NS  Vol.7 No.13 , December 2015
Group-Invariant Solutions for the Generalised Fisher Type Equation
Abstract: In this paper, we construct the group-invariant (exact) solutions for the generalised Fisher type equation using both classical Lie point and the nonclassical symmetry techniques. The generalised Fisher type equation arises in theory of population dynamics. The diffusion term and coefficient of the source term are given as the power law functions of the spatial variable. We introduce the modified Hopf-Cole transformation to simplify a nonlinear second Order Ordinary Equation (ODE) into a solvable linear third order ODE.
Cite this paper: Louw, K. and Moitsheki, R. (2015) Group-Invariant Solutions for the Generalised Fisher Type Equation. Natural Science, 7, 613-624. doi: 10.4236/ns.2015.713061.

[1]   Bokhari, A.H., Al-Rubaee, R.A. and Zaman, F.D. (2011) On a Generalised Fisher Equation. Communication in Nonlinear Science and Numerical Simulation, 16, 2689-2695.

[2]   Rosa, M., Bruzon, M.S. and Grandarias, M.L. (2015) A conservation Law for A Generalized Chemical Fisher Equation. Journal of Mathematical Chemistry, 53, 941-948.

[3]   Moitsheki, R.J. and Bradshaw-Hajek, B.H. (2013) Symmetry Analysis of a Heat Conduction Model for Heat Transfer in a Longitudinal Rectangular Fin of a Heterogeneous Material. Communications in Nonlinear Science and Numerical Simulation, 18, 2374-2387.

[4]   Fisher, R.A. (1937) The Wave of Advance of Advantageous Genes. Annals of Eugencies, 7, 355-369.

[5]   Broadbridge P., Bradshaw-Hajek B.H., Fulford G.R. and Aldis, G.K. (2002) Huxley and Fisher Equations for Gene Propagation: An Exact Solution. ANZIAM, 44, 11-20.

[6]   Scott, A.C. (1975) The Electrophysics of A Nerve Fiber. Reviews of Modern Physics, 47, 487-553.

[7]   Arrigo, D.J., Broadbridge, P. and Hill, J.M. (1994) Nonclassical Symmetry Reductions of the Linear Diffusion Equation with a Nonlinear Source. Journal of Applied Mathematics, 52, 1-24.

[8]   Dorodnitsyn, V.A. (1982) On Invariant Solutions of the Equation of Nonlinear Heat Conduction with a Source. USSR Computational Mathematics and Mathematical Physics, 22, 115-122.

[9]   Nucci, M.C. (2003) Nonclassical Symmetries as Special Solutions of Heir-Equations. Journal of Mathematical Analysis and Applications, 279, 168-179.

[10]   Bradshaw-Hajek, B.H. and Broadbridge, P. (2004) A Robust Cubic Reaction-Diffusion System for Gene Propagation. Mathematical and Computer Modelling, 39, 1151-1163.

[11]   Bradshaw-Hajek, B.H. and Moitsheki, R.J. (2015) Symmetry Solutions for Reaction-Diffusion Equations with Spatially Dependent Diffusivity. Applied Mathematics and Computation, 254, 30-38.

[12]   Olver, P.J. (1993) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York.

[13]   Bluman, G.W. and Anco, S.C. (2002) Symmetry and Integration Methods for Differential Equations. Springer-Verlag, New York.

[14]   Bluman, G.W., Cheviakov, A.F. and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential Equations. Springer-Verlag, New York.

[15]   Bluman, G.W. and Cole, J.D. (1969) The General Similarity Solution of the Heat Equation. Journal of Mathematics and Mechanics, 18, 1025-1042.

[16]   Polyanin, A.D. and Zaitsev, V.F. (2000) Handbook of Exact Solutions for Ordinary Differential Equations. CRC Press, Boca Raton, Florida, USA.