Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 543848, 10 pages

http://dx.doi.org/10.1155/2013/543848

Research Article

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

Asma Ali Elbeleze,¹ Adem Kılıçman,² and Bachok M. Taib¹

¹Faculty of Science and Technology, Universiti Sains Islam Malaysia, 71800 Nilai, Malaysia

²Department of Mathematics, Faculty of Science, University Putra Malaysia, 43400 UPM, Serdang, Selangor Darul Ehsan, Malaysia

Received 13 March 2013; Accepted 5 June 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Asma Ali Elbeleze et al. 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.

We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics 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.

1. 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.

(1) Fractional Klein-Gordon equations

 This model is obtained by replacing the order time derivative with the fractional derivative of order . 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]. (2) Fractional Burger's equation

 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 is the European call option price at asset price and at time , is the maturity, is the risk-free interest rate, and represents the volatility function of underlying asset. The payoff functions are

 where and are the value of the European call and put options, respectively, denotes the expiration price for the option, and the function gives the large value between and . 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 by using HPM, while [8] solved this equation by using HAM also when . 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 variational iteration method [15–29] is one of approaches to provide an analytical approximation solutions to linear and nonlinear problems. The fractional variational iteration method with Riemann-Liouville derivative was proposed by Wu and Lee [30] and applied to solve time fractional and space fractional diffusion equations. Furthermore Wu [31] explained a possible use of the fractional variational iteration method as a fractal multiscale method. Recently fractional variational iteration method has been used to obtain approximate solutions of fractional Riccati differential equation [32].

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 [33–35] 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.

2. Fractional Calculus

2.1. Fractional Derivative via Fractional Difference

Definition 1. The left-sides Riemann-Liouville fractional integral operator of order , of a function , , is a defined as

Definition 2. The modified Riemann-Liouville derivative [34, 35] is defined as

where , , and .

Definition 3. Let , denote a continuous (but not necessarily differentiable) function, and let denote a constant discretization span. Define the forward operator by the equality

Then the fractional difference of order , , 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].

For , some properties of the fractional modified Riemann-Liouville derivative.

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 is defined by the following series representation, valid in the whole complex plane [37]:

3. 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 is the restricted variation, that is, .

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 and is an initial approximation which can be freely chosen if it satisfies the initial and boundary conditions of the problem.

4. 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

Substituting , and in (1). Construction the iteration formula as follows:

Taking the initial value we can derive the first approximate as follows:

Thus, the approximate solution is

In Figures 1 and 2 we have shown the surface of corresponding to the values 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 while when the results of VIM and HPM somewhat diverge from the exact solution.

(a)

(b)

(c)

(d)

Figure 1: The surface shows the solution for (27) with initial condition (28): FVIM results are, respectively, (a) and (c) ; HPM [7] results are, respectively, (b) and (d) .

(a)

(b)

(c)

Figure 2: The surface shows the solution for (27) with initial condition (28): (a) FVIM when , (b) HPM [7] when , and (c) 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 we can derive the first approximate as follows:

So, the exact solution follows immediately. The exact solution is obtained by using two iterations and this is dependent on proper selection of initial guess .

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 is given by

By construction the iteration formula as follows:

Taking the initial value we can derive the first approximate as follows:

so that the solution of the problem is given by

where (z) 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 , this series has the closed form of the solution , which is an exact solution of the given Black-Scholes equation (36) for .

In Figure 3 we have shown the surface of corresponding to the value ( for FVIM&HPM and for FVIM ).

(a)

(b)

(c)

Figure 3: The surface shows the solution for (36) with initial condition (37): (a) FVIM (), (b) HPM [10] (), and (c) FVIM ().

5. 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.

Acknowledgments

The second author gratefully acknowledges that this research partially supported by Ministry of Higher Education (MOHE), Malaysia under the ERGS Grant 5527068.

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