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Numerical Study of One Dimensional Fishers KPP Equation with Finite Difference Schemes
Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, KSA
Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, KSA
- 1 Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, KSA
- 2 Department of Mathematics, Numerical Analysis, King Abdulaziz University, Jeddah, KSA
American Journal of Computational Mathematics·Volume 07 (2017)·Pages 70–83·Published 23 March 2017·DOI10.4236/ajcm.2017.71006
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Abstract
In this paper, we originate results with finite difference schemes to approximate the solution of the classical Fisher Kolmogorov Petrovsky Piscounov (KPP) equation from population dynamics. Fisher’s equation describes a balance between linear diffusion and nonlinear reaction. Numerical example illustrates the efficiency of the proposed schemes, also the Neumann stability analysis reveals that our schemes are indeed stable under certain choices of the model and numerical parameters. Numerical comparisons with analytical solution are also discussed. Numerical results show that Crank Nicolson and Richardson extrapolation are very efficient and reliably numerical schemes for solving one dimension fisher’s KPP equation.
KeywordsForward in Time and Centre in Space (FTCS)Lax WendroffTaylor’s SeriesCrank Nicolson and Richardson Extrapolation
- Fisher, R.A. (1936) The Wave of Advance of Advantageous Genes. Annals of Eugenics, 7, 355-369. https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
- Canosa, J.C. (1973) On a Nonlinear Diffusion Equation Describing Population Growth. IBM Journal of Research Development, 17, 307-313. https://doi.org/10.1147/rd.174.0307
- Arnold, R.A., Showalter, K. and Tyson, J.J. (1987) Propagation of Chemical Reactions in Space. Journal of Chemical Education, 64, 740-744. https://doi.org/10.1021/ed064p740
- Tuckwell, H.C. (1988) Introduction to Theoretical Neurobiology. Cambridge, UK: Cambridge University Press, Cambridge.
- Gazdag, J.G. and Canosa, J.C. (1974) Numerical Solutions of Fisher’s Equation. Journal of Applied Probability, 11, 445-457. https://doi.org/10.1017/S0021900200096236
- Abdullaev, U.G. (1994) Stability of Symmetric Travelling Waves in the Cauchy Problem for the KPP Equation. Differential Equations, 30, 377-386.
- Logan, D.J. (1994) An Introduction to Nonlinear Partial Differential Equations. Wiley, New York.
- Evans, D.J. and Sahimi, M.S. (1989) The Alternating Group Explicit (AGE) Iterative Method to Solve Parabolic and Hyperbolic Partial Differential Equations. Annals of Numerical Fluid Mechanics and Heat Transfer, 2, 283-389. https://doi.org/10.1615/AnnualRevHeatTransfer.v2.100
- Tang, S.T. and Weber, R.O. (1991) Numerical Study of Fisher’s Equations by a Petrov-Galerkin Finite Element Method. Journal of the Australian Mathematical Society Series B, 33, 27-38. https://doi.org/10.1017/S0334270000008602
- Khaled, K.A. (2001) Numerical Study of Fisher’s Diffusion Reaction Equation by the Sinc Collocation Method. Journal of Computational and Applied Mathematics, 137, 245-255.
- Ames, W.F. (1965) Nonlinear Partial Differential Equations in Engineering. Academic Press, New York.
- Ames, W.F. (1969) Finite Difference Methods for Partial Differential Equations. Academic Press, New York.
- Noye, J.N. (1981) Nonlinear Partial Differential Equations in Engineering. Conference in Queen’s College, University of Melbourne, North-Holland Publishing Company, Australia.
- Mittal, R.C. and Kumar, S.K. (2009) Numerical Study of Fisher's Equation by Wavelet Galerkin Method. International Journal of Computer Mathematics, 3, 287-298. https://doi.org/10.1080/00207160600717758