Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 365981, 8 pages

http://dx.doi.org/10.1155/2014/365981

Research Article

Local Fractional Laplace Variational Iteration Method for Solving Linear Partial Differential Equations with Local Fractional Derivative

Ai-Min Yang,1,2 Jie Li,1,3 H. M. Srivastava,⁴ Gong-Nan Xie,⁵ and Xiao-Jun Yang⁶

¹College of Science, Hebei United University, Tangshan 063009, China

²College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China

³School of Materials and Metallurgy, Northeast University, Shenyang 110819, China

⁴Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3R4

⁵School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an, Shaanxi 710048, China

⁶Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 4 April 2014; Accepted 28 April 2014; Published 17 July 2014

Academic Editor: Bing Xu

Copyright © 2014 Ai-Min Yang 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.

The local fractional Laplace variational iteration method was applied to solve the linear local fractional partial differential equations. The local fractional Laplace variational iteration method is coupled by the local fractional variational iteration method and Laplace transform. The nondifferentiable approximate solutions are obtained and their graphs are also shown.

1. Introduction

Fractional calculus [1] has successfully been used to study the mathematical and physical problems arising in science and engineering. Fractional differential equations are applied to describe the dynamical systems in physics and engineering (see [2, 3]). It is one of the hot topics for finding the solutions for the fractional differential equations for scientists and engineers. There are many analytical and numerical methods for solving them, such as the spectral Legendre-Gauss-Lobatto collocation method [4], the shifted Jacobi-Gauss-Lobatto collocation method [5], the variation iteration method [6], the heat-balance integral method [7], the Adomian decomposition method [8], the finite element method [9], and the finite difference method [10].

The above methods did not deal with some nondifferentiable problems arising in mathematics and physics (see [11–13]). Local fractional calculus (see [12–14] and the cited references) is the best choice to deal with them. Some methods for solving the local fractional differential equations were suggested, such as the Cantor-type cylindrical-coordinate method [15], the local fractional variational iteration method [16, 17], the local fractional decomposition method [18], the local fractional series expansion method [19], the local fractional Laplace transform method [20], and local fractional function decomposition method [21, 22]. More recently, the coupling schemes for local fractional variational iteration method and Laplace transform were suggested in [23]. However, the results are very little. In this paper, our aim is to use the local fractional Laplace variational iteration method to solve the linear local fractional partial differential equations. The structure of the paper is suggested as follows. In Section 2 the basic theory of local fractional calculus and local fractional Laplace transform are introduced. Section 3 is devoted to the local fractional Laplace variational iteration method. In Section 4, the four examples for the local fractional partial differential equations are given. Finally, the conclusions are considered in Section 5.

2. Local Fractional Calculus and Local Fractional Laplace Transform

In this section, we present the basic theory of local fractional calculus and concepts of local fractional Laplace transform.

The local fractional derivative of of order is defined as [12–15]

where

The local fractional integral of of order in the interval is defined as [12–14, 16–23]

where the partitions of the interval are , with , , , and , .

The local fractional series of nondifferentiable function used in this paper are presented as follows [12–14]:

The properties of local fractional derivatives and integral of nondifferentiable functions are given by [12, 13]

The local fractional Laplace transform is defined as [12, 20–22]

where is local fractional continuous and .

The inverse formula of local fractional Laplace transform is defined as [12, 20–22]

where and .

The local fractional convolution of two functions is defined as [12, 20–22]

The properties for local fractional Laplace transform used in the paper are given as [12]

3. Analysis of the Method

In this section, we introduce the idea of local fractional variational iteration method [16, 17], which is coupled by the local fractional variational iteration method and Laplace transform.

Let us consider the following nonlinear operator with local fractional derivative:

where the linear local fractional differential operator denotes and is a source term of the nondifferential function.

Following the local fractional Laplace variational iteration method [23], we have the local fractional functional formula as follows:

which leads to

Using the local fractional Laplace transform, from (12), we get

Taking the local fractional variation [21], we obtain

which leads to

where

Hence, from (15) and (16) we get

which yields

Hence, we get the new iteration algorithm as follows:

where the initial value is suggested as

Therefore, we have the local fractional series solution of (10)

so that

The above process is called the local fractional variational iteration method.

4. The Nondifferentiable Solutions for Linear Local Fractional Differential Equations

In this section, we present the examples for linear local fractional differential equations of high order.

Example 1. The local fractional differential equation is presented as

subject to the initial value

From (19) and (23) we obtain

where the initial value is given by

Therefore, the successive approximations are

Hence, we get

and its graph is shown in Figure 1.

Figure 1: Plot of with the fractal dimension .

Example 2. We report the following local fractional partial differential equation:

The initial value is given by

In view of (19) and (29) the local fractional iteration algorithm can be written as follows:

where the initial value is

Making use of (31) and (32), the successive approximate solutions are shown as follows:

Therefore, the nondifferentiable solution of (29) reads

and its plot is presented in Figure 2.

Figure 2: Plot of with the parameters and .

Example 3. The following local fractional partial differential equation

is considered and its initial value is

Making use of (19) and (35) the local fractional iteration algorithm reads

where the initial value is suggested as

From (38) we have the successive approximations as follows:

Hence, the approximate solution of (35) is given by

and its graph is presented in Figure 3.

Figure 3: Plot of with the parameters and .

5. Conclusions

Local fractional calculus was successfully applied to deal with the nondifferentiable problems arising in mathematical physics. In this work we considered the coupling method of the local fractional variational iteration method and Laplace transform to solve the linear local fractional partial differential equations and their nondifferentiable solutions were obtained. The results are efficient implement of the local fractional Laplace variational iteration method to solve the partial differential equations with local fractional derivative.

Conflict of Interests

The authors declare that they have no competing interests in this paper.

Acknowledgments

This work was supported by National Scientific and Technological Support Projects (no. 2012BAE09B00), the National Natural Science Foundation of China (no. 51274270), and the National Natural Science Foundation of Hebei Province (no. E2013209215).

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