scholarly journals Diffusion approximation for the one dimensional Boltzmann-Poisson system

2004 ◽  
Vol 4 (4) ◽  
pp. 1129-1142 ◽  
Author(s):  
N. Ben Abdallah ◽  
◽  
M. Lazhar Tayeb ◽  
Keyword(s):  
Diffusion Approximation ◽  
Poisson System ◽  
One Dimensional ◽  
The One

scholarly journals Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system

2011 ◽  
Vol 4 (4) ◽  
pp. 955-989 ◽  
Author(s):  
Blanca Ayuso ◽  
◽  
José A. Carrillo ◽  
Chi-Wang Shu ◽  
◽  
...  
Keyword(s):  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Poisson System ◽  
One Dimensional ◽  
The One

scholarly journals Computer-assisted Existence Proofs for One-dimensional Schrödinger-Poisson Systems

2020 ◽  
Vol 24 (3) ◽  
pp. 373-391
Author(s):  
Jonathan Wunderlich ◽  
Michael Plum
Keyword(s):  
Approximate Solution ◽  
Ritz Method ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Poisson System ◽  
Eigenvalue Bounds ◽  
One Dimensional ◽  
Existence Proofs ◽  
Norm Bound ◽  
The One

Motivated by the three-dimensional time-dependent Schrödinger-Poisson system we prove the existence of non-trivial solutions of the one-dimensional stationary Schrödinger-Poisson system using computer-assisted methods. Starting from a numerical approximate solution, we compute a bound for its defect, and a norm bound for the inverse of the linearization at the approximate solution. For the latter, eigenvalue bounds play a crucial role, especially for the eigenvalues "close to" zero. Therefor, we use the Rayleigh-Ritz method and a corollary of the Temple-Lehmann Theorem to get enclosures of the crucial eigenvalues of the linearization below the essential spectrum. With these data in hand, we can use a fixed-point argument to obtain the desired existence of a non-trivial solution "nearby" the approximate one. In addition to the pure existence result, the used methods also provide an enclosure of the exact solution.

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