On a numerical method for a homogeneous, nonlinear, nonlocal, elliptic boundary value problem

Abstract

In this work we develop a numerical method for the equation: − α ( ∫ 0 1 u ( t ) d t ) u ″ ( x ) + [ u ( x ) ] 2 n + 1 = 0 , x ∈ ( 0 , 1 ) , u ( 0 ) = a , u ( 1 ) = b . 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. From this analysis, we then prove the existence and uniqueness to the problem above by defining a continuous compact mapping, utilizing the a priori estimates and 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 error 풪 ( h ) . We conclude with some examples of the numerical process.