NEW EXACT SOLUTIONS FOR SOME FRACTIONAL ORDER DIFFERENTIAL EQUATIONS VIA IMPROVED SUB-EQUATION METHOD

. In this paper, improved sub-equation method is proposed to obtain new exact analytical solutions for some nonlinear fractional diﬀerential equations by means of modiﬁed Riemann Liouville derivative. The method is applied to time-fractional biological population model and space-time frac- tional Fisher equation successfully. Finally, simulations of new exact analytical solutions are presented graphically.

In this study, our aim is to carry out new exact analytical solutions of the time fractional biological population model and the space-time fractional Fisher equation by the aid of improved sub-equation method. The observed fractional derivative in the equations is symbolized by α-order modified Riemann Liouville derivative.
Here are aforesaid equations i) Time fractional biological population model is given by [10,16] D α t u = u 2 xx + u 2 yy + h u 2 − r 0 < α ≤ 1 (1) where h and r are any real constants. u and h u 2 − r denote population density and the amount of population due to death and birth, respectively. Eq. (1.1) has an important role to understand the dynamic process of population changes and it is also an assistant to achieve precision about it.
ii) Space-time fractional Fisher equations given by the following equation The original equation (u t = u xx − u (1 − u)) is a model for the spatial and temporal propagation of a virile gene in a infinite medium, proposed by Fisher [12].
(3) and (4) which are going to be used in the application of the improved sub-equation method, which are important tools for fractional calculus. In the rest of this section, basic steps of the method are going to be present. At the third section, application of the method and obtained results is going to take place. Let us assume a fractional order partial differential equation with is presented polynomial P including various order derivatives as P u, u t , u x , u y , D α t u, D α x u, D α y u, ... = 0 where x, y, t are independent variables and u (x, y, t) is an unknown function and polynomial P includes the highest order derivative and nonlinear term of u (x, y, t). Also , D α (·) , symbolizes modified Riemann Liouville fractional derivation.
Step 1. First of all, using a suitable fractional complex transform P polynomial given in Eq.(5) converts into nonlinear ordinary differential equation given below P (u, u , u , u , ...) = 0.
Step 2. Assuming, the solution form of Eq. (7) is During the process, constants a i (i = −n, ..., n) are going to be determined. Here, n is a positive integer and it is obtained using the homogeneous balance of the highest order derivative and the nonlinear term seen in Eq. (7).
where σ is a constant and the solutions of the equation (9) are obtained by Zhang [26] using the generalized Exp-function method as follow Step 3. In this step of the method, using necessary derivatives of Eq. (8) with Eq. (9) in Eq. (7), we obtain a new polynomial in terms of ϕ. This new polynomial is arranged according to the powers of ϕ k (k = 0, 1, 2, ..., −1, −2, ...), then all coefficients of ϕ k are set equal to zero.
Step 4. Finally, algebraic equations are obtained in the previous step for a i (−n ≤ i ≤ n), and σ are solved with the help of symbolic programming MAPLE. Then the solutions of fractional differential equation given in (5) are obtained using the newly obtained values, Riccati equation given in Eq. (9) and (10). 3. Applications. In this section of the paper, we are going to present the application of improved sub-equation method to time-fractional biological model and space-time fractional Fisher equation and obtain results for these fractional differential equations. The obtained results are compared with those available in the literature and also new results are presented graphically.
3.1. Time-fractional biological population model. As the first example, we consider time-fractional biological model given in (1), after applying travelling wave transformation to the Eq. (1.1) it is reduced to ordinary differential equation as The solution of Eq. (12) is the form (8) and here n is get from the homogeneous balance between the highest order derivative u and the nonlinear term u 2 follow; n + 1 = 2n ⇒ n = 1 (13) we obtain the solution of Eq. (12) as Substituting Eq. (14) together with its necessary derivatives into Eq. (12), then the algebraic equation is arranged according to the powers of the function ϕ k (ξ). Thus the following coefficients are obtained letting the coefficients to be zero in Eq. (15) , then solving the set of equations given above for a −1 , a 0 , a 1 and σ we obtain solution sets following as In this study, the solutions of differential equations are symbolized as u i j (i, j ∈ Z + ). Here, i denotes obtained set number and j is the solution number of Riccati equation, respectively. Thus, using set 1 and set 2 we obtain the following solutions; w is cons. σ = 0 (17) and with set 3 Finding exact solutions of fractional biological population model has attracted many researchers. One can see Refs. [1,27,6]. However, in the present study, we have obtained different exact analytical solution according to studies mentioned. Also, some of these solutions are similar with study of Zhang and Zhang [27]. the advantage of improved sub equation method than sub-equation method given is to obtain new solutions with the help of coefficient a −1 that encountered in (14). At the end of this example, graphical representation of exact analytical solution

Space-time fractional Fisher equation.
In the second example, we are going to consider space-time fractional Fisher equation given in (2). Applying the following wave transformation to the equation [13], we obtain following ordinary differential equation follow with the help of homogeneous balance, we calculate n = 1. Thus, we can rewrite the solution of Eq.(20) as Similar to the previous example, if we substitute the solution and its necessary derivatives into Eq. (20) and arrange, we acquire coefficients of algebraic equation according to power of ϕ k as By solving the coefficients given above with the aid of symbolic programming, we get the solution sets as given below (23) using solution sets and Eqs.(10) into (21), we get Using set 1 and 2 for c = k In the end, we obtained the results given below for σ = 0 and w is a constant  As a result of this example, to the best of our knowledge, the solutions which carry out in this paper, have not been seen in the literature.

4.
Conclusions. In summary, Improved sub-equation method has been employed for finding new exact analytical solutions of Time-fractional biological population model and Space-time fractional fisher equation. Applying the method, fractional order differential equations are converted into ordinary differential equations. Then, with the help of Riccati equation, generalized hyperbolic function and rational function solutions are obtained. Some of these experience solutions shown in figures are new in the literature. Consequently, the obtained results illustrate that the method is reliable and efficient method for solving a wide range of fractional order differential equations.