On a Neumann boundary value problem for the Painlevé II equation in two-ion electro-diffusion

A two-point Neumann boundary value problem for a two-ion electro-diffusion model reducible to the Painlevé II equation is investigated. The problem is unconventional in that the model equation involves yet-to-be-determined boundary values of the solution. In prior work by Thompson, the existence of a solution was established subject to an inequality on the physical parameters. Here, a two-dimensional shooting method is used to show that this restriction may be removed. A practical algorithm for the solution of the boundary value problem is presented in an appendix.