Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with...

Area of Science:

• Applied Mathematics
• Mathematical Physics

Background:

• Self-consistent expansions offer accurate approximations for strong coupling problems beyond perturbation theory.
• Previous applications include turbulence, polymer statistics, and anharmonic oscillators.

Purpose of the Study:

• To demonstrate the application of self-consistent expansions to singular perturbation problems in partial differential equations (PDEs).
• To enhance the calculation of anomalous dimensions in nonlinear diffusion using renormalization group (RG) methods.

Main Methods:

• Applying self-consistent expansions in conjunction with renormalization group methods.
• Utilizing the Callan-Symanzik equation for improved approximation of anomalous dimensions.
• Developing a field-theoretic framework for deterministic PDEs.

Main Results:

• The first-order self-consistent expansion improves approximations of anomalous dimensions in the strong coupling regime.
• Demonstrated application to Barenblatt's nonlinear diffusion equation for porous media filtration.
• Established a general field-theoretic framework for applying these methods to other dynamic systems.

Conclusions:

• Self-consistent expansions combined with RG methods are effective for singular perturbation problems in PDEs with incomplete similarity.
• These methods show potential for broader applications in areas like boundary layer theory and matched asymptotic expansions.