New results on a mixed convection boundary layer flow over a permeable vertical surface embedded in a porous medium

The objective of this paper is to prove the existence, non-existence and the sign of convex and convex-concave solutions of the third-order non-linear differential equation f ''' + ss f ' (f ' - 1) = 0, satisfying the boundary conditions f(0) = alpha is an element of R , f(0) = b< 0 and f ' -> lambda ast -> +infinity where lambda is an element of{0,1} and ss < 0. The problematic arises in the study of the Mixed Convection Boundary Layer flow over a permeable vertical surface embedded in a Porous Medium according to the mixed convection parameter b<0, the permeable parameter alpha is an element of R and the temperature parameter ss < 0.