APPROXIMATE SOLUTION OF ELLIPTIC BOUNDARY VALUE PROBLEMS BY SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Abstract

This chapter describes the approximate solution of elliptic boundary value problems by systems of ordinary differential equations. It presents a boundary value problem for a nonlinear perturbation of the Laplace equation. The components of a solution to this system approximate the solution to [P] along the lines y = yi, and consequently this approximation procedure has been named the method of lines. This type of approximation is used to investigate existence, uniqueness, and approximation of a problem similar to [P]. It is found that because [P(h)] is a boundary value problem for an infinite system of second order ordinary differential equations, any practical application of this system will require a reduction to a finite dimensional system of equations. The chapter presents results on monotone approximations of solutions of an elliptic boundary value problem on an unbounded domain in R2 obtained by the application of a few results on maximal and minimal solutions of infinite systems of ordinary differential equations.