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.
Highlights
The focus is on the generalised Fisher type equation arising in population dynamics
The analysis of the generalised Fisher equation has been carried out using Lie point symmetries and construction of conservation laws see e.g. [2])
The reaction-diffusion equations such as the generalized Fisher equation describe how the concentration of a substance is distributed in space changes, whereby the diffusion term causes the spread over the surface
Summary
The focus is on the generalised Fisher type equation arising in population dynamics. The analysis of the generalised Fisher equation has been carried out using Lie point symmetries It turns out that reaction diffusion equations such as the generalised Fisher equation admit the genuine nonclassical symmetries if the source term is given by a cubic In a recent work [3] [11], the authors assume a diffusivity which depends on space variable In this case, the diffusivity may be given as a power law function of space variable for the given reaction-diffusion equation to admit nonclassical symmetries.
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