LIE SYMMETRY ANALYSIS TO FISHER’S EQUATION WITH TIME FRACTIONAL ORDER∗

The aim of this letter is to apply the Lie group analysis method to the Fisher’s equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher’s(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher’s is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.


Introduction
Many phenomena in fluid mechanics, viscoelasticity, biology, physics, engineering and other areas of science can be successfully modeled by the fractional partial differential equations (FPDEs) in recent years. Several efficient methods have been presented to solve fractional partial differential equations of physical interest. It is necessary to point out that some methods to nonlinear FPDEs for constructing numerical, exact and explicit solutions, variational iteration method, fractional difference method, differential transform method, homotopy perturbation method, transform method, sub-equation method, Adomian decomposition method [1,2,4,5,8,9,12,[14][15][16][17]20,21,32,35,37,38] and so on. Recently, in [3,6,27,29,39], the Lie symmetry analysis is effectively applied to FPDEs, and some investigations are derived. Schrödinger equations, dynamics and rogue waves problems [22][23][24][25]36] are also hot topic recently. In [33], the author studies the invariance properties of the time fractional generalized fifth-order KdV equations by using the Lie group analy-sis method. At the same time, article [33] show that the FPDEs can be transformed into a nonlinear ODE of fractional order.
The motivation of this paper is to extend the application of Lie group analysis method to the nonlinear time-fractional Fisher's equation where 0 < α ≤ 1,D α t = ∂ α u/∂t α .When α = 1, Eq.(1.1) can be reduced to Fisher's equation of general meaning. Eq.(1) represents the evolution of the population due to the two competing physical processes, diffusion and nonlinear local multiplication. And it also describes a prototype mode for a spreading name and a model equation for the finite domain evolution of neutron population in a nuclear reactor. S. Momani and Z. Odibat derived the numerical solution of Eq.(1.1) by using Homotopy perturbation method in [18].
The paper is organized as follows. Riemann and Liouville definitions and formulas are given in section 2. In Section 3, we give an account of Lie symmetry analysis method for TFF briefly. In Section 4, we perform Lie group classification on the TFF equation, and investigate the symmetry reductions of the TFF equation. Through the symmetry reduction, we transform the FPDE into the fractional ordinary differential equations (FODE) with a new independent variable. In the meantime, some exact solutions are obtained. In Section 5 contains discussion of the obtained results.

Preliminaries
For the fractional derivative operators, there exist various definitions which are not necessarily equivalent to each other. In this paper, we consider the most common definition named after Riemann and Liouville, which is the natural generalization of the Cauchy formula for the n-fold primitive of a function f (x). The Riemann-Liouville(R-L) fractional derivative is defined as [28]: where n ∈ N , I µ f (t) is the R-L fractional integral of order µ , namely, and Γ(z) is the standard Gamma function.
Definition 2.1. The R-L fractional partial derivative is defined by If it exists, where ∂ n t is the usual partial derivative of integer order n [3,33].
Some useful formulas and properties are given in [10], here we only motion the following: The generalized Leibnitz rule [19,26] defined by Definition 2.3. In view of the generalization of the chain rule [13,33] for composite functions

Lie symmetry analysis to FPDEs
In this section, we consider the time-fractional differential equations as the form where ε ≪ 1 is a small parameter, and Here D x denotes total derivative (3.4) the vector field associated with the above group of transformations can be written as If the vector field Eq.(3.5) generates a symmetry of Eq.(3.1),then must satisfy the Lie's symmetry condition Conversely, the corresponding group transformations Eq.(3.2) to known operator Eq.(3.6) are found by solving the Lie equations It is not different to see that Eq.
(3.10) Furthermore, using the chain rule Eq.(2.8) and the generalized Leibnitz rule Eq.(3.10) withf (t) = 1, we can arrive at (3.12) It should be noted that we have µ = 0 when the infinitesimal η is linear of the variable u , because of the existence of the derivatives ∂ k η ∂u k , k ≥ 2 in above expression.Summarizing the reasonings above, we obtain the explicit form of η α,t (3.13) According to the Lie theory, we have

The time fractional Fisher's equation
In the preceding section, we have given some definitions and formulas of Lie symmetry analysis method on the FPDEs. In this section, we will deal with the invariance properties of the TFF equation. Then we give some exact and explicit solutions to the TFF equation.

Lie symmetry of time fractional Fisher's equation
By the Lie group theory, we can derive the corresponding system of symmetry equations as η 0 α − η xx − 6η + 12uη = 0.

(4.2)
Then we can get where c 1 and c 2 are arbitrary constants. Furthermore, the corresponding operator can be arrived at Similarly, the Lie algebra of infinitesimal symmetries of Eq.(1.1) is spanned by the two vector fields It is easy to check that the vector fields are closed under the Lie bracket, respectively, In order to get the similarity variables for V 2 , we have to solve the corresponding characteristic equations dx Thus, we derive group-invariant solution and group-invariant as follow It is not difficult to see that Eq.(1.1) is reduced to a nonlinear ordinary differential equation (NODE). We have a theorem as follow.
The theorem 2 has been proved in [33], here omit.

Exact and explicit solutions of the time-fractional Fisher's equation
We investigate the exact analytic solutions via power series method [7] and symbolic computations [31] for Eq.(1.1). Furthermore, we analyze the convergence of the power series solutions. Set from Eq.(4.12), we can have n(n − 1)a n θ n−2 . (4.14) Substituting Eqs.(4.13) and (4.14) into Eq.(4.9), we obtain (n + 2)(n + 1)a n+2 θ n − 6 ∞ n=0 a n θ n + 6 ∞ n=0 a n θ n ∞ n=0 a n θ n = 0. (4.16) Comparing coefficients in Eq.(4.15) when n = 0 , we get when n ≥ 1 ,we have a k a n−k −6a n . (4.18) Thus, each coefficient a n (n ≥ 1) for Eq.(4.13) are found by the arbitrary constants a i (i = 0, 1, 2). This means that the exact power series solution for Eq.(4.9) exists and its coefficients depend on the Eqs.(4.16) and (4.17).Therefore, it is obvious that the power series for Eq.(4.9) is an exact power series solution. Hence, the power series solution for Eq.(4.9) can be represented in the form: a k a n−k −6a n θ n+2 .
Consequently, we acquire the explicit power series solution for Eq.(1.1) as Remark 4.1. Above all, we could obtain power series solutions for some NFEDs.
To the best of our knowledge, the solutions obtained in this paper have not been reported in previous literature. Thus, these solutions are new.

Concluding remarks
In this research, we considered the symmetry analysis, explicit solutions to the time fractional Fisher's equations with Riemann-Liouville derivative. The time fractional Fisher's was reduced to a nonlinear ordinary differential equation(ODE) of fractional order. The reduced fractional ODE was solved using an explicit power series method. To summarize, Lie group analysis method is successfully to study the symmetry properties of Fisher's equation with time fractional order. However, the obtained point transformation groups of Eq.(1.1) are narrower than those for Fisher's equation for general meaning. It is shown that the technique introduced here is effective and easy to implement. This problem can be considered further.