Donald S. Cohen

A generalized diffusion model for growth and dispersal in a population

A reaction-diffusion model is presented in which spatial structure is maintained by means of a diffusive mechanism more general than classical Fickian diffusion. This generalized diffusion takes into account the diffusive gradient (or gradient energy) necessary to maintain a pattern even in a single diffusing species. The approach is based on a Landau-Ginzburg free energy model. A problem involving simple logistic kinetics is fully analyzed, and a nonlinear stability analysis based on a multi-scale perturbation method shows bifurcation to non-uniform states.

ROTATING SPIRAL WAVE SOLUTIONS OF REACTION-DIFFUSION EQUATIONS*

We resolve the question of existence of regular rotating spiral waves as a consequence of only the processes of chemical reaction and molecular diffusion. We prove rigorously the existence of these waves as solutions of reaction-diffusion equations, and we exhibit them by means of numerical computations in several concrete cases. Existence is proved via the Schauder fixed point theorem applied to a class of functions with sufficient structure that, in fact, important constructive properties such as asymptotic representations and frequency of rotation are obtained.

Sharp fronts due to diffusion and viscoelastic relaxation in polymers

A model for sharp fronts in glassy polymers is derived and analyzed. The major effect of a diffusing penetrant on the polymer entanglement network is taken to be the inducement of a differential viscoelastic stress. This couples diffusive and mechanical processes through a viscoelastic response where the strain depends upon the amount of penetrant present. Analytically, the major effect is to produce explicit delay terms via a relaxation parameter. This accounts for the fundamental difference between a polymer in its rubbery state and the polymer in its glassy state, namely the finite relaxation time in the glassy state due to slow response to changing conditions. Both numerical and analytical perturbation studies of a boundary value problem for a dry glass polymer exposed to a penetrant solvent are completed. Concentration profiles in good agreement with observations are obtained.

Free boundary problems in controlled release pharmaceuticals. I: diffusion in glassy polymers

This paper formulates and studies two different problems occurring in the formation and use of pharmaceuticals via controlled release methods. These problems involve a glassy polymer and a penetrant, and the central problem is to predict and control the diffusive behavior of the penetrant through the polymer. The mathematical theory yields free boundary problems which are studied in various asymptotic regimes.

Bifurcation of Localized Disturbances in a Model Biochemical Reaction

Asymptotic solutions are presented to the nonlinear parabolic reaction-diffusion equations describing a model biochemical reaction proposed by I. Prigogine. There is a uniform steady state which, for certain values of the adjustable parameters, may be unstable. When the uniform solution is slightly unstable, the two-timing method is used to find the bifurcation of new solutions of small amplitude. These may be either nonuniform steady states or time-periodic solutions, depending on the ratio of the diffusion coefficients. When one of the parameters is allowed to depend on space and the basic state is unstable, it is found that the nonuniform steady state which is approached may show localized spatial oscillations. The localization arises out of the presence of turning points in the linearized stability equations.

Shock formation in a multidimensional viscoelastic diffusive system

We examine a model for non-Fickian “sorption overshoot” behavior in diffusive polymer-penetrant systems. The equations of motion proposed by Cohen and White [SIAM J. Appl. Math., 51 (1991), pp. 472–483] are solved for two-dimensional problems using matched asymptotic expansions. The phenomenon of shock formation predicted by the model is examined and contrasted with similar behavior in classical reaction-diffusion systems. Mass uptake curves produced by the model are examined and shown to compare favorably with experimental observations.

Controlled drug release asymptotics

The solution of Higushis model for controlled release of drugs is examined when the solubility of the drug in the polymer matrix is a prescribed function of time. A time-dependent solubility results either from an external control or from a change in pH due to the activation of pH immobilized enzymes. The model is described as a one-phase moving boundary problem which cannot be solved exactly. We consider two limits of our problem. The first limit considers a solubility much smaller than the initial loading of the drug. This limit leads to a pseudo-steady-state approximation of the diffusion equation and has been widely used when the solubility is constant. The second limit considers a solubility close to the initial loading of the drug. It requires a boundary layer analysis and has never been explored before. We obtain simple analytical expressions for the release rate which exhibits the effect of the time-dependent solubility.

Slowly modulated oscillations in nonlinear diffusion processes

It is shown here that certain systems of nonlinear (parabolic) reaction-diffusion equations have solutions which are approximated by oscillatory functions in the form R(ξ - cτ)P(t^*) where P(t^*) represents a sinusoidal oscillation on a fast time scale t* and R(ξ - cτ) represents a slowly-varying modulating amplitude on slow space (ξ) and slow time (τ) scales. Such solutions describe phenomena in chemical reactors, chemical and biological reactions, and in other media where a stable oscillation at each point (or site) undergoes a slow amplitude change due to diffusion.

Multiplicity and stability of oscillatory states in a continuous stirred tank reactor with exothermic consecutive reactions A → B → C

Abstract We study the dimensionless heat and mass balance equations describing a continuous stirred tank reactor in which there is occurring consecutive first order exothermic reactions. Analytical asymptotic calculations and numerical calculations are presented for both steady states and oscillatory states. A formal perturbation theory, via multitime scale techniques, is derived to account for limit cycles in phase space. The theory produces formulas which are immediately interpretable physically and from which the stability of the oscillations is immediately resolved without recourse to further techniques. Furthermore, numerical calculations via our theory are performed which are far cheaper and easier than a direct numerical search and computation involving the long time study of the basic differential equations.

Field boundary problems in controlled release pharmaceuticals. II: swelling-controlled release

A problem in controlled release pharmaceutical systems is formulated and studied. The device modeled is a polymer matrix containing an initially immobilized drug. The release of the drug is achieved by countercurrent diffusion through a penetrant solvent with the release rate being determined by the rate of diffusion of the solvent in the polymer. The mathematical theory yields a free boundary problem which is studied in various asymptotic regimes.