Noise and stability in reaction-diffusion equations

Stability of patterns in reaction-diffusion equations

These are lecture notes associated with the hour-long talk “Stability of patterns in reaction-diffusion equations,” given at the BU/Keio Workshop in Dynamical Systems, during Sept 15-19, 2014, at Boston University. The abstract of the talk was: “Reaction-diffusion equations model a wide variety of chemical and biological processes. Such systems are well known for exhibiting patterns, such as tr...

Stability of Localized Structures in Non-local Reaction-diffusion Equations

The stability of non-homogeneous, steady state solutions of a scalar, non-local reaction-diffusion equation is considered. Sufficient conditions are provided that guarantee that the relevant linear operator possesses a countable infinity of discrete eigenvalues. These eigenvalues are shown to interlace the eigenvalues of a related local Sturm-Liouville operator. An oscillation theorem for the c...

Reaction-Diffusion Equations and Learning

Generalized reaction–diffusion equations

This Letter proposes generalized reaction–diffusion equations for treating noisy magnetic resonance images. An edge-enhancing functional is introduced for image enhancement. A number of super-diffusion operators are introduced for fast and effective smoothing. Statistical information is utilized for robust edge-stopping and diffusion-rate estimation. A quasi-interpolating wavelet algorithm is u...

Reaction-diffusion Equations

I have recently worked in the area of partial differential equations (PDE), specifically reaction-diffusion equations, drift-diffusion equations, and fluid dynamics. These equations model physical processes such as combustion, mixing, or turbulence, and my main interest has been in long term dynamics of their solutions as well as in the formation of singularities. I consider myself an applied a...