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

1 College of Science, Hebei United University, Tangshan 063009, China 2 College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China 3 School of Materials and Metallurgy, Northeast University, Shenyang 110819, China 4Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4 5 School 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


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][12][13]). Local fractional calculus (see [12][13][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 cylindricalcoordinate 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.

Analysis of the Method
In this section, we introduce the idea of local fractional variational iteration method [16,17], which is coupled by Discrete Dynamics in Nature and Society 3 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 wherẽ Hence, from (15) and (16) we get which yields̃{ 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.

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.
subject to the initial value From (19) and (23) we obtaiñ where the initial value is given bỹ Therefore, the successive approximations arẽ Hence, we get and its graph is shown in Figure 1.

Example 2.
We report the following local fractional partial differential equation: (29) 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: Discrete Dynamics in Nature and Society Therefore, the nondifferentiable solution of (29) reads and its plot is presented in Figure 2.
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: and its graph is presented in Figure 3.

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.