TY - JOUR

T1 - A Numerical method for a nonlocal elliptic boundary value problem

AU - Cannon, John R.

AU - Galiffa, Daniel J.

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 2008

Y1 - 2008

N2 - In 2005 Corrêa and Filho established existence and uniqueness results for the nonlinear PDE: -δu = g(x,u)α; ∫Ω f(x,u) β;, which arises in physical models of thermodynamical equilibrium via Coulomb potential, among others [3]. In this work we discuss a numerical method for a special case of this equation: -α(∫1 0 u(t)dt u" = f(x), 0 < x < 1, u(0) = a, u(1) = b. We first consider the existence and uniqueness of the analytic problem using a fixed point argument and the contraction mapping theorem. Next, we evaluate the solution of the numerical problem via a finite difference scheme. From there, the existence and convergence of the approximate solution will be addressed as well as a uniqueness argument, which requires some additional restrictions. Finally, we conclude the work with some numerical examples where an interval-halving technique was implemented.

AB - In 2005 Corrêa and Filho established existence and uniqueness results for the nonlinear PDE: -δu = g(x,u)α; ∫Ω f(x,u) β;, which arises in physical models of thermodynamical equilibrium via Coulomb potential, among others [3]. In this work we discuss a numerical method for a special case of this equation: -α(∫1 0 u(t)dt u" = f(x), 0 < x < 1, u(0) = a, u(1) = b. We first consider the existence and uniqueness of the analytic problem using a fixed point argument and the contraction mapping theorem. Next, we evaluate the solution of the numerical problem via a finite difference scheme. From there, the existence and convergence of the approximate solution will be addressed as well as a uniqueness argument, which requires some additional restrictions. Finally, we conclude the work with some numerical examples where an interval-halving technique was implemented.

UR - http://www.scopus.com/inward/record.url?scp=67349275441&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67349275441&partnerID=8YFLogxK

U2 - 10.1216/JIE-2008-20-2-243

DO - 10.1216/JIE-2008-20-2-243

M3 - Article

AN - SCOPUS:67349275441

VL - 20

SP - 243

EP - 261

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

SN - 0897-3962

IS - 2

ER -