Troesch's equation with charge separation

Abstract

We consider the following generalization of Troesch's classical two-point-BVP in order to take into account charge separation, which is required by physical considerations: $w^{\prime \prime }=L\sinh (Lw),w(0)=W_0<0<W_1=w(1)$. In contrast with the classical case, the solution of this generalized equation possesses boundary layer at both ends. We determine an explicit, approximate solution and verify its precision in several ways. The analytic solution hinges on a result concerning a class of overdetermined first order BVPs. Presenting this matter constitutes the second objective of the article.