Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

Article history: Received: 8 July 2019 Accepted: 12 April 2020 Available Online: 17 December 2020 In this paper, exact analytical solutions of the biological population model, the EW and the modified EW equations with a conformable derivative operator have been examined by means of the trial solution algorithm and the complete discrimination system. Dark, bright and singular traveling wave solutions of the equations have been obtained by algorithm. Also, revealed singular periodic solutions have been listed. All solutions were verified by substituting them into their corresponding equation via Mathematica package program.


Introduction
Due to the applications in nonlinear optics, biology, population dynamics, biomathematics, and other areas, fractional order differential equations have been considered more often. Differential equations of noninteger order construct more accurate models for the phenomena they describe. Hence, to find analytical and numerical solutions of these equations, some effective methods have been introduced and applied so far. Even so, a general method could not be proposed so scientists are still trying to develop new approaches. Finding exact analytical solutions of noninteger order differential equations play an essential role in describing the behavior of the considered model. This study implements the trial solution algorithm with the aid of complete discrimination system to establish exact solutions of non-integer order differential equations. To this purpose, we first consider time fractional biological population model [1][2][3][4]. The equation describes population dynamics and gives ideas about complex interactions. Then, space-time fractional equal width (EW) equation [5] which describes complex physical phenomena in many fields has been considered. Finally, space-time fractional modified equal width equation [6] that describes the wave propagation with dispersion processes for one-dimensional nonlinear form was considered. Exact analytical solutions of the considered equations have been obtained successfully. All of these solutions have been confirmed by substituting them into their corresponding equation with the aid of Mathematica. Also, some solutions have been plotted with Maple to depict the structure of the solution equations. These models are quite important for mathematical physics and finding exact analtyical solutions of them may help the further analytical studies.

Conformable fractional derivative and trial solution algorithm
Fractional derivative reveals more suitable models for real world problems than integer order derivative. For this reason, many researchers have paid attention to develop new definitions of fractional derivative such as Caputo-Fabrizio [7] and Riemann-Liouville [8]. Also, a new definition, conformable derivative is proposed to overcome some setbacks of the existing derivatives, see Khalil et al. [9].
Conformable fractional derivative for a function f of order  is defined as  [9]. Some properties of the conformable fractional derivative are summarized as [9,10]: Also, fractional versions of the Laplace transform, Taylor power series expansions and integration by parts are given by Abdeljawad, see [10].
Some authors [11][12][13][14][15][16] have studied fractional differential equations by different methods. Besides these, to investigate analytical solutions of PDEs, Liu proposed an approach called trial equation method which aims to reduce the examined equation to the solvable differential equations. Also Liu proposed complete discrimination system (CDS) to find exact solutions of PDEs, see [17][18][19]. The method is studied by some authors [20][21][22][23] to investigate analytical solutions of integer and non-integer order partial differential equations.
Solution steps of the trial solution algorithm can be outlined as follows [17][18][19]: Step 1. We can consider a PDE of . u Using a wave transformation, it can be reduced to an ODE.
Step 2.Trial equation can be chosen as according to the structure of the reduced equation.
Using the balance procedure, the value of n can be determined.
Step 3. Rewriting these equations into integral form and using the CDS for polynomial yield the exact solution of the considered PDE. We will apply this procedure for the fractional models.

The time fractional biological population model
Consider the time fractional biological population model: hrare constants.
u is the population density and 2 () h u r − denotes the population supply as a result of births and deaths. [1][2][3][4]. Under the transformation where , ckare constants and 2 1, i =− Eq.(6) turns into the following ODE:

The space-time fractional equal width (EW) equation
Consider the space-time fractional EW equation [ .

U a a U a U a U
With the same procedure, corresponding system of algebraic equation

The space-time fractional modified EW equation
Now we consider the space-time fractional modified EW equation [6] 33 0, where 0 t  and 0 1.

 
Using the transformation (16) If one uses the same procedure above, corresponding system

Conclusion
The trial solution algorithm with aid of CDS was implemented to find exact traveling wave solutions of the fractional biological population model, the fractional EW and the fractional modified EW equations in sense of conformable derivative. Using this algorithm, significant dark, bright and also singular periodic traveling wave solutions of these equations were obtained. All solutions were checked by Mathematica and some of them were plotted by Maple. Finding exact analytical solutions of the models may play a quite important role for explaining the physical phenomenon they characterize.