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| {{distinguish2|the [[Fisher equation]] in [[financial mathematics]]}}
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| [[File:FKPPwiki.jpg|thumb|Numerical simulation of the Fisher–KPP equation. In colors: the solution ''u''(''t'',''x''); in dots : slope corresponding to the theoretical velocity of the traveling wave.]]
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| In mathematics, '''Fisher's equation''', also known as the '''Fisher–Kolmogorov equation''' and the '''Fisher–KPP equation''', named after [[Ronald Fisher|R. A. Fisher]] and [[Andrey Kolmogorov|A. N. Kolmogorov]], is the [[partial differential equation]]
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| :<math> \frac{\partial u}{\partial t}=u(1-u)+\frac{\partial^2 u}{\partial x^2}.\, </math> | |
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| Fisher proposed this equation to describe the spatial spread of an advantageous allele and explored its travelling wave solutions.<ref name="Fisher-1930-GTN">Fisher, R. A., ''The genetical theory of natural selection''. Oxford University Press, 1930. Oxford University Press, USA, New Ed edition, 2000, ISBN 978-0-19-850440-5, variorum edition, 1999, ISBN 0-19-850440-3</ref> For every wave speed ''c'' ≥ 2, it admits [[travelling wave]] [[Solution (disambiguation)|solution]]s of the form
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| :<math> u(x,t)=v(x \pm ct)\equiv v(z),\, </math>
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| where <math>\textstyle v</math> is increasing and
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| :<math> \lim_{z\rightarrow-\infty}v\left( z\right) =0,\quad\lim_{z\rightarrow\infty }v\left( z\right) =1. </math>
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| That is, the solution switches from the equilibrium state ''u'' = 0 to the equilibrium state ''u'' = 1. No such solution exists for ''c'' < 2.<ref>R. A. Fisher. [http://digital.library.adelaide.edu.au/dspace/handle/2440/15125 "The wave of advance of advantageous genes"], ''Ann. Eugenics'' '''7''':353–369, 1937.</ref><ref name="Kolmogorov-1937-SDE">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</ref><ref name="Grindrod-1996-TAR">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.</ref> The wave shape for a given wave speed is unique.
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| For the special wave speed <math>c=\pm 5/\sqrt{6}</math>, all solutions can be found in a closed form,<ref>Ablowitz, Mark J. and Zeppetella, Anthony,
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| ''Explicit solutions of Fisher's equation for a special wave speed'', Bulletin of Mathematical Biology 41 (1979) 835–840</ref> with
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| :<math> v(z) = \left( 1 + C \mathrm{exp}\left(\pm{z}/{\sqrt6}\right) \right)^{-2} </math>
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| where <math>C</math> is arbitrary, and the above limit conditions are satisfied for <math>C>0</math>.
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| It is perhaps the simplest example of a semilinear [[reaction-diffusion equation]]
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| :<math> \frac{\partial u}{\partial t}=\Delta u+F\left( u\right) , </math>
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| which can exhibit traveling wave solutions that switch between equilibrium states given by <math> f(u) = 0</math>. Such equations occur, e.g., in [[ecology]], [[physiology]], [[combustion]], [[crystallization]], [[plasma physics]], and in general [[phase transition]] problems.
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| Proof of the existence of traveling wave solutions and analysis of their properties is often done by the [[phase space method]].
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| ==Traveling wave solutions==
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| </math>
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| {{Gallery
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| |File:Fisher Kolmogorov equation traveling wave plot8.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot11.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot13.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot14.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot16.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot17.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot19.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot20.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot23.gif|
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| |File:Fisher Kolmogorov equation traveling wave plot33.gif|
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| }}
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| ==See also==
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| *[[List of plasma (physics) articles]]
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| *[[Allen–Cahn equation]]
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| ==References==
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| <!--This article uses the Cite.php citation mechanism. If you would like more information on how to add references to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php -->
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| {{Reflist}}
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| == External links ==
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| *[http://mathworld.wolfram.com/FishersEquation.html Fisher's equation] on [[MathWorld]].
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| *[http://eqworld.ipmnet.ru/en/solutions/npde/npde1101.pdf Fisher equation] on EqWorld.
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| {{DEFAULTSORT:Fisher's Equation}}
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| [[Category:Partial differential equations]]
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Hi there! :) My name is Orville, I'm a student studying Art from Kemmern, Germany.
My website: how to get free fifa 15 coins
