scholarly journals A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

2017 ◽  
Vol 315 ◽  
pp. 195-207 ◽  
Author(s):  
Luoping Chen ◽  
Bin Zheng ◽  
Guang Lin ◽  
Nikolaos Voulgarakis
Keyword(s):  
Collocation Method ◽  
Elliptic Equations ◽  
Random Coefficients ◽  
Semilinear Elliptic Equations ◽  
Stochastic Collocation ◽  
Stochastic Collocation Method ◽  
Semilinear Elliptic

A Stochastic Collocation Method for Delay Differential Equations with Random Input

2014 ◽  
Vol 6 (4) ◽  
pp. 403-418 ◽  
Author(s):  
Tao Zhou
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Numerical Approach ◽  
Random Coefficients ◽  
Stochastic Collocation ◽  
Stochastic Collocation Method ◽  
Time Space ◽  
Delay Differential ◽  
The Given

AbstractIn this work, we concern with the numerical approach for delay differential equations with random coefficients. We first show that the exact solution of the problem considered admits good regularity in the random space, provided that the given data satisfy some reasonable assumptions. A stochastic collocation method is proposed to approximate the solution in the random space, and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations. Convergence property of the proposed method is analyzed. It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space. Numerical examples are given to illustrate the theoretical results.

scholarly journals Nonexistence Results of Semilinear Elliptic Equations Coupled with the Chern-Simons Gauge Field

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Hyungjin Huh
Keyword(s):  
Gauge Field ◽  
Elliptic Equations ◽  
Semilinear Elliptic Equations ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Nontrivial Solutions ◽  
Semilinear Elliptic ◽  
Chern Simons

We discuss the nonexistence of nontrivial solutions for the Chern-Simons-Higgs and Chern-Simons-Schrödinger equations. The Derrick-Pohozaev type identities are derived to prove it.

scholarly journals Large solutions of semilinear elliptic equations with nonlinear gradient terms

1999 ◽  
Vol 22 (4) ◽  
pp. 869-883 ◽  
Author(s):  
Alan V. Lair ◽  
Aihua W. Wood
Keyword(s):  
Positive Solutions ◽  
Compact Support ◽  
Elliptic Equations ◽  
Nonnegative Function ◽  
Semilinear Elliptic Equations ◽  
Decay Condition ◽  
Large Solutions ◽  
Semilinear Elliptic ◽  
The Given

We show that large positive solutions exist for the equation(P±):Δu±|∇u|q=p(x)uγinΩ⫅RN(N≥3)for appropriate choices ofγ>1,q>0in which the domainΩis either bounded or equal toRN. The nonnegative functionpis continuous and may vanish on large parts ofΩ. IfΩ=RN, thenpmust satisfy a decay condition as|x|→∞. For(P+), the decay condition is simply∫0∞tϕ(t)dt<∞, whereϕ(t)=max|x|=tp(x). For(P−), we require thatt2+βϕ(t)be bounded above for some positiveβ. Furthermore, we show that the given conditions onγandpare nearly optimal for equation(P+)in that no large solutions exist if eitherγ≤1or the functionphas compact support inΩ.

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