Exact Solutions for Some Fractional Differential Equations

The extended Jacobi elliptic function expansionmethod is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations.


Introduction
Fractional differential equations attracted attention in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics [1][2][3].Also, they are employed in social sciences such as food supplement, climate, finance, and economics.
In [34], Jumarie proposed a modified Riemann-Liouville derivative.With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation.
In this paper, we used extended Jacobi elliptic function expansion method [35][36][37] to establish exact solutions for three nonlinear space-time fractional differential equations in the sense of Jumarie's modified Riemann-Liouville derivative, namely, the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation.Also, we included figures to show the properties of some Jacobi elliptic function solutions of these fractional differential equations.

Jumarie's Modified Riemann-Liouville Derivative and the Extended Jacobi Elliptic Function Expansion Method
In this section, we first give the definition and some properties of the modified Riemann-Liouville derivative which are used further in this paper.The Jumarie modified Riemann-Liouville derivative of order  is defined by the expression [34] 2 Advances in Mathematical Physics where  :  → ,  → () denote a continuous (but not necessarily differentiable) function.Some properties of the fractional modified Riemann-Liouville derivative were summarized and three useful formulas of them are [34] Next, let us consider nonlinear partial fractional differential equation

𝑃 (𝑢, 𝐷
where  is an unknown function and  is a polynomial of .
In this equation, the partial fractional derivatives involving the highest order derivatives and the nonlinear terms are included.
Li and He [38] presented a fractional complex transform to convert fractional differential equations into ordinary differential equations (ODEs), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus.By using the traveling wave variable where ,  are nonzero arbitrary constants and  is the wave speed, we can rewrite (3) as the following nonlinear ODE: where the prime denotes the derivation with respect to .If possible, we should integrate (5) term by term one or more times.
Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE.For this purpose, using the extended Jacobi elliptic function expansion method, () can be expressed as a finite series of Jacobi elliptic functions, sn , that is, the ansatz: The parameter  is determined by balancing the linear term(s) of highest order with the nonlinear one(s).And where cn  and dn  are the Jacobi elliptic cosine function and the Jacobi elliptic function of the third kind, respectively, with the modulus  (0 <  < 1).Therefore, the highest degree of   /  is taken as Substituting ( 6)-( 8) into (5) and comparing the coefficients of each power of sn  in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to ,   ( = 0, 1, . . ., ), and   ( = 1, 2, . . ., ).Solving this system, with the aid of Mathematica, then ,   ( = 0, 1, . . ., ), and   ( = 1, 2, . . ., ) can be determined.Substituting these results into (6), then some new Jacobi elliptic function solutions of (3) can be obtained.We can get other kinds of Jacobi doubly periodic wave solutions.

Since lim
degenerates, respectively, as the following form.

Applications of the Method
In this section, we present three examples to demonstrate the effectiveness of our approach to solve nonlinear fractional partial differential equations.

Space-Time Fractional Generalized Reaction Duffing Equation.
We have applied the extended Jacobi elliptic function expansion method to construct the exact solutions of spacetime fractional generalized reaction duffing equation [39,40] in the form where , , , and  are all constants.Equation ( 12) reduces many well-known nonlinear fractional wave equations such as the following.
(i) Fractional Klein-Gordon equation: (ii) Fractional Landau-Ginzburg-Higgs equation: (iii) Fractional  4 equation: (iv) Fractional duffing equation: (v) Fractional Sine-Gordon equation: For our purpose, we introduce the following transformations: where  is a wave variable and  and  are constants; all of them are to be determined.Substituting ( 18) into ( 12), ( 12) is reduced into an ODE: where   = /.Suppose that the solution of ( 19) can be expressed by Considering the homogeneous balance between the highest order derivative   and the highest order nonlinear term  3 in (19), we obtain  = 1.So Substituting ( 21) into (19) and comparing the coefficients of each power of sn  in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to ,  0 ,  1 , and  1 .Solving this system with Mathematica, we get the following results.

Case 1.
Consider Advances in Mathematical Physics Case 2. Consider Case 3. Consider Thus, we obtain the following solutions of (12).

Triangular Periodic Solutions.
When the modulus  approaches to zero in (26), (27), we can obtain trigonometric function solutions of space-time fractional generalized reaction duffing equation, respectively:

Space-Time Fractional Bidirectional Wave Equations.
Let us apply our method to the space-time fractional bidirectional wave equations in the form [41,42] where  represents the distance along the channel,  is the elapsed time, the variable (, ) is the dimensionless horizontal velocity, V(, ) is the dimensionless deviation of the water surface from its undisturbed position, and , , , and  are real constants.When  = 1, (30) is the generalization of bidirectional wave equations, which can be used as a model equation for the propagation of long waves on the surface of water with a small amplitude by Bona and Chen [43].
For our purpose, we use the following transformation: where  and  are nonzero constants.Substituting (31) into (30), we obtain where   = /.Suppose that the solutions of ( 32) can be expressed by Balancing the highest order derivative terms and nonlinear terms in (32), we can obtain  1 =  2 = 2.So we have Proceeding as in the previous case, we get the following results.

Case 1.
Consider Advances in Mathematical Physics 7 Case 2. Consider Thus, we obtain the following solutions of (30).

Solution 1.
Consider Solution 2. Consider where 3.2.1.Soliton Solutions.When the modulus  approaches to 1 in ( 37), (38), we can obtain solitary wave solutions of the space-time fractional bidirectional wave equations, respectively: where Advances in Mathematical Physics

The Space-Time Nonlinear Fractional SRLW Equation.
We consider the space-time nonlinear fractional SRLW equation [44,45] which arises in several physical applications including ion sound waves in plasma.For our purpose, we use the following transformation: where , , and  0 are constants with ,  ̸ = 0. Substituting ( 43) into (42), we obtain Suppose that the solutions of ( 44) can be expressed by Considering the homogeneous balance between the highest order derivative   and the highest order nonlinear term  2 in (44), we obtain  = 2.So we have Proceeding as in the previous cases, we get the following results.

Case 1.
Consider where where where Thus, we obtain the following solutions of (42).

Conclusion
In this paper, we used the extended Jacobi elliptic function expansion method for solving fractional differential equations and applied it to find exact solutions of the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation.With the aid of Mathematica, we successfully obtained some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions for these equations.This method is effective and can also be applied to other fractional differential equations.