Quasi Exact Solution of the Fisher Equation

ABSTRACT

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.

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.

Cite this paper

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

G. Dattoli, E. Palma, E. Sabia and S. Licciardi, "Quasi Exact Solution of the Fisher Equation,"

References

[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

[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