This paper is concerned with existence, uniqueness and behavior of the solutions of the autonomous third order nonlinear differential equation f ′′′ +(m + 2) f f ′′ −(2m + 1) f ′2 = 0 on R + with the boundary conditions f (0) = −γ, f ′ (∞) = 0 and f ′′ (0) = −1. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing-up coordinates to obtain a plane dynamical system.

This paper reviews results about free convection near a vertical flat plate embedded in some saturated porous medium. We focus on a third order autonomous differential equation that gives a special class of solutions called similarity solutions. Two cases are under consideration: in the first one we prescribe the temperature on the plate and in the second one we prescribe the heat flux on it. We will also see that the same equation appears in other industrial processes.

In this paper we investigate the similarity solutions of a plane mixed convection boundary layer flow near a semi-vertical plate, with a prescribed power law function of the distance from the leading edge for the temperature, that is embedded in a porous medium. We show the existence and uniqueness of convex and concave solutions for positive values of the power law exponent.

In this paper, we investigate the similarity solutions for a steady laminar incompressible boundary layer equations governing the magnetohydrodynamic (MHD) flow near the forward stagnation point of two-dimensional and axisymmetric bodies. This leads to the study of a boundary value problem involving a third order autonomous ordinary differential equation. Our main results are the existence, uniqueness and nonexistence for concave or convex solutions. MSC: 34B15, 34C11, 76D10 PACS: 47.35.Tv, 47.65.d, 47.15.Cb