness of this solution when the data are smooth. It is necessary to impose a nonvanishing condition for the incoming velocities in order to control the lifetime of particles in the domain.

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

2007 ◽  
Vol 137 (2) ◽  
pp. 349-371 ◽  
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.

scholarly journals Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

2015 ◽  
Vol 28 (1) ◽  
pp. 49-67 ◽  
Author(s):  
M. D. Korzec ◽  
P. Nayar ◽  
P. Rybka
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Initial Conditions ◽  
Periodic Boundary Conditions ◽  
Global Attractors ◽  
Sixth Order ◽  
Two Dimensional ◽  
One Dimensional ◽  
Crystalline Surface ◽  
Dimensional Version

Abstract A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface h(x, y, t) that undergoes faceting is considered with periodic boundary conditions, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes $$u_1=h_{x}$$ u 1 = h x and $$u_2=h_y$$ u 2 = h y to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in $$\dot{H}^2_{per}$$ H ˙ p e r 2 , we consider the solution operator $$S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}$$ S ( t ) : H ˙ p e r 2 → H ˙ p e r 2 , to gain our results. We prove the necessary continuity, dissipation and compactness properties.

THE DISTRIBUTION OF PERIODIC AND APERIODIC PATTERN EVOLUTIONS IN RINGS OF DIFFUSIVELY COUPLED MAPS

2001 ◽  
Vol 11 (10) ◽  
pp. 2647-2661 ◽  
Author(s):  
PEDRO G. LIND ◽  
JOÃO A. M. CORTE-REAL ◽  
JASON A. C. GALLAS
Keyword(s):  
Boundary Conditions ◽  
Parameter Space ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
One Dimensional ◽  
Quadratic Maps ◽  
Coupled Maps

This paper reports histograms showing the detailed distribution of periodic and aperiodic motions in parameter-space of one-dimensional lattices of diffusively coupled quadratic maps subjected to periodic boundary conditions. Particular emphasis is given to the parameter domains where lattices support traveling patterns.

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