This project studies the nonhomogeneous steady-state solutions of the Gray-Scott model, a system of nonlinear partial differential equations that has received attention in the past decade in the context of pattern formation and morphogenesis. Morphogenesis, or ‘birth of shape’, is the biological term for the initial formation of patterns that occur in development as cells begin to differentiate...

The most popular numerical method for solving systems of reaction–diffusion equations continues to be a low order finite–difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high–order methods one can speed up such simulations for 2D and 3D problems by factors of 10–100. A short Matlab code (2/3D) that can ...

Metastable dynamics, which qualitatively refers to physical processes that involve an extremely slow approach to their nal equilibrium states, is often associated with singularly perturbed convection-di usion-reaction equations. A problem exhibits metastable behavior when the approach to equilibrium occurs on a time-scale of order O(eC" ), where C > 0 and " is the singular perturbation paramete...

Fractional diffusion equations are useful for applications where a cloud of particles spreads faster than the classical equation predicts. In a fractional diffusion equation, the second derivative in the spatial variable is replaced by a fractional derivative of order less than two. The resulting solutions spread faster than the classical solutions and may exhibit asymmetry, depending on the fr...

The macroscopic behavior of dissipative stochastic partial differential equations usually can be described by a finite dimensional system. This article proves that a macroscopic reduced model may be constructed for stochastic reaction-diffusion equations with cubic nonlinearity by artificial separating the system into two distinct slow-fast time parts. An averaging method and a deviation estima...

The principle of virtual dissipation for irreversible processes in open systems is given a new formulation where variations are unconstrained everywhere including the boundary. As a complementary development a new chemical thermodynamics of open systems initiated earlier is given a simplified derivation and new results are presented. A new evaluation of entropy production for fully nonlinear re...

We develop perturbative expansions to obtain solutions for the initial-value problems of two important reaction-diffusion systems, viz., the Fisher equation and the time-dependent Ginzburg-Landau (TDGL) equation. The starting point of our expansion is the corresponding singular-perturbation solution. This approach transforms the solution of nonlinear reaction-diffusion equations into the soluti...