In the last century, many authors visualized the study on properties of fractional derivatives. Various types of methods have been used from that period. In recent decades, non-linear fractional differential equations (FDEs) have been charmed at large scale. New researchers are being attracted by this new concept due to its versatile applications. Variant applications of non-linear fractional differential equations are noticeable in various sciences such as physics, engineering, biological network, landscape evolution, fluid flow, elasticity, quantum mechanics, medical science and some other branches of pure and applied mathematics. Several mathematicians proposed different types of fractional derivatives. The most popular ones are Riemann-Liouville, Caputo, Grunwald-Letnikov, Hadamard, Erdelyi, Kober, Marchaud, and Riesz. One can see that these are theoretically much easier to handle and satisfy the classical properties. The exact solutions are important to describe the physical phenomenon of non-linear fractional partial differential equations. Again we see that fractional differential equations are the generalizations of classical differential equations of integer order. There are many techniques to obtain exact traveling wave solutions such as modified kudryashov method , fractional sub-equation method , residential power series method , fractional Riccati expansion method , the fractional sub-equation method , the first integral method , improved kudryashov method , the variational iteration method , Jumarie’s modified Riemann-Liouville derivative  and other sciences (See     ). In this work, the modified kudryashov method  is used for solving the space-time fractional Zakharov Kuznetsov Benjamin Bona Mahony (ZKBBM) and Kolmogorov Petrovskii Piskunov (KPP) equation in the sense of modified Riemann-Liouville derivative  of order α, defined by the following expression
Some properties for the proposed modified Riemann-Liouville derivative are listed as follows:
We regard non-linear fractional partial differential equation with independent variables x, t and dependent variable u is given by
The main steps of this method are following as:
Step 1. We use the variable transformation
where K, L and M are non-zero arbitrary constants for transforming (2.1.1) to the following non-linear fractional ordinary differential equation with independent variable is
Here prime denotes the derivative with respect to xi.
Step 2. We seek for the exact solution of Equation (2.1.3)
where are constants to be determined such that and is the solution of the equation
Equation (2.1.5) has the solution
where is a number.
Step 3. We determine the positive integer N in Equation (2.1.4) by considering the homogenous balance between the highest derivative and the non-linear term in (2.1.3).
Step 4. Substitute (2.1.4) along with (2.1.5) into (2.1.3), we calculate all necessary derivatives of the function . As a result of this substitution, we get a polynomial of . In this polynomial we gather all terms of same powers of and equating them to zero, we obtain a system of algebraic equations which can be solved by the Maple to get the unknown parameters , K and L. Consequently, we obtain the exact solutions of (2.1.1).
3. Applications of the Method
3.1. Solutions for the Space-Time Fractional Zakharov Kuznetshov Benjamin Bona Mahony (ZKBBM) Equation
In this section, we will manage the modified Kudryashov method to find the space-time fractional Zakharov Kuznetshov Benjamin Bona Mahony (ZKBBM) equation
Introducing the wave transformation
(3.1.1) becomes the following ODE:
We consider that Equation (3.1.2) has the travelling wave solution of the form
where are constant to be determined by considering the homogenous balance between and u2 appearing in Equation (3.1.2), we have such that . We get the solution of (3.1.2) as
where are constants to be determined afterwards and satisfies the equation
Equation (3.1.5) has the solution
where is a number.
Substituting (3.1.4) and (3.1.5) into (3.1.2) and collecting the coefficients of each power of Qi and then equating each of the coefficients to zero, a system of algebraic equations is found and the solution of this system gives the following traveling wave solutions.
Setting (3.1.7) into (3.1.4), we reach at the following traveling wave solution
Similarly setting (3.1.9) into (3.1.4), the following traveling wave solution is obtained
3.2. Solutions for Kolmogorov Petrovskii Piskunov (KPP) Equation
Next we use the modified Kudryashov method to find the space-time fractional Kolmogorov Petrovskii Piskunov (KPP) equation
where are arbitrary constants.
Making the wave transformation , (3.1.1) becomes the following ODE:
Further we consider
where are constants to be determined by considering the homogenous balance between the highest order derivatives and the non-linear terms appearing in Equation (3.2.2), we have such that . Putting the values of n, we get the solution of (3.2.2)
where are constants to be determined.
Solving in similar procedure of example 1, we get the following families of values, then obtain traveling wave solutions.
For (3.2.5), the traveling wave solution of (3.2.2) is
For (3.2.7), the traveling wave solution of (3.2.2) is obtained as
3.3. Result and Direction
Here the three dimensional graph (Figure 1) of Equation (3.1.8) shows a bell shaped soliton profile. Similarly Figure 2 displays a bell shaped three dimensional plot of Equation (3.1.10). Again, the traveling wave three dimensional sketch (Figure 3) of Equation (3.2.6) indicates kink shape. Likewise Figure 4
Figure 1. Profile of for within the interval .
Figure 2. Profile of for within the interval .
Figure 3. Profile of for within the interval .
Figure 4. Profile of for within the interval .
shows kink shape of Equation (3.2.8). We solve these equations with the help of Maple.
In our study, we solve such two equations (ZKBBM and KPP) which are very important for describing physical phenomenon. The first equation is used for modeling long surface gravity wave of small amplitude and the later one describes the phase transition problems. Here, the obtained analytical solutions are exact and the solutions expressed by the rational functions depict the propagations of all kinds of traveling wave. The process of finding exact results is very easy and effective. Therefore, it can also be applied to other fractional partial differential equations.
 Zayed, E.M.E. and Alurrfi, K.A.E. (2015) The Modified Kudryashov Method for Solving Some Seventh Order Nonlinear PDEs in Mathematical Physics. World Journal of Modeling and Simulation, 11, 308-319.
 Alzaidy, J.F. (2013) Fractional Sub-Equation Method and Its Applications to the Space-Time Fractional Differential Equations in Mathematical Physics. British Journal of Mathematics & Computer Science, 3,153-163.
 Freihet, A.A. and Zuriqat, M. (2019) Analytical Solution of Fractional Burgers-Huxley Equations via Residual Power Series Method. Lobachevskii Journal of Mathematics, 40, 174-182.
 Liu, X.H. (2018) The Traveling Wave Solutions of Space-Time Fractional Differential Equation Using Fractional Riccati Expansion Method. Journal of Applied Mathematics and Physics, 6, 1957-1967.
 Salam, M.A. and Habiba, U. (2019) Application of the Improved Kudryashov Method to Solve the Fractional Non-Linear Partial Differential Equations. Journal of Applied Mathematics and Physics, 7, 912-920
 Wu, G.-C. (2012) Applications of the Variational Iteration Method to Fractional Diffusion Equations: Local versus Nonlocal Ones. International Review of Chemical Engineering (I.RE.CH.E.), 4, 505-510.
 Al Masalmeh, M. (2017) Series Method to Solve Conformable Fractional Riccati Differential Equations. International Journal of Applied Mathematical Research, 6, 30-33.
 Bhrawy, A.H. and Zaky, M.A. (2015) A Method Based on the Jacobi Tau Approximation for Solving Multi-Term Time-Space Fractional Partial Differential Equations. Journal of Computational Physics, 281, 886-895.
 Ege, S.M. and Misirli, E. (2014) The Modified Kudryashov Method for Solving Some Fractional-Order Nonlinear Equations. Advances in Difference Equations, 2014, Article No. 135.