A Numeric–Analytic Method for Fractional Order Nonlinear PDE’s With Modified Riemann-Liouville Derivative by Means of Fractional Variational Iteration Method

Introduction It is known that various problems in electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, material science and signal processing can be successfully modeled by linear or nonlinear fractional order differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers [1-8]. The variational iteration method (VIM), which proposed by JiHuan He [9], was successfully applied to autonomous ordinary and partial differential equations and other fields. Ji-Huan He [10] was the first to apply the variational iteration method to fractional differential equations. Fractional convection–diffusion equation with nonlinear source term solved by Momani and Yildirim [11], space–time fractional advection–dispersion equation by Yildirim and Kocak [12], fractional Zakharov–Kuznetsov equations by Yildirim and Gulkanat [13], integro-differential equation by El-Shahed [14], non-Newtonian flow by Siddiqui et al. [15], fractional PDEs in fluid mechanics by Yildirim [16], fractional Schrödinger equation [17,18]. Different fractional partial differential equations have been studied and solved including the space-time fractional diffusion-wave equation [19-21], the fractional advection-dispersion equation [22,23], the fractional telegraph equation [24], the fractional KdV equation [25] and the linear inhomogeneous fractional partial differential equations [26]. Recently, a new modified Riemann-Liouville left derivative is suggested by G. Jumarie [27-31]. The most recently, a new application of Fractional Variational Iteration Method (FVIM) for solving non-linear fractional coupledKDV equations with modified Riemann-Liouville derivative performed by Merdan et al. [32]. Roul [33] applied the fractional VIM for obtaining the exact and approximate analytical solutions of time fractional biological population models.


Introduction
It is known that various problems in electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, material science and signal processing can be successfully modeled by linear or nonlinear fractional order differential equations. Methods of solutions of problems for fractional differential equations have been studied extensively by many researchers [1][2][3][4][5][6][7][8].
The variational iteration method (VIM), which proposed by Ji-Huan He [9], was successfully applied to autonomous ordinary and partial differential equations and other fields. Ji-Huan He [10] was the first to apply the variational iteration method to fractional differential equations.
The aim of this paper is to extend the application of the variational iteration method method to solve fractional KDV, K(2,2), mKDV equation and some fractional partial equations in fluid mechanics with modified Riemann-Liouville derivative. This paper is organized as follows: In Basic definitions section, we gave definitions related to the fractional calculus theory briefly. In Fractional variational iteration method section, we define the solution procedure of the fractional variational iteration method. To show in efficiency of this method, we give the implementation of the FVIM for the fractional KDV, K (2,2), mKDV equation and some fractional partial equations in fluid mechanics with modified Riemann-Liouville derivative and numerical results in Applications section. The conclusions are then given at the last.

Definition 1
Fractional derivative is defined as the following limit form [27][28][29] This definition is close to the standard definition of derivatives (calculus for beginners), and as a direct result, the α th derivative of a constant, 0 1 α < < , is zero.

Definition 2
The left-sided Riemann-Liouville fractional integral operator of order 0, The properties of the operator J α can be found in [1,2,34].

Definition 5
The integral with respect to ( ) dx α [27][28][29] is defined as the solution of the fractional differential equation

Definition 6
Assume that the continuous function : , has a fractional derivative of order kα , for any positive integer k and any α , 0 1 α < ≤ ; then the following equality holds, which is On making the substitution h x → and 0 x → we obtain the fractional Mc-Laurin series

Fractional Variational Iteration Method
To describe the solution procedure of the fractional variational iteration method, we consider the following fractional differential equation [34]: According to the VIM, we can build a correct functional for Eq. (4) as follows , , Using Eq. (5), we obtain a new correction functional It is obvious that the sequential approximations , 0 , u x t will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function 0 u . The initial values are usually used for choosing the zeroth approximation 0 u . With λ determined, then several approximations , 0 k u k ≥ follows immediately [35]. Consequently, the exact solution may be procured by using

Applications
In this section, we present the solution of nonlinear fractional partial differential equations as the applicability of FVIM.

Example 1
Consider the fractional KDV equation where 0 1 With initial conditions (15) Construction the following functional: We have Similarly, we can get the coefficients of n u δ to zero: Taking the initial value For the special case 1 α = is [36] ( ) ( ) With initial conditions (24) Construction the following functional: We have Similarly, we can get the coefficients of n u δ to zero: The generalized Lagrange multiplier can be identified by the above equations, 1. x t u x t Taking the initial value ( ) For the special case 1 α = is [36] ( ) ( ) Similarly, we can get the coefficients of n u δ to zero: The generalized Lagrange multiplier can be identified by the above equations, 1.

Example 4
In this example we consider one-dimensional linear inhomogeneous fractional Burgers equation [37] where 0 1, We have ( ) Similarly, we can get the coefficients of n u δ to zero: The generalized Lagrange multiplier can be identified by the above equations, 1.
Taking the initial value By the similar operations, we have

Example 5
Consider the following one -dimensional linear inhomogeneous fractional wave equation [37] where 0 1, Similarly, we can get the coefficients of n u δ to zero: The generalized Lagrange multiplier can be identified by the above equations, 1.
By the similar operations, we have Which is easily confirmed. This formally proved right in [18].

Example 6
We consider the following one -dimensional linear inhomogeneous fractional Klein-Gordon equation [37] where 1 2,    The VIM has been successfully applied to derive explicit numerical solutions for nonlinear problems and ordinary, partial, fractional, integral equations. In this paper, we have discussed modified variational iteration method having integral w.r.t. ( ) d α τ used for the first time by Jumarie. The obtained results indicate that this method is powerful and