Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T06:31:05.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical solution of Fisher's equation

Published online by Cambridge University Press:  14 July 2016

Jenö Gazdag  and
José Canosa
José Canosa
Affiliation:
IBM Corporation, Palo Alto, California
Get access Rights & Permissions [Opens in a new window]

Abstract

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration.

Keywords

NONLINEAR DIFFUSIONEPIDEMIC WAVESNUMERICAL SIMULATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 445 - 457
Copyright
Copyright © Applied Probability Trust 1974 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Gazdag, J. (1973) Numerical convective schemes based on accurate space derivatives. J. Comp. Phys. 13, 100–113.CrossRefGoogle Scholar
[2] Cooley, J. W., Lewis, P. A. W. and Welch, P. D. (1969) The finite Fourier transform. IEEE Trans. AU–17, 77–85.Google Scholar
[3] Fisher, R. A. (1936) The wave of advance of advantageous genes. Ann. Eugen. 7, 355–369.Google Scholar
[4] Kolmogoroff, A., Petrovsky, I. and Piscounoff, N. (1937) Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Bull, de l'Univ. d'Etat à Moscou (Ser. Intern) A1, 1–25.Google Scholar
[5] Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Oxford University Press. 178–183.Google Scholar
[6] Kendall, D. G. (1948) A form of wave propagation associated with the equation of heat conduction. Proc. Camb. Phil. Soc. 44, 591–594.CrossRefGoogle Scholar
[7] Kendall, D. G. (1965) Mathematical models of the spread of infection. In Symposium on Mathematics and Computer Science in Biology and Medicine, London, 1965. 213–225.Google Scholar
[8] Mollison, D. (1972) Possible velocities for a simple epidemic. Adv. Appl. Prob. 4, 233–257.Google Scholar
[9] Mollison, D. (1972) The rate of spatial propagation of simple epidemics. Proc. 6th Berkeley Symp. Math. Statist. Prob. 3, 579–614.Google Scholar
[10] Canosa, J. (1973) On a nonlinear diffusion equation describing population growth. IBM J. Res. Develop. 17, 307–313.CrossRefGoogle Scholar
[11] Kendall, D. G. Private communication (May 1972).Google Scholar
[12] Jeffrey, A. and Kakutani, T. (1972) Weak nonlinear dispersive waves: a discussion centred around the Korteweg-de Vries equation. SIAM Rev. 14, 582–643.Google Scholar