Daniel J Galiffa, Penn State Erie (The Behrend College)


On a Numerical Method for a Homogeneous, Nonlinear, Nonlocal Elliptic Boundary Value Problem

Abstract: In this work we develop a numerical method for a nonlocal equation that is related to various other physical models. We begin by establishing a priori estimates and the existence and uniqueness of the solution to the nonlinear auxiliary problem via the Schauder fixed point theorem. We then prove the existence of the solution to the nonlinear auxiliary problem by defining a continuous compact mapping and utilizing a priori estimates. From this analysis, we then prove the existence and uniqueness of the problem defined above via the Brouwer fixed point theorem. Next, we analyze a discretization of the above problem and show that a solution to the nonlinear difference problem exists and is unique, and that the numerical procedure converges with a sufficient error. We conclude with some examples of the numerical process.