A Numerical method for a nonlocal elliptic boundary value problem

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.