Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluidmechanics and in financial models.The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.


Introduction
The topic of fractional calculus (theory of integration and differentiation of an arbitrary order) was started over 300 years ago.Recently, fractional differential equations have attracted many scientists and researchers due to the tremendous use in fluid mechanics, mathematical biology, electrochemistry, and physics.For example, differential equations with fractional order have recently proved to be suitable tools to modeling of many physical phenomena [1] and the fluid-dynamic traffic model with fractional derivative [2], and nonlinear oscillation of earthquake can be modeled with fractional derivatives [3].
There are several types of time fractional differential equations.
The linear and nonlinear Klein-Gordon equations are used to modeling many problems in classical and quantum mechanics and condensed matter physics.
For example, nonlinear sine Klein-Gordon equation models a Josephson junction [4,5]. ( In general, fractional Burger's model is derived from well-known Burger's equation model by replacing the ordinary time derivatives to fractional order time derivatives.Reference [6] has investigated unsteady flows of viscoelastic fluids with fractional Burger's model and fractional generalized Burger's model through channel (annulus) tube and solutions for velocity field.
(3) Fractional Black-Scholes European option pricing equations In financial model the fractional Black-Scholes equation is obtained by replacing the order of derivative with a fractional derivative order [10].
where V(, ) is the European call option price at asset price  and at time ,  is the maturity, () is the riskfree interest rate, and (, ) represents the volatility function of underlying asset.The payoff functions are where V  (, ) and V  (, ) are the value of the European call and put options, respectively,  denotes the expiration price for the option, and the function max(, 0) gives the large value between  and 0. The Black-Scholes equation is one of the most significant mathematical models for a financial market.This equation is used to submit a reasonable price for call or put options based on factors such as underlying stock volatility and days to expiration.
Formerly, [7] investigated approximate analytical solution of fractional nonlinear Klein-Gordon equation (1) when 0 <  ≤ 1 by using HPM, while [8] solved this equation by using HAM also when 1 ≤  < 2. Reference [9] solved the coupled Klein-Gordon equation with time fractional derivative by ADM.References [10,11] solved fractional Black-Scholes equations by using HPM using Sumudu and Laplace transforms, respectively.Reference [12] gave the exact solution of fractional Burgers equation, while [13] used DTM to find the approximate and exact solution of space-and time fractional Burgers equations.Reference [14] solved this equation by using VIM.
The objective of this paper is to extend the application of the fractional variational iteration method to obtain analytical approximate solution for some fractional partial differential equations.These equations include fractional Klein-Gordon equation (1), Burgers equation (2), and fractional Black-Scholes equations (3).
Motivated and inspired by the ongoing research in this field, we will consider the following time fractional differential equation: with initial condition where   /  is modified Riemann-Liouville derivative [33][34][35] of order  defined in Section 2, () and (, ) are continuous functions, [](, ) are linear and nonlinear operators, and (, ) is unknown function.
To solve the problem ( 1)-( 2), we consider the FVIM in this work.This method is based on variational iteration method [19,36] and modified Riemann-Liouville derivatives proposed by Jumarie.This paper is organized as follows.in Section 2 some basic definitions of fractional calculus theory are given.In Section 3, the solution procedure of the fractional iteration method is given; we present the application of the FVIM for some fractional partial differential equations in Section 4. The conclusions are drawn in Section 5.

Fractional Derivative via Fractional Difference
Definition 1.The left-sides Riemann-Liouville fractional integral operator of order  ≥ 0, of a function  ∈   ,  ≥ −1, is a defined as Definition 2. The modified Riemann-Liouville derivative [34,35] is defined as where  ∈ [0, 1],  − 1 ≤  < , and  ≥ 1. Definition 3. Let  :  → ,  → () denote a continuous (but not necessarily differentiable) function, and let ℎ > 0 denote a constant discretization span.Define the forward operator FW(ℎ) by the equality Then the fractional difference of order , 0 <  < 1, of () is defined by the expression and its fractional derivative of order  is defined by the limit Equation ( 11) is defined as Jumarie fractional derivative of order  which is equivalent to (8).For more details we refer the reader to [35].
Fractional Leibnitz product law: fractional Leibnitz formulation: The fractional integration by parts formula: Definition 4. Fractional derivative of compounded function [34,35] is defined as Definition 5 (see [34,35]).The integral with respect to ()  is defined as the solution of the fractional differential equation Lemma 6 (see [34,35]).Let () denote a continuous function; then the solution of (2) is defined as that is, For example, with () =   in (7), one obtains Definition 7. The Mittag-Leffler function   () with  > 0 is defined by the following series representation, valid in the whole complex plane [37]:

Fractional Variational Iteration Method
To describe the solution procedure of fractional variational iteration method, we consider the time-fractional differential equations ( 1)-( 3).
According to variational iteration method we construct the following correction function: where  is the general Lagrange multiplier which can be defined optimally via variational theory [22] and ũ(, ) is the restricted variation, that is, ũ(, ) = 0.By using (7), we obtain a new correction functional Making the above functional stationary the following conditions can be obtained: Now, we can get the coefficients of  to zero: So, the generalized Lagrange multiplier can be identified as Then we obtain the following iteration formula by substituting ( 25) in (23): where 0 <  ≤ 1 and  0 (, ) is an initial approximation which can be freely chosen if it satisfies the initial and boundary conditions of the problem.

Applications
In this section, we have applied fractional variational iteration method (FVIM) to fractional partial differential equations.
Example 8.In this example we consider the following fractional nonlinear Klein-Gordon differential equation: subject to initial condition  (, 0) = 1 + sin () .
Substituting ( = 0,  = 0 and  = 1) in (1).Construction the iteration formula as follows: Taking the initial value  0 (, ) = 1 + sin() we can derive the first approximate  1 (, ) as follows: Thus, the approximate solution is In Figures 1 and 2 we have shown the surface of (, ) corresponding to the values  = 0.01, 0.5, 1 for FVIM and HPM; the two figures indicate that the differences among VIM and HPM, and the exact solution in Example 8 are negligible when  = 0.5, 1 while when  = 0.01 the results of VIM and HPM somewhat diverge from the exact solution.Example 9. We consider the one-dimensional linear inhomogeneous fractional Burger equation subject to initial condition By construction the iteration formula as follows: Taking the initial value  0 (, ) = 0 we can derive the first approximate  1 (, ) as follows: So, the exact solution (, ) =  2 +  2 follows immediately.The exact solution is obtained by using two iterations and this is dependent on proper selection of initial guess  0 (, ).
Example 10.We consider the following fractional Black-Scholes option pricing equation [38] as follows: where  is the risk-free interest rate subject to initial condition The exact solution for special case  = 1 is given by By construction the iteration formula as follows: Taking the initial value  0 (, ) = max(  −1, 0) we can derive the first approximate  1 (, ) as follows: so that the solution (; ) of the problem is given by where   () is Mittag-Leffler function in one parameter.Equation (41) represents the closed form solution of the fractional Black-Scholes equation (36).Now for the standard case  = 1, this series has the closed form of the solution (; ) = max(  − 1, 0) − + max(  , 0)(1 −  − ), which is an exact solution of the given Black-Scholes equation (36) for  = 1.
In Figure 3 we have shown the surface of (, ) corresponding to the value ( = 1 for FVIM&HPM and for FVIM  = 0.01).

Conclusion
Variational iteration method has been known as a powerful method for solving many fractional equations such as partial differential equations, integrodifferential equations, and so many other equations.In this paper, based on the variational iteration method and modified Riemann-Liouville derivative, we have presented a general framework of fractional variational iteration method for analytical and numerical treatment of fractional partial differential equations in fluid mechanics and in financial models.All of the examples concluded that the fractional variational iteration method is powerful and efficient in finding analytical approximate solutions as well as numerical solutions.For example, the results of Examples 8 and 10 illustrate that the present method is in excellent agreement with those of HPM and exact solution, where the obtained solution is shown graphically.Further, in Example 9 we got the exact solution in two iterations.The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.Maple has been used for presenting graph of solution in the present paper.