Abstract
In 2005 Correa and Filho established existence and uniqueness results for the nonlinear PDE: −Δu = g(x,u) (∫ 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.
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