  The classical numerical treatment of boundary value problems defined on
infinite intervals is to replace the boundary conditions at infinity by
suitable boundary conditions at a finite point, the so-called truncated
boundary. A truncated boundary allowing for a satisfactory accuracy of the
numerical solution has to be determined by trial and errors and this seems to
be the weakest point of the classical approach. On the other hand, the free
boundary approach overcomes the need for a priori definition of the truncated
boundary. In fact, in a free boundary formulation the unknown free boundary can
be identified with a truncated boundary and being unknown it has to be found as
part of the solution. In this paper we consider a different way to overcome the
introduction of a truncated boundary, namely finite differences schemes defined
on quasi-uniform grids. A quasi-uniform grid allows us to describe the infinite
domain by a finite number of intervals. The last node of such grid is placed on
infinity so right boundary conditions are taken into account exactly. We apply
the proposed approach to the Falkner-Skan model and to a problem of interest in
foundation engineering. The obtained numerical results are found in good
agreement with those available in literature. Moreover, we provide a simple way
to improve the accuracy of the numerical results using Richardson's
extrapolation. Finally, we indicate a possible way to extend the proposed
approach to boundary value problems defined on the whole real line.
