Abstract and Applied Analysis

Volume 2013 (2013), Article ID 202650, 6 pages

http://dx.doi.org/10.1155/2013/202650

Research Article

A Local Fractional Variational Iteration Method for Laplace Equation within Local Fractional Operators

Yong-Ju Yang,¹ Dumitru Baleanu,2,3,4 and Xiao-Jun Yang5,6

¹School of Mathematics and Statistics, Nanyang Normal University, Nanyang 4730 61, China

²Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University, 06530 Ankara, Turkey

³Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

⁴Institute of Space Sciences, Magurele, RO-077125 Bucharest, Romania

⁵Institute of Software Science, Zhengzhou Normal University, Zhengzhou 450044, China

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

Received 15 November 2012; Accepted 27 February 2013

Academic Editor: Syed Tauseef Mohyud-Din

Copyright © 2013 Yong-Ju 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 variational iteration method for local fractional Laplace equation is investigated in this paper. The operators are described in the sense of local fractional operators. The obtained results reveal that the method is very effective.

1. Introduction

As it is known, the partial differential equations [1, 2] and fractional differential equations [3–5] appear in many areas of science and engineering. As a result, various kinds of analytical methods and numerical methods were developed [6–8]. For example, the variational iteration method [9–15] was applied to solve differential equations [16–18], integral equations [19], and numerous applications to different nonlinear equations in physics and engineering. Also, the fractional variational iteration method [20–23] and the fractional complex transform [24–27] were discussed recently. The efficient techniques have successfully addressed a wide class of nonlinear problems for differential equations; see [28–36] and the references therein. We notice that the developed methods are very convenient, efficient, and accurate.

Recently, the local fractional variational iteration method [37] is derived from local fractional operators [38–48]. The method, which accurately computes the solutions in a local fractional series form or in an exact form, presents interest to applied sciences for problems where the other methods cannot be applied properly.

In this paper, we investigate the application of local fractional variational iteration method for solving the local fractional Laplace equations [49] with the different fractal conditions.

This paper is organized as follows.

In Section 2, the basic mathematical tools are reviewed. Section 3 presents briefly the local fractional variational iteration method based on local fractional variational for fractal Lagrange multipliers. Section 4 presents solutions to the local fractional Laplace equations with differential fractal conditions.

2. Mathematical Fundamentals

In this section, we present few mathematical fundamentals of local fractional calculus and introduce the basic notions of local fractional continuity, local fractional derivative, and local fractional integral of nondifferential functions.

2.1. Local Fractional Continuity

Lemma 1 (see [42]). Let be a subset of the real line and a fractal. If is a bi-Lipschitz mapping, then there is, for constants and ,

such that for all ,

As a direct result of Lemma 1, one has [42]

such that

where is fractal dimension of .

Suppose that there is [38–43]

with , for and , then is called local fractional continuous at and it is denoted by

Suppose that the function is satisfied the condition (5) for , and hence it is called a local fractional continuous on the interval , denoted by

2.2. Local Fractional Derivatives and Integrals

Suppose that , then the local fractional derivative of of order at is given by [37–43]

where .

There is [38–40]

if

for any .

Local fractional derivative of high order is written in the form [38–40]

and local fractional partial derivative of high order is [38–40]

Let a function satisfy the condition (7). Local fractional integral of of order in the interval is given by [37–43]

where , , and , , , , is a partition of the interval . For other definition of local fractional derivative, see [44–48].

There exists [38–40]

if

for any .

Local fractional multiple integrals of is written in the form [40]

if (7) is valid for .

3. Local Fractional Variational Iteration Method

In this section, we introduce the local fractional variational iteration method derived from the local fractional variational approach for fractal Lagrange multipliers [40].

Let us consider the local fractional variational approach in the one-dimensional case through the following local fractional functional, which reads [40]

where is taken in local fractional differential operator and .

The local fractional variational derivative is given by [40]

where is local fractional variation signal and .

The nonlinear local fractional equation reads as

where and are linear and nonlinear local fractional operators, respectively.

Local fractional variational iteration algorithm can be written as [37]

Here, we can construct a correction functional as follows [37]:

where is considered as a restricted local fractional variation and is a fractal Lagrange multiplier; that is, [37, 40].

Having determined the fractal Lagrangian multipliers, the successive approximations , , of the solution will be readily obtained upon using any selective fractal function . Consequently, we have the solution

Here, this technology is called the local fractional variational method [37]. We notice that the classical variation is recovered in case of local fractional variation when the fractal dimension is equal to 1. Besides, the convergence of local fractional variational process and its algorithms were taken into account [37].

4. Solutions to Local Fractional Laplace Equation in Fractal Timespace

The local fractional Laplace equation (see [38–40] and the references therein) is one of the important differential equations with local fractional derivatives. In the following, we consider solutions to local fractional Laplace equations in fractal timespace.

Case 1. Let us start with local fractional Laplace equation given by

and subject to the fractal value conditions

A corrected local fractional functional for (24) reads as

Taking into account the properties of the local fractional derivative, we obtain

Hence, from (25)-(26) we get

As a result, from (27) we can derive

We have such that the fractal Lagrange multiplier reads as

From (24) we take the initial value, which reads as

By using (25) we structure a local fractional iteration procedure as

Hence, we can derive the first approximation term as

The second approximation can be calculated in the similar way, which is

Proceeding in this manner, we get

Thus, the final solution reads as

Case 2. Consider the local fractional Laplace equation as

subject to fractal value conditions given by

Now we can structure the same local fractional iteration procedure (31).

By using (36)-(37) we take an initial value as

The first approximation term reads as

In the same manner, the second approximation is given by

Finally, we can obtain the local fractional series solution as follows:

Thus, the expression of the final solution is given by

As is known, the Mittag-Leffler function in fractal space can be written in the form

Hence, the fractal dimensions of both and are equal to .

5. Conclusions

Local fractional calculus is set up on fractals and the local fractional variational iteration method is derived from local fractional calculus. This new technique is efficient for the applied scientists to process these differential and integral equations with the local fractional operators. The variational iteration method [9–19, 27] is derived from fractional calculus and classical calculus; the fractional variational iteration method [20–22, 27] is derived from the modified fractional derivative, while the local fractional variational iteration method [37] is derived from the local fractional calculus [37–43]. Other methods for local fractional ordinary and partial differential equations were considered in [27].

In this paper, two outstanding examples of applications of the local fractional variational iteration method to the local fractional Laplace equations are investigated in detail. The reliable obtained results are complementary with the ones presented in the literature.

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