  We propose a method for the treatment of two--point boundary value problems
given by nonlinear ordinary differential equations. The approach leads to
sequences of roots of Hankel determinants that converge rapidly towards the
unknown parameter of the problem. We treat several problems of physical
interest: the field equation determining the vortex profile in a
Ginzburg--Landau effective theory, the fixed--point equation for Wilson's exact
renormalization group, a suitably modified Wegner--Houghton's fixed point
equation in the local potential approximation, a Riccati equation, and the
Thomas--Fermi equation. We consider two models where the approach does not
apply in order to show the limitations of our Pad\'{e}--Hankel approach.
