Nonlinear fractional differential equations (NFDEs) are universal models of the classical differential equations of integer order. In recent years, the fractional order derivative and integral is becoming a hot spot of international research; it can more accurately describe the nonlinear phenomena in physics. Such as chemical kinematics, chemical physics and geochemistry, communication, phy- sics, biology, engineering, mathematics, diffusion processes in porous media, in vibrations in a nonlinear string, power-law non-locality, and power-law long- term memory can use NFDEs as models to express these problem      . In the last few years, it has become an important issue and matter of interest for researchers about the study of analytical and numerical solutions of fractional differential equations (FDEs). There are a lot of effective methods which can be used to study soliton, such as the fractional functional sub-equation method  , the fractional modified trial equation method  , the first integral method  , the fractional functional variable method  , the extended tanh-function method  , the (G’/G)-expansion method   and so on.
The present article aims to find out the modified KDV-Zakharov Kuznetsov    equation’s exact solutions by using named method of undeter- mined coefficients. The following is the organization of this paper. Some basic definitions and mathematical preliminaries of the fractional calculus are intro- duced in the next section. Investigated method of undetermined coefficients applied to solve fractional differential equations based on a fractional complex transform is presented in Section 3. In Section 4, we apply method of undeter- mined coefficients to the space-time nonlinear fractional modified KDV-ZK equation. Finally, we give some conclusions.
2. Basic Definitions
Fractional calculus is a generalization of classical calculus. There are a lot of approaches developed over years to generalize the concept of fractional order derivative, such as, Riemann-Liouville, Grünwald-Letnikow, Caputo  , Kolwankar- Gangal, Oldham and Spanier, Miller and Ross, Cresson have presented many methods, and Jumnarie put forward a modified Riemann-Liouville derivative   .
In the section, the some properties and definitions of the modified Riemann- Liouville derivative that will be applied in the sequel of the work were given.
The following is the modified Riemann-Liouville derivative defined by Jumarie  
Remark1. denote a continuous but not necessarily differentiable function.
The probability calculus, fractional Laplace problems, and fractional varia- tional calculus successfully applied Jumarie’s modified Riemann-Liouville derivative. To summarize a few useful formulae by Jumarie’s modified Riemann- Liouville derivative in   , we give some properties as follows
Remark 2. J. H. He et al. in  modified the chain rule given by Equation (5) to the formula
where is called the sigma indexes (see  ). Therefore, Equation (5) is mo- dified to the forms
3. Method of Undetermined Coefficients
In the section, we introduce the generally steps of method of undetermined coefficients
Step 1: We set a nonlinear fractional order partial differential equation as follows
where is an unknown function about two independent variables, modified Riemann-Liouville derivative of , and is a polyno- mial of and its partial fractional derivatives, in which includes the highest order derivatives and the nonlinear terms.
Step 2: By using the traveling wave transformation
where and are non zero arbitrary constants. And by using the chain rule
where and are called the sigma index. The sigma index usually is determined by gamma function  . In general, we can take where is a constant.
Substituting (10) along with (2) and (11) into (9), we can rewrite Equation (9) in the following nonlinear ordinary differential equation
where the prime denotes the derivative with respect to . For the convenience of calculation, we should obtain a new equation by integrating Equation (12) term by term one or more times.
Step 3: By the following form  , assume that solution of the Equation (14) can be represented
where is nonzero constant, is obtained by balancing the highest order term and nonlinear term of Equation (9) or Equation (12).
Step 4: Substituting the constant and into Equation (14), we can obtain the solution of the fractional order Equation (9).
4. The (3 + 1) Dimensional Space-Time Fractional mKDV-ZK Equation
In this current sub-section, we apply method of undetermined coefficients to solve the (3 + 1) dimensional space-time fractional mKDV-ZK equation of the form,
where , , and are nonzero constants, is a parameter describing the order of the fractional space-time-derivative. When , , , , Equation (14) is called the fractional modified KDV equation
when Equation (14) is called the modified KDV-ZK equation
The modified KDV-ZK equation is applied in many physical areas. Existence of the solutions for this equation has been considered in several papers, see references in   . Next, we will obtain the non-topological soliton and dark soliton solutions to Equation (14) by method of undetermined coefficients   .
Therefore, we use the following transformations,
Where are nonzero constants.
Substituting Equation (17) with Equation (2) and Equation (11) into Equation (14), we have
where . By once integrating and setting the constants of integration
to zero, we obtain
4.1. The Non-Topological Soliton Solution
To get the non-topological soliton solution of Equation (19), we can make the assumption,
where are nonzero constants coefficients. The is unknown at this point and will be determined later. From the Equation (20)-(21), we obtain
Thus, substituting the ansatz (23)-(27) into Equation (21), yields to
Now, from Equation (25), equating the exponents and leads to
From Equation (25), setting the coefficients of and terms to zero, we obtain
by using Equation (27) and after some calculations, we have
We find, from setting the coefficients of terms in Equation (25) to zero
also we get
From Equation (29), it is important to note that
Thus finally, the 1-soliton solution of Equation (14) is given by:
4.2. The Dark Soliton Solution
In order to start off with the solution hypothesis, we use the solitary wave ansatz of the form
where are the free parameters. Also the is unknown at this point and will be determined later.
From Equations (35)-(36), we obtain
Substituting Equations (35)-(39) into Equation (19), gives
Now, from Equation (40), equating the exponents of and gives,
Setting the coefficients of and terms in Equation (40) to zero, we have
then, we get
Again, from Equation (40) setting the coefficients of terms to zero,
and from Equation (45) we have
Equation (46) prompts the constraint
Thus finally, the dark soliton solution for the (3 + 1) dimensional space-time fractional mKDV-ZK equation is given by:
In this article, we have got the new solutions for the (3 + 1) dimensional space- time fractional mKDV-ZK equation by using the method of undetermined coefficients. Up to now, we could not find that these solutions were reported in other papers. In order to solve many systems of nonlinear fractional partial differential equations in mathematical and physical sciences, such as, the space-time fractional mBBM equation, the time fractional mKDV equation, the nonlinear fractional Zoomeron equation and so on, we can use the method of undetermined coefficients recommended herein would be general to a certain extent.
This work is in part supported by the Natural Science Foundation of China (Grant Nos. 11271008, 61072147).
 Baskonus, H.M. and Bulut, H. (2015) On the Numerical Solutions of Some Fractional Ordinary Differential Equations by Fractional Adams-Bashforth-Moulton Method. Open Mathematics, 13, 547-556.
 Bulut, H., Belgacem, F.B.M. and Baskonus, H.M. (2015) Some New Analytical Solutions for the Nonlinear Time-Fractional KdV-Burgers-Kuramoto Equation. Advances in Mathematics and Statistical Sciences, 118-129.
 Hammouch, Z. and Mekkaoui, T. (2012) Travelling-Wave Solutions for Some Fractional Partial Differential Equation by Means of Generalized Trigonometry Functions. International Journal of Applied Mathematical Research, 1, 206-212.
 Guo, S., Mei, L., Li, Y. and Sun, Y. (2012) The Improved Fractional Sub-Equation Method and Its Applications to the Space-Time Fractional Differential Equations in Fluid Mechanics. Physics Letters A, 376, 407-411.
 Gurefe, Y., Sonmezoglu, A. and Misirli, E. (2011) Application of the Trial Equation Method for Solving Some Nonlinear Evolution Equations Arising in Mathematical Physics. Pramana-Journal of Physics, 77, 1023-1029.
 Taghizadeh, N., Mirzazadeh, M. and Samiei Paghaleh, A. (2012) The First Integral Method to Nonlinear Partial Differential Equations. Applications and Applied Mathematics: An International Journal, 7, 117-132.
 Sun, H.G., Zhang, Y., Chen, W. and Reeves, D.M. (2014) Use of a Variable-Index Fractional-Derivative Model to Capture Transient Dispersion in Heterogenerous Media. Journal of Contaminant Hydrology, 157, 47-58.
 Biswas, A. and Zerrad, E. (2009) 1-Soliton Solution of the Zakharov-Kuznetsov Equation with Dual-Power Law Nonlinerity. Communications in Nonlinear Science and Numerical Simulation, 14, 3574-3577.
 Jumarie, G. (2012) An Approach to Differential Geometry of Fractional Order via Modified Riemann-Liouville Derivative. Acta Mathematica Sinica, English Series, 28, 1741.
 Matebese, B.T., Adem, A.R., Khalique, C.M. and Biswas, A. (2011) Solutions to Zakharov-Kuznetsov Equation with Power Law Nonlinearity in (1+3)-Dimensions. Physics of Wave Phenomena, 19, 148-154.
 Lslam, M.H., Khan, K., Ali Akbar, M. and Salam, M.A. (2014) Exact Traveling Wave Solutions of Modified Kdv-Zakharov-Kuznetsov Equation and Viscous Burger Equation. Springer Plus, 3, 105.
 Guner, O., Bekir, A., Moraru, L. and Biswas, A. (2015) Bright and Dark Soliton Solutions of the Generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahoney Nonlinear Evolution Equation, Dark Soliton and Periodic Wave Solutions of Nonlinear Evolution Equations. Proceedings of the Romanian Academy, 16, 422-429.
 Bakodah, H.O., Al Qarni, A.A., Banaja, M.A., Zhou, Q., Moshokoa, S.P. and Biswas, A. (2017) Bright and Dark Thirring Optical Solitons with Improved Adomian Decomposition Scheme. Optik, 130, 1115-1123.