- Fisher's equation
In mathematics,

**Fisher's equation**, also known as the**Fisher-Kolmogorov equation**, named after R. A. Fisher and A. N. Kolmogorov, is thepartial differential equation :$frac\{partial\; u\}\{partial\; t\}=u(1-u)+frac\{partial^2\; u\}\{partial\; x^2\}.,$

For every wave speed "c" ≥ 2, it admits

travelling wave solutions of the form:$u(x,t)=v(pm\; x\; +\; ct),,$

where $extstyle\; v$ is increasing and

:$lim\_\{z\; ightarrow-infty\}vleft(\; z\; ight)\; =0,quadlim\_\{z\; ightarrowinfty\; \}vleft(\; z\; ight)\; =1.$

That is, the solution switches from the equilibrium state "u" = 0 to the equilibrium state "u" = 1. No such solution exists for "c" < 2. [

*R. A. Fisher. [*] A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V. M. Tikhomirov, editor, "Selected Works of A. N. Kolmogorov I", pages 248--270. Kluwer 1991, ISBN 90-277-2796-1. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937] Peter Grindrod. "The theory and applications of reaction-diffusion equations: Patterns and waves." Oxford Applied Mathematics and Computing Science Series. The Clarendon Press Oxford University Press, New York, second edition, 1996 ISBN 0-19-859676-6; ISBN 0-19-859692-8.]*http://digital.library.adelaide.edu.au/dspace/handle/2440/15125 "The wave of advance of advantageous genes"*] , "Ann. Eugenics"**7**:353–369, 1937.For the special wave speed $c=5/sqrt\{6\}$, all solutions can be found in a closed form, [

*Ablowitz, Mark J. and Zeppetella, Anthony,"Explicit solutions of Fisher's equation for a special wave speed", Bulletin of Mathematical Biology 41 (1979) 835-840*] with:$v(z)\; =\; left(\; 1\; +\; C\; mathrm\{exp\}left(-\{z\}/\{sqrt6\}\; ight)\; ight)^\{-2\}$

where $C$ is arbitrary, and the above limit conditions are satisfied for $C>0$.

In particular, the wave shape for a given wave speed is not necessarily unique.

This equation was originally derived for the simulation of propagation of a gene in a population Fisher, R. A., "The genetical theory of natural selection". Oxford University Press, 1930. Oxford University Press, USA, New Ed edition, 2000, ISBN 978-0198504405, variorum edition, 1999, ISBN 0-19-850440-3

] . It is perhaps the simplest model problem for

reaction-diffusion equation s:$frac\{partial\; u\}\{partial\; t\}=Delta\; u+fleft(\; u\; ight)\; ,$

which exhibit traveling wave solutions that switch between equilibrium states given by "f"("u") = 0. Such equations occur, e.g., in

combustion ,crystallization ,plasma physics , and in generalphase transition problems.Proof of the existence of traveling wave solutions and analysis of their properties is often done by the

phase space method .**References****External links*** [

*http://mathworld.wolfram.com/FishersEquation.html Fisher's equation*] onMathWorld .

* [*http://eqworld.ipmnet.ru/en/solutions/npde/npde1101.pdf Fisher equation*] on EqWorld.**ee also***

Allen-Cahn equation

*Wikimedia Foundation.
2010.*