  The method of self-consistent expansions is a powerful tool for handling
strong coupling problems that might otherwise be beyond the reach of
perturbation theory, providing surprisingly accurate approximations even at low
order. First applied in its embryonic form to fully-developed turbulence, it
has subsequently been successfully applied to a variety of problems that
include polymer statistics, interface dynamics and high order perturbation
theory for the anharmonic oscillator. Here we show that the self-consistent
expansion can be applied to singular perturbation problems arising in the
theory of partial differential equations. We demonstrate its application to
Barenblatt's nonlinear diffusion equation for porous media filtration, where
the long-time asymptotics exhibits anomalous dimensions which can be
systematically calculated using the perturbative renormalization group. We find
that even the first order self-consistent expansion improves the approximation
of the anomalous dimension obtained by the first order perturbative
renormalization group, especially in the strong coupling regime. We also
develop a field-theoretic framework for deterministic partial differential
equations to facilitate the application of self-consistent expansions to other
dynamic systems, and illustrate its application using the example of
Barenblatt's equation. The scope of our results on the combination of
renormalization group and self-consistent expansions is limited to partial
differential equations whose long-time asymptotics is controlled by incomplete
similarity. However, our work suggests that these methods could be applied to a
broader suite of singular perturbation problems such as boundary layer theory,
multiple scales analysis and matched asymptotic expansions, for which excellent
approximations using renormalization group methods alone are already available.
