Improved Fractional Subequation Method and Exact Solutions to Fractional Partial Differential Equations

In this paper, the improved fractional subequation method is applied to establish the exact solutions for some nonlinear fractional partial differential equations. Solutions to the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation, and the (3 + 1)-spacetime fractional Zakharov-Kuznetsov equation are obtained, respectively.

In order to deal with nondifferentiable functions, Jumarie [38] has proposed a modification of the Riemann-Liouville definition which appears to provide a framework for a fractional calculus which is quite parallel to the classical calculus.
Jumarie's modified Riemann-Liouville derivative of order α for a function f is defined as follows: Some useful properties of modified Riemann-Liouville derivative are given below: D α which holds for nondifferentiable functions. Equations (2), (3), (4) which are important tools for fractional calculus. Based on these merits, the modified Riemann-Liouville derivative was successfully applied to the probability calculus, fractional Laplace problems, and fractional variational calculus.
In this paper, we aim to find new exact solutions of some important partial fractional differential equations under Jumarie's definition by improved fractional subequation method.
In what follows, we introduce the aforementioned fractional partial differential equations. They are the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the spacetime fractional regularized long-wave equation, and the (3 + 1)-space-time fractional Zakharov-Kuznetsov equation.
Suppose time t > 0, D α t u is time modified Riemann-Liouville derivative of order α for a function u, 0 < α ≤ 1, the parameters k 1 , k 2 , ⋯, k 7 are any real constants.
The generalized time fractional biological population model is given by D α t u + k 1 u 2 À Á xx + k 2 u 2 À Á yy + k 3 u x + k 4 u y + k 5 u 2 + k 6 u + k 7 = 0, ð5Þ where u = uðx, y, tÞ is an unknown function. When k 1 = −1, k 2 = −1, k 3 = k 4 = k 6 = 0, k 5 = −h ≠ 0, k 7 = rh ≠ 0, Equation (5) is the time fractional biological population model [39]: where u = uðx, y, tÞ denotes the population density and h ðu 2 − rÞ represents the amount of population due to death and birth. Moreover, hðu 2 − rÞ leads to Verhulst law. Equation (5) has an important role to understand the dynamic process of population changes, and it is also an assistant to achieve precision about it. The generalized time fractional compound KdV-Burgers equation is given by where u = uðx, tÞ is an unknown function. When k 1 = k 3 = 0, Equation (7) becomes the time fractional mKdV equation when k 1 = k 3 = 0 Equation (7) becomes the time fractional KdV equation when k 2 = k 4 = 0, Equation (7) becomes the time fractional Burgers equation [40] when k 1 = 0, Equation (7) becomes the time fractional mKdV-Burgers equation when k 2 = 0, Equation (7) becomes the time fractional KdV-Burgers equation The space-time fractional regularized long-wave equation is given by [41]: where u = uðx, tÞ is an unknown function, D α x u is the modified Riemann-Liouville derivative of order α for a function u, and D 2α x u = D α x ðD α x uÞ. The regularized long-wave equation, which describes approximately the unidirectional propagation of long waves in certain nonlinear dispersive systems, was proposed by Benjamin et al. in 1972. The regularized long-wave equation is considered an alternative to the KdV equation, which is modeled to govern a large number of physical phenomena such as shallow waters and plasma waves.
where u = uðx, y, z, tÞ is an unknown function; D α x u, D α y u, and D α z u are the modified Riemann-Liouville derivatives of the function u; D 2α The Zakharov-Kuznetsov equation was first derived for analysing weakly nonlinear ion acoustic waves in heavily magnetized lossless plasma and geophysical flows in two dimensions. The ZK equation is one of the two wellestablished canonical two-dimensional extensions of the KdV equation. The ZK equation governs the behavior of weakly nonlinear ionacoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field.
Motivated by the above results, in this paper, we use the improved subequation method to find new exact solutions of the generalized time fractional biological population model, the generalized time fractional compound KdV-Burgers equation, the space-time fractional regularized long-wave equation,

A Brief Description of the Improved Fractional Subequation Method
In this section, basic steps of the improved subequation method [42] are presented. Consider the following nonlinear fractional differential equation, where t, x 1 , x 2 , ⋯, x m are independent variables, uðt, x 1 , x 2 , ⋯, x m Þ is an unknown function, and P is a polynomial of u, u 2 , ⋯ and their partial fractional derivatives. Also, D α ð⋅Þ symbolizes the modified Riemann-Liouville fractional derivative.
Step 1. First of all, using a suitable fractional complex transform, Equation (15) converts into nonlinear ordinary differential equation given below: where u′, u″, u′″, ⋯ denotes the derivations with respect to ξ. We used to utilize real transformation, but actually, complex transformations are more useful. They make some equations easier to simplify.
Step 2. Suppose that the solution of ordinary differential equation (17) is where constants a i ði = −n, ⋯, −1, 1, ⋯, nÞ 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 Equation (17). φ = φðξÞ is the solution of the Riccati equation where σ is a constant, and the solutions of Equation (19) are obtained by Zhang et al. [34] as follows: In the previous literatures, u = a 0 + ∑ n i=1 a i φ i was considered. In this paper, we assume u = ∑ −1 i=−n a i φ i + a 0 + ∑ n i=1 a i φ i , and we can get a solution that has both hyperbolic tangent function and hyperbolic cotangent function or both tangent function and cotangent function.
Step 4. Finally, the system of algebraic equations is obtained in the previous step for a i ð−n ≤ i ≤ nÞ, and σ is solved by the Maple package. By substituting the newly obtained values into Equation (20), we get the exact solutions for the nonlinear fractional differential equation (15).
Applying a suitable fractional complex transform of the improved fractional subequation method and the chain rule, nonlinear fractional differential equations with the modified Riemann-Liouville derivative can be converted into nonlinear ordinary differential equations. Then, using the solutions of a Riccati equation, we can find exact analytical solutions expressed by triangle functions, hyperbolic functions, or power functions.

Applications of the Improved Fractional Subequation Method
In this section, the improved fractional subequation method is utilized to solve some nonlinear fractional differential equations introduced in Section 1.

The Generalized Time Fractional Biological Population
Model. The generalized time fractional biological population model is given by where u = uðx, y, tÞ is an unknown function.
We considered two cases.
3 Journal of Function Spaces then Equation (21) is reduced to the ordinary differential equation as follows: When k 4 ≠ 0, Equation (23) has only constant solutions. When k 4 = 0, Equation (23) has solutions in the form of (18). u′ is obtained from the homogeneous balance between the highest order derivative and the nonlinear term u 2 . We obtain the solution of Equation (21) as follows: Substituting Equation (24) together with its necessary derivatives into Equation (21), the algebraic equation is arranged according to the powers of the function φ k ðξÞ. Then, the following coefficients are obtained: Let the coefficients be zero. By solving the set of equations given above for a −1 , a 0 , a 1 , a, b, and σ, we obtain solution sets as follows: Set 1 Set 2 Set 3 then Equation (21) is reduced to the ordinary differential equation as follows: The solution of Equation (30) is in the form of (18). n = 1 is taken from the homogeneous balance between the highest order derivative u′ and the nonlinear term u 2 . We obtain the solution of Equation (30) as Equation (24). Substituting Equation (24) together with its necessary derivatives into Equation (30), the algebraic equation is arranged according to the powers of the function φ k ðξÞ. Then, the following coefficients are obtained: Journal of Function Spaces Let the coefficients be zero. By solving the set of equations given above for a −1 , a 0 , a 1 , a, b, and σ, we obtain solution sets as follows: Set 4 Set 5 Set 6 where μ = aa −1 k 3 k 5 + aa −1 k 4 k 5 ffiffiffiffiffiffiffiffiffiffiffiffiffi −k 1 /k 2 p . We find that a −1 , a 0 , a 1 , and σ are equal in set 1 and set 4, set 2 and set 5, and set 3 and set 6, respectively. In this study, the solutions of differential equations are symbolized as u ði,jÞðx,y,tÞ , ði, j ∈ Z + Þ, where i denotes obtained set number and j is the solution number of the Riccati equation, respectively. Thus, using set 1 to set 6, we obtain the solution of Equation (21) as u i,j ðx, y, tÞ, ði = 1, 2, ⋯, 6 ; j = 1, 2, ⋯, 5Þ. u i,j ðx, y, tÞ is the following: When k 1 k 2 > 0, λ = 4k 5 k 7 − k 2 6 < 0, we have σ < 0, then
Clearly, we get more solutions of the time fractional biological population model (40) than the literature [42].
Let the coefficients of φ k ðξÞ be zero. By solving the set of equations given above for a −1 , a 0 , a 1 , a, b, and σ, we obtain solution sets as follows:         16 Journal of Function Spaces