Reaction diffusion equation

Reaction diffusion equations (RD equations) describe processes in which a local interaction and, in addition, diffusion occurs. An example from chemistry are models for the Belousov-Zhabotinsky reaction  (BZ reaction), in which spatial patterns arise because a locally oscillating chemical reaction is coupled with a diffusion process. An example from biology are the spatial spreading processes of animals and plants. Here the interaction term often has the form of a logistic Kolmogorov equation .

RD equations are partial differential equations of the second degree, which are rate equations in form (for derivation see there). They describe the change in a variable X over time (e.g. amount of substance , abundance , concentration , etc.):

${\displaystyle {\frac {\partial }{\partial t}}u=f(u)+D\cdot {\frac {\partial ^{2}}{\partial x^{2}}}u}$.

• The functions of time and place represent the quantities whose dynamics are described. Several substances that interact with each other can be taken into account by giving a vector form and interpreting the equation as a matrix equation. ${\displaystyle u}$ ${\displaystyle t}$ ${\displaystyle x}$ ${\displaystyle u}$
• The function describes the proportion of the reaction. Without the reaction fraction, the RD equation would have the form of the heat conduction equation . ${\displaystyle f(u)}$
• The term comes from Fick's 2nd law and describes diffusion. ${\displaystyle D\cdot {\tfrac {\partial ^{2}}{\partial x^{2}}}u}$
  • ${\displaystyle D}$is the diffusion coefficient .

If there is also a directed transport process ( convection ), the above reaction-diffusion equation must be extended by a convection term, analogous to the convection-diffusion equation .

Reaction diffusion equations are used in technical chemistry and mechanical engineering . Different systems are considered in which reaction, diffusion and convection occur together ( macrokinetics ). Examples are the design of chemical reactors or technical combustion processes . In developmental biology , reaction diffusion equations have played a dominant role in the mathematical theory of morphogenesis since Alan Turing , see Turing mechanism . Systems with one activating and two inhibiting components play an important role in modeling the structure formation processes of localized particle-like structures, so-called dissipative solitons . B. in oscillating chemical reactions of the type of the Belousov-Zhabotinsky reaction and semiconductor - gas discharge systems can be observed. Also Chemical waves and propagation of nerve pulses are described by reaction-diffusion equations.

Special cases

Special versions of the RD equations are distinguished depending on the form of the reaction component:

• ${\displaystyle f(u)=u\cdot (1-u)}$the Fisher equation, it is used in population dynamics (without the diffusion term it would be the differential equation for the logistic function ). A more general variant is the KPP equation for the , and for . The Fisher equation and the Newell-Whitehead equation are special cases of the KPP equation. ${\displaystyle f(1)=f(0)=0}$ ${\displaystyle f(u')>0}$ ${\displaystyle {\frac {df}{du}}(u')<{\frac {df}{du}}(0)}$ ${\displaystyle 0<u'<1}$
• ${\displaystyle f(u)=u^{2}\cdot (1-u)}$, Seldowitsch equation (Zeldovich equation) for example in combustion processes.
• ${\displaystyle f(u)=u(1-u^{2})}$Newell-Whitehead equation or amplitude equation used in Rayleigh-Bénard convection .
• ${\displaystyle f(u)=u\cdot (1-u)\cdot (u-\alpha )}$(with one parameter ) Nagumo equation for the propagation of nerve pulses in an axon ${\displaystyle 0<\alpha <1}$

Another example is the porous media equation and the Burgers equation .

See also

• Belousov-Zhabotinsky reaction
• Morphogenesis
• Hantz reactions
• Liesegang rings

literature

• Dilip Kondepudi, Ilya Prigogine : Modern Thermodynamics. From Heat Engines to Dissipative Structures. John Wiley & Sons, Chichester et al. 1998, ISBN 0-471-97393-9 .
• JD Murray : Mathematical Biology. 2 volumes. 3rd edition, corrected printing. Springer, New York NY et al. 2008, ISBN 978-0-387-95223-9 (Vol. 1), ISBN 978-0-387-95228-4 (Vol. 2), ( Interdisciplinary applied mathematics 17-18).
• Andreas W. Liehr: Dissipative Solitons in Reaction Diffusion Systems. Mechanism, Dynamics, Interaction. Volume 70 of Springer Series in Synergetics, Springer, Berlin Heidelberg 2013, ISBN 978-3-642-31250-2 .
• BA Grzybowski: Chemistry in Motion: Reaction-Diffusion Systems for Micro- and Nanotechnology 2009.

Individual evidence

1. ↑ BH Gilding u. a. (Ed.), Traveling waves in nonlinear diffusion-convection equation reaction, Birkhäuser 2004, p. 2