SOLUTION OF SPACE TIME FRACTIONAL GENERALIZED KdV EQUATION , KdV BURGER EQUATION AND BONA-MAHONEY-BURGERS EQUATION WITH DUAL POWER-LAW NONLINEARITY USING COMPLEX FRACTIONAL TRANSFORMATION

SOLUTION OF SPACE TIME FRACTIONAL GENERALIZED KdV EQUATION, KdV BURGER EQUATION AND BONA-MAHONEY-BURGERS EQUATION WITH DUAL POWER-LAW NONLINEARITY USING COMPLEX FRACTIONAL TRANSFORMATION UTTAM GHOSH, SANTANU RAUT, SUSMITA SARKAR AND SHANTANU DAS Department of Applied Mathematics, University of Calcutta, Kolkata, India Mathabhanga College, Cooch Behar, West Bengal, India Reactor Control Systems Design Section, E & I Group, BARC, Mumbai, India Copyright © 2018 Ghosh, Raut, Sarkar and Das. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract: Fractional calculus is a rising subject in the current research field. The researchers of different disciplines are using fractional calculus models to investigate different practical problems. In this paper, we found the exact solutions of space-time fractional generalized KdV equation, KdV Burger equation and Benjamin-Bona-MahoneyBurgers equation with dual power-law nonlinearity. The solutions are expressed in terms of hyperbolic, trigonometric and rational functions.


Introduction
It is well known that to formulate the real world phenomenon most of the places non-linear integer order or fractional differential equations arise [1][2][3].The authors are using the fractional differential models to formulate the systems which have memory [4].To solve the non-linear classical or fractional differential equations which appear in the physical systems we have to consider linear approximation and therefore the solution loses some information.To solve those 115 SOLUTION OF SPACE TIME FRACTIONAL GENERALIZED KdV EQUATION non-linear classical and fractional differential equations different research groups have developed many methods.The Adomian Decomposition Method [5][6][7], Homotopy Perturbation Method (HPM) [8][9][10], sub-equation method, Generalized Tanh method [11], Generalized exp method [12] etc.Some researchers extended the methods used to solve the integer order differential equations for solving the fractional order differential equations.The fractional subequation method is one of them [13][14].The Tanh method is an important method to find exact solution of the non-linear differential equations.It was introduced by Huiblin and Kelin [15] to find the travelling wave solutions of non-linear differential equations.Wazwaz [16] used this method to find soliton solutions of the Fisher equation.Fan [17] modified the Tanh method to solve KdV-Burgers and Boussinesq equation.The Fractional sub-equation method [13][14] and Generalized Tanh-method [11] are both based on the Homogeneous balance principal.He [19][20] developed the complex fractional transformation method to convert the fractional order differential equation to integer order differential equations.
In this paper we used the complex fractional transformation and generalized Tanh method to solve three non-linear space-time fractional differential equations arises in fluid dynamics and plasma dynamics.The space-time fractional generalized KdV equation, KdV-Burger equation and Bona-Mahoney-Burgers equation with dual power-law nonlinearity first converted to integer order differential equation using the complex fractional transformation and then those equations are solved using generalized Tanh method.In this method the solutions are expressed in terms the hyperbolic, trigonometric and the rational functions.
Organization of the paper is as follows: In section-2 we gave the review of fractional calculus, complex fractional transformation and generalized Tanh method.Section-3,4 and 5 is devoted to solve the space -time fractional generalized KdV equation, KdV-Burger equation and Bona-Mahoney-Burgers equation respectively and numerical presentation of the solutions are given for all the solutions for different values of order of derivative.Finally a conclusion is drawn.

Tanh method
Here we discussed some basic definitions of the fractional derivatives complex fractional transformation and the generalized tanh method.

a) Fractional derivative
There are different definitions of fractional derivative in which Riemann-Liouville (R-L) fractional derivative is one of the widely used definition of fractional derivative.For any continuous and integrable function () fxit is defined as, In terms of this definition, derivative of the constant where as in the classical sense the fractional derivative of the constant should have been equal to zero.
After that M. Caputo [1][2] generalized the fractional order derivative in the new way and overcome this difficulty.He proposed the definition in the following form: Let  be a positive number and n be a positive integer satisfy ft exists (that is n  th order ordinary derivative exists).Then -th order fractional derivative is defined by, 00 0 ( ) 1 ( ) where 0 RL tt J  is the R-L integral operator, defined as follows The Caputo derivative fractional derivative a constant becomes zero and it is analogous to standard classical calculus.But it is applicable only when the function accepts differentiability.
Then Jumarie [21] gave a new definition which is applicable for continuous (but not necessarily differentiable) functions.

Let ()
fx is defined in 0 xa  , (0) f is finite.Jumarie [21] defines the fractional derivatives of a continuous function () where the function ( , ) u x t is continuous but not necessarily differentiable.A constant function's fractional derivative is zero with Jumarie fractional derivative.

b) The fractional Complex Transformation
The complex fractional transformation [18][19][20] reduces the fractional order ordinary or partial differential can be reduce to the integer order ordinary or fractional order differential equation.
Let a fractional order ordinary differential equation be of the form, where D  as Jumarie fractional derivative operator.We introduce the complex fractional transformation where p is constant.Using fractional Taylor series of Jumarrie type we get . Thus, we can write, Then by using the complex fractional transformations, , Then (1.7) reduces to the integer order partial differential equation as

(c) Generalized Tanh Method
In this method the exact solutions of non-linear partial differential equations are expressed in terms of hyperbolic, trigonometric and rational functions.Let us consider a non-linear partial differential equation of the form,

 
L , , , , ,..... 0 satisfied by ( , , ) u x y t .Now we use the travelling wave transformation lx my ct     , where ( ; ) lmare the wave vector and c is the velocity of propagating waves.Then the equation (2.9) reduces to the ordinary differential equation, that is Fan and Hon [11] introduced a generalized method called Tanh method which is based on the a priori assumption that the travelling wave solutions can be expressed as the power series expansion of solutions of the non-linear Riccati differential and the homogeneous balance principle.
as constants.Then the highest power similarly that of u is 2 n  .Then we balance the highest power of  in the highest order derivative term and the non-linear term to determine n.Then put the reduce form of (2.13) in (2.10) and comparing the coefficients of  the constants can be determined.

Solution of KdV Equation using Generalized Tanh Method and complex fractional transformation
Let us consider the well known space-time fractional generalized KdV equation where ,  are the arbitrary constants.Now we consider a transformation that is Then the equation (3.1) reduces to the ordinary differential form, 3) once with respect to , we get the following    where  satisfies the equation (2.11).Hence the equation (3.5) reduces to the form, Equating the order of the highest degree of  from the non-linear term and the highest order derivative term from (3.8) we get 23 nn  , i.e.
Comparing the coefficient of 0 1 2 ,,    and 3  in the above equation (3.9), we get Solving the above equations, we get, Hence the general solution of the equation (3.1) can be written as, Hence we obtain the generalized solution of the space time fractional generalized KdV equation in terms of hyperbolic, trigonometric and the rational functions.

Solution of KdV Burger Equation using Generalized Tanh Method and complex fractional transformation
Let us consider the well known KdV Burger Equation Integrating (4.2) once both side with respect to  we get, Here also the integral constant is taken as zero because the soliton solutions are localized solutions and hence u and its derivatives will vanish when    .
Equating the order of the highest degree  in both sides of the above we get 22 nn  , i.e. 2 n  .Thus () u  becomes in the form   Comparing the coefficient of 0 Solving the above equations in (4.
Thus we obtained the exact solution of space time fractional generalized KdV Burger equation using the transformation of He [5,6 ] and the Generalized Tanh method.

Solution of Benjamin-Bona-Mahoney-Burgers equation with dual powerlaw nonlinearity using Generalized Tanh Method and complex fractional transformation
Here we consider the Benjamin-Bona-Mahoney-Burgers equation with dual power-law Here a represents the strength of defection or drifting.Strengths of the two nonlinear terms are measured by 2, 3 bb while the exponent n stands for the power law of nonlinearity parameter.The parameters , ck are the dissipative diffraction coefficient.Putting 1 n  Eq. (5.1) becomes Now use the transformation (3.2) in equation (5.2) and then it reduces to the following form, (5.4) The equation (5.4) can be written as where , , , A B C D given below (Note that term D here should be distinguished from derivative operator-that we have used) Now again change the independent variable   where  satisfies the equation (2.11).Hence the equation (5.5) becomes Let the solution of the equation (5.5) be expressed as   Then the equation (5.2) can be expressed as Equating the order of the highest degree of  in both sides of the above we get 23 nn    and so we have Comparing the coefficient of 0 1 2 3 , , and     from the equation (5.8) we get, 0 0 0 1 (5.9) Solving the equations in (5.9) we get Hence the general solution of the equation (5.2) can be written as,

Numerical presentation of the solutions
In this section, we gave the graphical presentation of the solution obtained in the previous sections for different values of the order of fractional derivative ( ).

Conclusion
In this paper we found the solutions of space-time fractional generalized KdV equation, KdV Burger equation, and Benjamin-Bona-Mahoney-Burgers equation with dual power-law nonlinearity using complex fractional transformation and the generalized Tanh hyperbolic method.Using this method the solutions are expressed in terms the hyperbolic, trigonometric and the rational functions.The obtained solutions are new type which is also predicted from the graphical presentation of the solution.Using these type formulations other non-linear fractional differential equations can be easily solved.From the numerical presentation of the solution it is clear that with the change of order of derivative solution pattern changes.

4 )
The integral constant is taken as zero because the soliton solutions are localized solutions and hence u and it's derivatives will vanish when   .UTTAM GHOSH, SANTANU RAUT, SUSMITA SARKAR AND SHANTANU DASThe equation (3.4) can be written as Now change the independent variable these values in the equation (3.6), we get a  and l and m are connected by the following relation 121 SOLUTION OF SPACE TIME FRACTIONAL GENERALIZED KdV EQUATION

1 )
Now we consider the transformation defined in (3.2) .Then the equation transforms into 32

3 )
Integrating equation(5.3)  with respect to  and using the boundary condition of solitary type solution we get 124 UTTAM GHOSH, SANTANU RAUT, SUSMITA SARKAR AND SHANTANU DAS in the equation (5.6) we get

Figure- 1 1 
Figure-1 and figure-2 respectively represents the graphical presentation of the first (in terms Tanh) and third (in terms tan) solution of (3.10) respectively for 4, 1, 1, 1 lm       .In

Figure- 3
Figure-3 and 4 represents the graphical representation of the first and third solution (4.8).In figure-3 for 1   the pattern of the solution is soliton type and with the decrease of  a new type of graphical presentation arises.Here also the range of solution increases with the decrease of order of derivative.In figure-4 multiple inverted soliton like solution arises for 1   .