scholarly journals Blow Up of Quintic Power 1-D Nonlinear Schroedinger Equation

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Blow Up ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Heat Equations ◽  
Adaptive Grids ◽  
Time Singularity ◽  
One Dimensional ◽  
Blowing Up ◽  
Similarity Scaling ◽  
The One

This work studies an adaptive finite difference approximation to the one dimensional nonlinear Schroedinger equiation with quintic power, with special emphasis on the case when the solution blows up with finite blowing-up time $T_\infty.$ The adaptivity is utilizing similarity scaling adaptive grids studied by Berger and Kohn to study numerical solution of semilinear heat equations with finite blowing-up time.Furthermore, we reports an asymptotic behavior of the blow-up solution approaching $T_\infty$ time singularity.

AN AUDIO-FREQUENCY CIRCUIT MODEL OF THE ONE-DIMENSIONAL SCHROEDINGER EQUATION AND ITS SOURCES OF ERROR

1955 ◽  
Vol 33 (8) ◽  
pp. 483-491
Author(s):  
J. H. Blackwell ◽  
D. R. Fewer ◽  
L. J. Allen ◽  
R. S. Cass
Keyword(s):  
Finite Difference ◽  
Schroedinger Equation ◽  
Circuit Model ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Audio Frequency ◽  
Previous Theory ◽  
One Dimensional ◽  
The Past ◽  
The One

An audio-frequency circuit model of the one-dimensional Schroedinger equation has been constructed, based on original suggestions by G. Kron. The previous theory of such devices has been re-examined thoroughly and errors due to the basic finite-difference approximation separated from those due to electrical causes. It is found that the former type of error is likely to be dominant in practice and that in the past discrepancies due to errors of this kind have actually been ascribed to experimental causes.

scholarly journals A fully discrete approximation of the one-dimensional stochastic heat equation

2018 ◽  
Vol 40 (1) ◽  
pp. 247-284 ◽  
Author(s):  
Rikard Anton ◽  
David Cohen ◽  
Lluis Quer-Sardanyons
Keyword(s):  
Exact Solution ◽  
Heat Equation ◽  
Stochastic Heat Equation ◽  
Finite Difference Approximation ◽  
Discrete Approximation ◽  
Difference Approximation ◽  
One Dimensional ◽  
Fully Discrete ◽  
Fully Discrete Approximation ◽  
The One

Abstract A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in $L^q(\varOmega )$, for all $q\geqslant 2$, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

scholarly journals Existence and nonexistence of entire solutions to the logistic differential equation

2003 ◽  
Vol 2003 (17) ◽  
pp. 995-1003 ◽  
Author(s):  
Marius Ghergu ◽  
Vicentiu Radulescu
Keyword(s):  
Blow Up ◽  
Entire Solutions ◽  
One Dimensional ◽  
Nonexistence Result ◽  
Nonnegative Solutions ◽  
Existence And Nonexistence ◽  
Logistic Problem ◽  
Whole Space ◽  
The Mean ◽  
The One

We consider the one-dimensional logistic problem(rαA(|u′|)u′)′=rαp(r)f(u)on(0,∞),u(0)>0,u′(0)=0, whereαis a positive constant andAis a continuous function such that the mappingtA(|t|)is increasing on(0,∞). The framework includes the case wherefandpare continuous and positive on(0,∞),f(0)=0, andfis nondecreasing. Our first purpose is to establish a general nonexistence result for this problem. Then we consider the case of solutions that blow up at infinity and we prove several existence and nonexistence results depending on the growth ofpandA. As a consequence, we deduce that the mean curvature inequality problem on the whole space does not have nonnegative solutions, excepting the trivial one.

FREE-BOUNDARY PROBLEM OF THE ONE-DIMENSIONAL EQUATIONS FOR A VISCOUS AND HEAT-CONDUCTIVE GASEOUS FLOW UNDER THE SELF-GRAVITATION

2013 ◽  
Vol 23 (08) ◽  
pp. 1377-1419 ◽  
Author(s):  
MORIMICHI UMEHARA ◽  
ATUSI TANI
Keyword(s):  
Blow Up ◽  
A Priori Estimates ◽  
A Priori ◽  
Global Solvability ◽  
The Self ◽  
Fundamental Result ◽  
System Of Equations ◽  
One Dimensional ◽  
Unique Existence ◽  
The One

In this paper we consider a system of equations describing the one-dimensional motion of a viscous and heat-conductive gas bounded by the free-surface. The motion is driven by the self-gravitation of the gas. This system of equations, originally formulated in the Eulerian coordinate, is reduced to the one in a fixed domain by the Lagrangian-mass transformation. For smooth initial data we first establish the temporally global solvability of the problem based on both the fundamental result for local in time and unique existence of the classical solution and a priori estimates of its solution. Second it is proved that some estimates of the global solution are independent of time under a certain restricted, but physically plausible situation. This gives the fact that the solution does not blow up even if time goes to infinity under such a situation. Simultaneously, a temporally asymptotic behavior of the solution is established.

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