We consider spatially discrete bistable reaction-diffusion equations that admit wave front solutions. Depending on the parameters involved, such wave fronts appear to be pinned or to glide at a certain speed. We study the transition of traveling waves to steady solutions near threshold and give conditions for front pinning (propagation failure). The critical parameter values are characterized a...

In a series of papers, Saxena, Mathai, and Haubold (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of ge...

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...

In this paper, firstly we introduce the concept of norm-to-weak continuous cocycle in Banach space and give a technical method to verify this kind of continuity, then we obtain some abstract results for the existence of pullback attractors about this kind of cocycle, using the measure of noncompactness. As an application, we prove the existence of pullback attractors in H 1 0 of the cocycle ass...

A collection of codes (in MATLAB & Fortran 77), and examples, for solving reaction-diffusion equations in one and two space dimensions is presented. In areas of the mathematical community spectral methods are used to remove the stiffness associated with the diffusive terms in a reactiondiffusion model allowing explicit high order timestepping to be used. This is particularly valuable for two (a...

In this paper, we introduce a generalization of travelling waves for evolution equations. We are especially interested in reaction-diffusion equations and systems in heterogeneous media (with general operators and general geometry). Our goal is threefold. First we give several definitions, for transition waves, fronts, pulses, global mean speed of propagation, etc. Next, we discuss the meaning ...

We present a model describing population dynamics in an environment. The model is a nonlinear, nonlocal, reaction diffusion equation with Neumann boundary conditions. An inverse method, involving minimization of a least-squares cost functional, is developed to identify unknown model parameters. Finally, numerical results are presented which display estimates of these parameters using computatio...

We have developed several results on the existence and asymptotic behavior of nlild solutions to rextiondiffusion systems that have infinite delirys in the nonlinear reaction terms. We find that the semiflow generated by a cooperative and irreducible reaction-tliffilsio system with infinite delay is not compact but set-condcrrsing, ancl not strongly order-preserving but quasi strongly ordc~r-1)...