KPP equation

The Kolmogorov-Petrovsky-Piscounov equation ( KPP equation according to Andrei Kolmogorow , Iwan Petrowski and N. Piscounov 1937) is a nonlinear partial differential equation in the form of a reaction-diffusion equation .

A special case is Fisher's equation (after Ronald Aylmer Fisher 1937) of population dynamics , a continuous variant of the logistic equation (see also logistic function ).

Bulk

The KPP equation has the form:

${\displaystyle \partial _{t}u=\partial _{x}^{2}u+g(u)}$

with a nonlinear function which satisfies: , and for (the interval [0,1] is often the interval of definition of the variable , if this indicates a concentration). ${\displaystyle g(u)}$ ${\displaystyle g(1)=g(0)=0}$ ${\displaystyle g(u')>0}$ ${\displaystyle {\tfrac {dg}{du}}(u')<{\tfrac {dg}{du}}(0)}$ ${\displaystyle 0<u'<1}$ ${\displaystyle u}$

Fisher's equation is a special case of the form:

${\displaystyle \partial _{t}u=\partial _{x}^{2}u+u\cdot (1-u)=\partial _{x}^{2}u+u-u^{2}}$

Sometimes a term is given instead of the reaction term , similar to the logistic equation. ${\displaystyle g(u)=u\cdot (1-u)}$ ${\displaystyle g(u)=r\cdot u\cdot (1-u)}$

This is a semi-linear parabolic equation of the second order. It is used to model various processes in nature, such as population dynamics or chemical reactions .

The differential equation consists of a diffusion term and a nonlinear reaction term . ${\displaystyle \partial _{x}^{2}u}$ ${\displaystyle u-u^{2}}$

If one uses a location-independent function , one obtains the ordinary differential equation ${\displaystyle f(t)=u(x,t)}$

${\displaystyle \partial _{t}f=f-f^{2}}$.

At this it can be seen that the model an exponential growth is modeled, but which a saturation term includes. This stands z. B. in the population dynamics for the limited food supply or in chemical reactions for the saturation of the concentration . ${\displaystyle \partial _{t}f=f}$ ${\displaystyle -f^{2}}$

Reaction fronts

If the equation is used to model a locally localized starting reaction, it is clear that a reaction front is developing. As can be shown, this has a minimal velocity of propagation.

Using the usual approach for waves

${\displaystyle u(x,t)=f(x-vt)=f(w)}$,

so after insertion one obtains the ordinary differential equation of the second order

${\displaystyle \partial _{w}^{2}f+v\partial _{w}f+f-f^{2}=0}$.

After linearization and assuming that the "concentration" f can only assume values ​​between 0 and 1, the equation for the eigenvalues ​​is obtained

${\displaystyle \lambda _{1,2}={\frac {-v\pm {\sqrt {v^{2}-4}}}{2}}}$.

Since these must be real for stable waves , it must apply. ${\displaystyle v\geq 2}$

Generalizations

The Fisher equation can be generalized to:

${\displaystyle \partial _{t}u=\partial _{x}^{2}u+(1-u)u^{m}}$

with a positive integer . ${\displaystyle m}$

In the case of the Fisher equation then applies . ${\displaystyle m=1}$

See also

• Nernst-Planck equation

Individual evidence

1. ↑ BH Gilding u. a. (Ed.), Traveling waves in nonlinear diffusion-convection equation reaction, Birkhäuser 2004, p. 2
2. ↑ F. Hamel, N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. , Vol. 52, 1999, pp. 1255-1276, doi : 10.1002 / (SICI) 1097-0312 (199910) 52:10 <1255 :: AID-CPA4> 3.0.CO; 2-W .