Fisher-Kolmogorov equation

Not to be confused with the Fisher's equation.

Fisher-Kolmogorov equation named after R. A. Fisher and A. N. Kolmogorov, is a nonlinear partial differential equation of the form[1][2]


\frac{\partial u}{\partial t}=\frac{\alpha}{k}*u(1-u^q)+\frac{\partial^2 u}{\partial x^2}.\,

F-K equation has application in biology.

Analytic solution

Fisher Kolmogorov eq plot


u(x,t)=(1+\alpha*\exp(k*(x-c*t)))^s

In whichs=-\frac{2}{q}

k =\frac{q}{\sqrt{2*(q+2)}}

c = \frac{q+4}{\sqrt{(2*q+4)}}

\alpha=1

and;

{u(x, t) = -1/2+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)},

{u(x, t) = -1/2+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)},

{u(x, t) = -1/2-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)},

{u(x, t) = -1/2-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)},

{u(x, t) = 1/2-(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x-(3/4)*t)},

{u(x, t) = 1/2-(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x-(3/4)*t)},

{u(x, t) = 1/2+(1/2)*tanh(_C1-(1/4)*\sqrt(2)*x+(3/4)*t)},

{u(x, t) = 1/2+(1/2)*tanh(_C1+(1/4)*\sqrt(2)*x+(3/4)*t)}

Traveling wave plot

 
 
 
 
 
 
 
 
 
 

Reference

  1. ↑ Graham W. Griffiths, William E. Schiesser, Traveling Wave Analysis of Partial Differential Equations, Chapter 8 Fisher-Kolmogorov Equation, p 135-146 Academy Press
  2. ↑ G. Adomian,Fisher-Kolmogorov equation,Applied Mathematics Letters,Volume 8, Issue 2, March 1995, Pages 51–52
  1. Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential Equations Academy Press
  2. Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
  3. Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
  4. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
  5. Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
  6. Dongming Wang, Elimination Practice,Imperial College Press 2004
  7. David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
  8. George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759