Hyperbolic reaction-diffusion equations

From a mathematical viewpoint, the origin of the infinitely fast spreading of local disturbances in the diffusion equation can be traced to its parabolic character. This can be addressed in an ad hoc manner by adding a small term rdffp to the diffusion equation or the reaction-diffusion equation to make it hyperbolic. From the diffusion equation (2.1) we obtain the telegraph equation, a damped wave equation. 

The fundamental solution of this equation with a point source at x = 0 and t = 0 is given by 

Adding the term r9 p to (2.3), we obtain hyperbolic reaction-diffusion equations (HRDEs)  

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

---

Remark 2.2 Nomenclature in this field is unfortunately not uniform, and some authors use the term hyperbolic reaction-diffusion equations for reaction-telegraph equations. 

Turing Instabilities in Hyperbolic Reaction-Diffusion Equations... 

We first use hyperbolic reaction-diffusion equations, see Sect. 2.2.1, to study the effect of inertia on Turing instabilities [206]. Specifically, we consider two-variable HRDEs,... 

The Turing condition C4 = 0 for hyperbolic reaction-diffusion equations leads to exactly the same conditions as for the standard reaction-diffusion equation, namely (10.42) and (10.40). In other words, the Turing condition is independent of and Ty. If inertia in the transport is modeled by HRDEs, then the inertia has no effect whatsoever on the Turing instability to stationary patterns. 

Mendez, V., Fort, J., Farjas, J. Speed of wave-front solutions to hyperbolic reaction-diffusion equations. Phys. Rev. E 60(5), 5231-5243 (1999). http //dx.doi.org/10.1103/ PhysRevE.60.5231... 

This shows that the usual ideas associated with propagating waves in electromagnetism or fluid dynamics do not describe the behaviors found here. These differences could be expected because of the mathematical structure of reaction-diffusion equations, which owing to their parabolic character propagate information with infinite velocity. On the contrary, in the case of classical wave equations or hyperbolic equations there is a well-defined domain of influence and a characteristic velocity of propagation of information. ... 

The hyperbolic systems derived from a mathematical or macroscopic viewpoint overcome the pathological feature of the reaction-diffusion equation, but they suffer from other drawbacks (i) Hyperbolic equations typically do not preserve positivity. 

Vlad, M.O., Ross, J. Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport... 

Now consider the other extreme condition where diffusion is rapid relative to chemical reaction [i.e., hT( 1 — a) is small]. In this situation the effectiveness factor will approach unity for both the poisoned and unpoisoned reactions, and we must retain the hyperbolic tangent terms in equation 12.3.124 to properly evaluate Curve C in Figure 12.11 is calculated for a value of hT = 5. It is apparent that in this instance the activity decline is not nearly as sharp at low values of a as it was at the other extreme, but it is obviously more than a linear effect. The reason for this result is that the regions of the catalyst pore exposed to the highest reactant concentrations do not contribute proportionately to the overall reaction rate because they have suffered a disproportionate loss of activity when pore-mouth poisoning takes place. 

This equation gives the differential yield of V for a porous catalyst at a point in a reactor. For equal combined diffusivities and the case where hT approaches zero (no diffusional limitations on the reaction rate), this equation reduces to equation 9.3.8, since the ratio of the hyperbolic tangent terms becomes y/k2 A/ki v As hT increases from about 0.3 to about 2.0, the selectivity of the catalyst falls off continuously. The selectivity remains essentially constant when both hyperbolic tangent terms approach unity. This situation corresponds td low effectiveness factors and, in tliis case, equation 12.3.149 becomes... 

Recall that there are a number of reactions where homogeneous catalysis involves two phases, liquid and gas, for example, hydrogenation, oxidation, carbonylation, and hydroformylation. The role of diffusion becomes important in such cases. In Chapter 6, we considered the role of diffusion in solid catalyzed fluid-phase reactions and gas-liquid reactions. The treatment of gas-liquid reactions makes use of an enhancement factor to express the enhancement in the rate of absorption due to reaction. A catalyst may or may not be present. If there is no catalyst, we have a simple noncatalytic gas-liquid heterogeneous reaction in which the reaction rate is expressed by simple power law kinetics. On the other hand, when a dissolved catalyst is present, as in the case of homogeneous catalysis, the rate equations acquire a hyperbolic form (similar to LHHW models discussed in Chapters 5 and 6). Therefore, the mathematical analysis of such reactions becomes more complex. 

---