The structure of solutions for a third order differential equation in boundary layer theory

In this paper, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium. We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.