Scientific Research and Essays

In this article, the fractional derivatives in the sense of Caputo, and the homotopy analysis method (HAM) are used to construct the approximate solutions for the nonlinear fractional variant Bussinesq equations with respect to time fractional derivative. The HAM contains a certain auxiliary h parameter which provides us a simple way to adjust and control the convergence region and rate of convergence of the series solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equation.


INTRODUCTION
In recent years, there has been a great deal of interest in fractional differential equations.First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use.Nearly 30 years ago, the paradigm began to shift from pure mathematical formulations to applications in various fields.During the last decade fractional calculus has been applied to almost every field of science, engineering, and mathematics.Several fields of application of fractional differentiation and fractional integration are already well established, some others have just started.
nonlinear fractional order systems.Recently, many system have been identified that display fractional-order dynamics, such as viscoelastic systems, quantum evolution of complex system, and the control of fractionalorder dynamic systems (Ahmed et al., 2006(Ahmed et al., , 2007)).Gepreel and Omran (2012) and Khaled and Mohamed (2013) have used the complex transformation to calculate the exact solution for some nonlinear fractional differential equations.In this work, a new algorithm for solving the fractional order variant Bussinesq equation (Lu, 2005)  In this article, we present an alternative approach based on HAM to the system of FDEs (Oldham and Spanier, 1974) with initial conditions (Miller and Ross, 1993), to obtain the analytic approximations solutions for fractional order variant Bussinesq equation and demonstrate the effectiveness of the HAM algorithm.We use the h -curve to control the convergence of series solutions.

BASIC DEFINITIONS
Here, we give some definitions and properties of the fractional calculus.Several definitions of fractional calculus have been proposed in the last two centuries.The fractional calculus is a name for the theory of integrals and derivatives of arbitrary order.There are many books (Oldham and Spanier, 1974;Miller and Ross, 1993;Luchko and Gorenflo, 1998;Podlubny, 1999) that develop fractional calculus and various definitions of fractional integration and differentiation, such as Grunwald-Letnikov's definition, Riemann-Liouville definition, Caputo's definition and generalized function approach.
The Riemann-Liouville fractional integral operator is the well-known Gamma function.Some of the properties of the operator  J , which we will need here, are as follows: .!

HOMOTOPY ANALYSIS METHOD FOR SYSTEM OF FDES
Let us consider the following system of differential equation

If the auxiliary parameters
Then differentiating Equation 7m times with respect to q , setting 0  q and dividing by m! , we have the mth -order deformation equation The mth -order deformation Equations 14 are linear and they can be easily solved, especially by means of symbolic computation software such as Mathematic, Maple, Mat Lab.

APPLICATION
To demonstrate the effectiveness of the method, we consider the system of nonlinear partial fractional (Oldham and Spanier, 1974) with the initial conditions (Miller and Ross, 1993).We let Where, Then, the mth -order deformation equations become Where, The system (Equation 21) has the following general solutions  , the homotopy analysis method equivalent the homotopy perturbation method and the Adomain decomposition method and take the following form:-(32) and ...
integral constant and the nonlinear operators are defined as

.
this case the approximate solution of time-fractional Equation (1) according to the HAM takes the following form
exact solutions of the variant Bussinesq equations partial differential Equation (1) obtained when replacing in Equations 31 and 32.These solutions (Equations 35 and 36) are exactly the same solutions obtained in Equation 28.The h - curves have been plotted in Figure1 for

Figure 2 .Figure 3 .CONCLUSION
Figure 2. The approximate solution (Equation 34) shown in the figure b) in comparison with the exact solution (36) shown in (a) when and . 1 0   t

Figure 4 .
Figure 4.The approximate solution (34) shown in the figure (b) in comparison with the exact solution (36) shown in (a) when