Quasi Exact Solution of the Fisher Equation


We propose an accurate non numerical solution of the Fisher Equation (FE), capable of reproducing the known analytical solutions and those obtained from a numerical analysis. The form we propose is based on educated guesses concerning the possibility of merging diffusive and logistic behavior into a single formula.

Share and Cite:

G. Dattoli, E. Palma, E. Sabia and S. Licciardi, "Quasi Exact Solution of the Fisher Equation," Applied Mathematics, Vol. 4 No. 8A, 2013, pp. 7-12. doi: 10.4236/am.2013.48A002.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] R. A. Fisher, “The Wave of Advance of Advantageous Genes,” Annals of Genetics, Vol. 7, No. 4, 1937, p. 353.
[2] A. Kolmogorov, I. Petrovskii and N. Piscounov, “Etude de L’équation de la Diffusion Avec Croissance de la Quan tité de Matière et Son Application a un Problem Bio logique,” In: V. M. Tikhomirov, Ed., Selected Works of A. N. Kolmogorov I, Kluwer, Dordrecht, 1991, p. 248.
[3] B. H. Gilding and R. Kersner, “Travelling Waves in Non linear Diffusion Convection Reaction,” Birkhauser, Basel, 2004. doi:10.1007/978-3-0348-7964-4
[4] J. Vandermeer, “How Populations Grow: The Exponen tial and Logistic Equations,” Nature Education Know ledge, Vol. 1, No. 8, 2010, p. 1.
[5] D. Babusci, G. Dattoli and M. Delfranco, “Lectures on Mathematical Methods for Physics,” RT/2010/58/ENEA. http://www.frascati.enea.it/biblioteca
[6] E. E. Holmes, M. A. Lewis, J. A. Banks and R. R. Veit, “Partial Differential Equations in Ecology: Spatial Inter actions and Population Dynamics,” Ecology, Vol. 75, No. 1, 1994, pp. 17-29. doi:10.2307/1939378
[7] M. J. Ablowitz and A. Zeppetella, “Explicit Solutions of Fisher’s Equation for a Special Wave Speed,” Bulletin of Mathematical Biology, Vol. 41, No. 6, 1979, pp. 835-840.
[8] N. A. Kudryashov, “On Exact Solutions of Families of Fisher Equations,” Theoretical and Mathematical Physics, Vol. 94, No. 2, 1993, pp. 211-218.
[9] V. G. Danilov, V. P. Maslov and K. A. Volosov, “Mathe matical Modelling of Heat and Mass Transfer Processes,” Kluwer, Dordrecht, 1995.
[10] E. D. Kocacoban, A. B. Koc, A. Kurnaz and Y. Keskin, “A Better Approximation to the Solution of Burger Fisher Equation,” Proceedings of the World Congress on En gineering 2011, London, 6-8 July 2011.
[11] G. Strang, “On the Construction and Comparison of Dif ference Schemes,” SIAM Journal on Nu merical Analysis, Vol. 5, No. 3, 1968, pp. 506-517.
[12] G. Dattoli, P. L. Ottaviani, A. Torre and L. Vazquez, “Evolution Operator Equations: Integration with Algebraic and Finite Difference Methods. Application to Physical Problems in Classical and Quantum Mechanics and Quantum Field Theory,” La Rivista Del NuovoCimento, Vol. 20, No. 2, 1997, pp. 3-133.
[13] S. Blanes, F. Casas and A. Murua, “Symplectic Splitting Operator Methods Tailored for the Time-Dependent Schrodinger Equation,” Journal of Chemical Physics, Vol. 124, 2006, Article ID: 234105. doi:10.1137/0705041
[14] L. Giuggioli and V. M. Kenkre, “Analytic Solutions of a Nonlinear Convective Equation in Population Dynamics,” Physica D: Nonlinear Phenomena, Vol. 183, No. 3-4, 2003, pp. 245-259. doi:10.1016/S0167-2789(03)00176-3

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.