COMPARISON OF DIRECT TO SHOOTING ENCLOSURES FOR AN INVERSE-MONOTONE BOUNDARY-VALUE PROBLEM WITH LOCALLY STEEP SOLUTION

For an inverse-monotone boundary value problem with the nonlinear ODE -epsilon u ''+sinh(u)=1, epsilon>0, u(0)-u(1)=0 applications of the fo llowing enclosure methods are presented and discussed: (i) on the basi s of a piecewise replacement of sinh(u) by polynomials, the constructi on of monotone sequences of upper and lower bounds for u; (ii) on the basis of Lohner's enclosure algorithms for solutions of ODEs, simple a nd multiple shooting methods. Existence of a classical solution follow s from literature and (independently) from the execution of (ii). Wher eas (i) requires the inverse-monotonicity of the problem, this is not so for (ii). For small epsilon, the unique solution of the BVP is stro ngly repellent.