Similarity solutions of mixed convection boundary-layer flows in a porous medium

The similarity differential equation $f'''+ff''+\beta f'(f'-1)=0$ with $\beta\textgreater{}0$ is considered. This differential equation appears in the study of mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. In order to prove the existence of solutions satisfying the boundary conditions $f(0)=a\geq0$, $f'(0)=b\geq0$ and $f'(+\infty)=0$ or $1$, we use shooting and consider the initial value problem consisting of the differential equation and the initial conditions $f(0)=a$, $f'(0)=b$ and $f''(0)=c$. For $0\textless{}\beta\leq1$, we prove that there exists a unique solution such that $f'(+\infty)=0$, and infinitely many solutions such that $f'(+\infty)=1$. For $\beta\textgreater{}1$, we give only partial results and show some differences with the previous case.