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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.

In this paper, 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 (see e.g. [

A significant amount of work has been done in the process of studying the reaction-diffusion equations. In particular, from the classical Lie symmetry analysis point of view (see e.g. [

This paper is arranged as follows. In Section 2, we provide the mathematical models for problems arising in population dynamics. In Section 3, we provide a brief account of the symmetry methods. In Sections 4 and 5, we provide the nonclassical and classical Lie point symmetry reductions, respectively. In Section 6, we briefly provide remarks on the conservation laws of the equation in question. The discussions and concluding remarks are given in Section 7.

For a diploid population having two available alleles at the locus in question (

where

Differentiating Equation (2) with respect to t, then the three genotype equations (1) collapse into a single equation that describes the change in frequency of the new mutant gene

where

constant in space, Equation (3) reduces to the Fitzhugh-Nagumo equation. When deriving the models with the continuous method, there is an extra convective term due to the assumption that total population density is not uniform spatially.

If one is to consider the conditions that Fisher had examined, so that the allele in question is completely recessive. This implies that the genotypes

where

which models the propagation of impulses along nerve axons. In this paper we focus on the reaction diffusion equation of the form

We refer to Equation (6) as the governing equation. The variable u may be viewed as representing population. The diffusivity

In this section, we restrict discussions to symmetry analysis of second order differential equations. Given for example, a second order differential equation of the form

where the subscripts denote all possible first and second derivatives of u with respect to t and x. Finding classical Lie point symmetries of Equation (7) implies seeking infinitesimal transformations of the form

generated by the vector field

Note that the transformations in (8) are equivalent to the one-parameter Lie group of transformations that leaves the Equation (7) unchanged or invariant. The action of X is extended to all derivatives appearing in the equation in question through the appropriate prolongation. The infinitesimal criterion for invariance of the given equation is given by

Equation (10) yields an overdetermined system of linear homogeneous equation which can be solved algorithmically. Note that the solution of Equation (10) yield the classical Lie point symmetries admitted by Equation (7). Full theory of determination of Lie point symmetries may be obtain in among other texts [

known as the invariant surface condition (ISC), that is given

then one obtain a system of nonlinear determining equations which may yield the nonclassical symmetry generators [

In this section, we consider nonclassical symmetry reductions of the generalised Fisher type equation given in Equation (6). Here,

In this subsection, we consider Equation (6) with

1:

The solution of these determining equations yields the admitted nonclassical symmetry generator given by

The associated ISC is given by

Using governing equation and the ISC (14) simultaneously one may eliminate

Employing a change of variables

We introduce the “modified” Hopf-Cole transformation given by

to simplify Equation (16). Upon substituting (17) into Equation (16), it turns out that

Hence

and so Equation (16) transforms to a linear third ODE

The solution to Equation (19) is given by

Substituting backwards for

Solving for the arbitrary functions,

where

If the source term is given by

In this case, the exact (group-invariant) solution will be given by,

Suppose that one considers a more general case,

Following the steps above, the admitted genuine nonclassical symmetry is given by

The associated ISC is given by

Eliminating

Introducing the transformation

In this case, it turns out that the “modified” Hopf-Cole transformation should be given by

as such the transformed Equation (29) becomes a solvable linear third order ODE

In terms of the original variables the exact solution is given by

where

or

and

If

We consider the case where

It is easy to show that this Lie algebra is closed. Reductions are possible by any linear combination of these symmetries. Usually one may determine the optimal system of subalgebras of these classical Lie point symmetries to determine reductions which are not connected by any point transformation. However here we restrict analysis to three cases only. Note that symmetry generator

The corresponding characteristic equations corresponding to this scaling symmetry are given by

and the functional form of thew group-invariant solutions is given by

The transformation,

In terms of the original variables the exact (group-invariant) solution for Equation (38) is given by

The time translation symmetry leads to the steady state problems with the model given by the modified Emden- Fowler equation of the form

The transformation

which is hard to solve exactly. Note that ODE (41) admits the scaling symmetry which may be used to reduce the order this equation by one.

To construct the exact (group-invariant) solution we consider the linear combination of symmetries

The basis of invariants is given by

and

Let

where F satisfies the second order ODE

Equation (43) admits the following rotation symmetry,

We implement the method of differential invariants to reduce the order of Equation (43) by one. The first prolongation of X is given by

The characteristic equations are given by,

The invariants are therefore,

A further simplification can be made to Equation (47), whereby

Suppose

wherein an integration constant vanished for simplicity.

In terms of the original variables we obtain a group-invariant (particular) solution given by

where

In this case, Equation (25) admits a two dimensional Lie symmetry algebra spanned by the base vectors

The time translation led to the steady state problem given by the Emden-Fowler type ODE

Which is harder to solve exactly.

Reduction by this symmetry generator led to a functional form of the group-invariant solution given by

which is in fact separation of variables. Here F satisfies the first order ODE

which has a solution

where

It is worth noting that Equation (25) has the conservation laws given by the conserved vectors

provided

In this paper, we have used both classical and nonclassical symmetry methods to construct the exact solutions. Some new group-invariant (exact) solutions for reaction diffusion equation with spatially dependent diffusivity and the coefficient of the source term have been constructed using both classical and nonclassical symmetry techniques. Figures 1-6 depict the change in mutant population with respect to either time or space or both. The effects of the parameters appearing in the plotted exact solutions on the population are displayed. We have introduced the modified Hopf-Cole transformation to transform a nonlinear second order ODE to a simpler to solve linear third order ODE. To the best of our knowledge, this transformation has never been used in the recorded literature.

KirstenLouw,Raseelo J.Moitsheki, (2015) Group-Invariant Solutions for the Generalised Fisher Type Equation. Natural Science,07,613-624. doi: 10.4236/ns.2015.713061