Similarity solutions of differential equations for boundary layer approximations in porous media

This paper is concerned with the ordinary differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{f}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi + mf\,{{f}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi - \alpha {f}\ifmmode{'}\else$'$\fi^{2} = 0,$$\end{document} on (0, + ∞), subject to the boundary conditions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(0) = a,\quad {f}\ifmmode{'}\else$'$\fi(0) = b,\quad {f}\ifmmode{'}\else$'$\fi(\infty ) = {\mathop {\lim }\limits_{t \to \infty } }{f}\ifmmode{'}\else$'$\fi(t) = 0,$$\end{document} in which a and b are reals, m > 0 and α < 0. Such problem, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m = \frac{{\alpha + 1}}{2},\;a = 0\;{\text{ and }}\;b = 1,$$\end{document} arises in the study of the free convection, along a vertical flat plate embedded in a porous medium.