Advances in Mathematical Physics

Volume 2014 (2014), Article ID 161580, 7 pages

http://dx.doi.org/10.1155/2014/161580

Research Article

Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators

Sheng-Ping Yan,¹ Hossein Jafari,² and Hassan Kamil Jassim³

¹School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

²Department of Mathematical Sciences, University of South Africa, Pretoria, South Africa

³Department of Mathematics, University of Mazandaran, Babolsar 47415-416, Iran

Received 26 May 2014; Accepted 9 June 2014; Published 30 June 2014

Academic Editor: Xiao-Jun Yang

Copyright © 2014 Sheng-Ping Yan 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 perform a comparison between the local fractional Adomian decomposition and local fractional function decomposition methods applied to the Laplace equation. The operators are taken in the local sense. The results illustrate the significant features of the two methods which are both very effective and straightforward for solving the differential equations with local fractional derivative.

1. Introduction

Many problems of physics and engineering are expressed by ordinary and partial differential equations, which are termed boundary value problems. We can mention, for example, the wave, the Laplace, the Klein-Gordon, the Schrodinger, the Advection, the Burgers, the Boussinesq, and the Fisher equations, and others [1].

Several analytical and numerical techniques were successfully applied to deal with differential equations, fractional differential equations, and local fractional differential equations [1–10]. The techniques include the heat-balance integral [11], the fractional Fourier [12], the fractional Laplace transform [12], the harmonic wavelet [13, 14], the local fractional Fourier and Laplace transform [15], local fractional variational iteration [16–18], the local fractional decomposition [19], and the generalized local fractional Fourier transform [20] methods.

In this paper, we investigate the application of local fractional Adomian decomposition method and local fractional function decomposition method for solving the local fractional Laplace equation [21, 22] 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 Adomian decomposition method and the local fractional function decomposition method. Section 4 presents solutions to the local fractional Laplace equation with differential fractal conditions.

2. Mathematical Fundamentals

We recall in this section the notations and some properties of the local fractional operators [15–20, 23, 24].

Definition 1 (see [15–20, 23, 24]). The function is local fractional continuous at , if it is valid for

with , for and . For , it is so called local fractional continuous on the interval , denoted by .

We notice that there are existence conditions of local fractional continuities that operating functions are right-hand and left-hand local fractional continuities. Meanwhile, the right-hand local fractional continuity is equal to its left-hand local fractional continuity. For more details, see [20].

Definition 2 (see [15–20, 23, 24]). The local fractional derivative of at is defined as

where .

Local fractional derivative of high order is written in the form

And local fractional partial derivative of high order is written in the form

Definition 3 (see [15–20, 23, 24]). A partition of the interval is denoted by , , and with and . Local fractional integral of in the interval is given by

If the functions are local fractional continuous, then the local fractional derivatives and integrals exist. Some properties of local fractional derivative and integrals are given in [20].

Definition 4. Let be -periodic. For and , the local fraction Fourier series of is defined as (see [15, 25])

where

are local fractional Fourier coefficients.

Definition 5. Let . The Yang-Laplace transforms of are given by [15, 22]

where the latter integral converges and .

Definition 6. The inverse formula of the Yang-Laplace transforms of is given by [15, 22]

where ; fractal imaginary unit and .

3. Analytical Methods

In order to illustrate two analytical methods, we investigate the nonlinear local fractional equation of order as follows:

with constants and with boundary and initial conditions

3.1. Local Fractional Adomian Decomposition Method

We rewrite (10) in the following form:

Applying the inverse operator to both sides of (12) yields

where the term is to be determined from the fractal initial conditions.

Now, we decompose the unknown function as a sum of components defined by the series:

The components are obtained by the recursive formula:

3.2. Local Fractional Function Decomposition Method

According to the decomposition of the local fractional function, with respect to the system , the following functions coefficients can be given by

where

Substituting (16) into (10) implies that

Suppose that the Yang-Laplace transforms of functions and are and , respectively. Then, we obtain

That is,

Hence, we have

Let

Hence, we get

Then, making use of (8) and (9) and rearranging integration sequence, we have the following several formulas about and .

If , then

where .

Then, we get

In case and , see [26].

4. Solutions of Local Fractional Laplace Equation in Fractal Time-Space

In this section, two examples for Laplace equation are presented in order to demonstrate the simplicity and the efficiency of the above methods.

The local fractional Laplace equation (see [21]) is one of the important differential equations with local fractional derivatives. In the following, we consider solutions to local fractional Laplace equations in fractal time-space.

Example 7. Consider the following local fractional Laplace equation:

subject to the fractal value conditions

According to formula (15), we have

where

Hence, from (29) we obtain

where

Making use of (31), we present

Proceeding in this manner, we get

Thus, the final solution reads as follows:

Now, we solve Example 7 by using the local fractional function decomposition method.

We suppose that

which leads to

Contrasting (28) with (36), we directly get , , and and

Conclusively, we get

Thus, we obtain

and its graph is shown in Figure 1.

Figure 1: Exact solution for local fractional Laplace equation with fractal dimension .

Example 8. We consider the following local fractional Laplace equation:

subject to the fractal value conditions

Now we can structure the same local fractional iteration procedure (15). Hence, we have

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

Thus, the final solution reads as follows:

Now, we solve Example 8 by using the local fractional function decomposition method.

We suppose that

which leads to

Contrasting (28) with (36), we directly get , , and and

Conclusively, we get

Thus, we obtain

and its graph is given in Figure 2.

Figure 2: The plot of solution to local fractional Laplace equation with fractal dimension .

5. Conclusions

In this work solving the Laplace equations using the local fractional function decomposition method with local fractional operators is discussed in detail. Two examples of applications of the local fractional Adomian decomposition method and local fractional function decomposition method to the local fractional Laplace equations are investigated in detail. The reliable obtained results are complementary with the ones presented in the literature.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of the paper.

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