Research Article | Open Access

Sheng-Ping Yan, Hossein Jafari, Hassan Kamil Jassim, "Local Fractional Adomian Decomposition and Function Decomposition Methods for Laplace Equation within Local Fractional Operators", *Advances in Mathematical Physics*, vol. 2014, Article ID 161580, 7 pages, 2014. https://doi.org/10.1155/2014/161580

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

**Academic Editor:**Xiao-Jun Yang

#### 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.

*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.

#### 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.

#### References

- A. M. Wazwaz,
*Partial Differential Equations: Methods and Applications*, Elsevier, Balkema, The Netherlands, 2002. - W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,”
*Journal of Mathematical Physics*, vol. 30, no. 1, pp. 134–144, 1989. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Zhao and C. Li, “Fractional difference/finite element approximations for the time-space fractional telegraph equation,”
*Applied Mathematics and Computation*, vol. 219, no. 6, pp. 2975–2988, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. Momani, Z. Odibat, and A. Alawneh, “Variational iteration method for solving the space- and time-fractional KdV equation,”
*Numerical Methods for Partial Differential Equations*, vol. 24, no. 1, pp. 262–271, 2008. View at: Publisher Site | Google Scholar | MathSciNet - N. Laskin, “Fractional Schrödinger equation,”
*Physical Review E: Statistical, Nonlinear, and Soft Matter Physics*, vol. 66, no. 5, Article ID 056108, 2002. View at: Publisher Site | Google Scholar | MathSciNet - Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,”
*Nonlinear Analysis. Real World Applications*, vol. 11, no. 5, pp. 4465–4475, 2010. View at: Publisher Site | Google Scholar | MathSciNet - S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 488–494, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. E. Tarasov, “Fractional heisenberg equation,”
*Physics Letters A*, vol. 372, no. 17, pp. 2984–2988, 2008. View at: Publisher Site | Google Scholar | MathSciNet - A. K. Golmankhaneh, A. K. Golmankhaneh, and D. Baleanu, “On nonlinear fractional KleinGordon equation,”
*Signal Processing*, vol. 91, no. 3, pp. 446–451, 2011. View at: Publisher Site | Google Scholar - Z. Li, W. Zhu, and L. Huang, “Application of fractional variational iteration method to time-fractional Fisher equation,”
*Advanced Science Letters*, vol. 10, pp. 610–614, 2012. View at: Publisher Site | Google Scholar - J. Hristov, “Heat-balance integral to fractional (half-time) heat diffusion sub-model,”
*Thermal Science*, vol. 14, no. 2, pp. 291–316, 2010. View at: Publisher Site | Google Scholar - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus Models and Numerical Methods*, vol. 3 of*Series on Complexity, Nonlinearity and Chaos*, World Scientific, Boston, Mass, USA, 2012. View at: Publisher Site | MathSciNet - C. Cattani, “Harmonic wavelet solution of Poisson's problem,”
*Balkan Journal of Geometry and Its Applications*, vol. 13, no. 1, pp. 27–37, 2008. View at: Google Scholar | MathSciNet - C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,”
*Computers & Mathematics with Applications*, vol. 50, no. 8-9, pp. 1191–1210, 2005. View at: Publisher Site | Google Scholar | MathSciNet - X. J. Yang,
*Local Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, China, 2011. - X. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at: Publisher Site | Google Scholar - W.-H. Su, D. Baleanu, X.-J. Yang, and H. Jafari, “Damped wave equation and dissipative wave equation in fractal strings within the local fractional variational iteration method,”
*Fixed Point Theory and Applications*, vol. 2013, no. 89, 2013. View at: Publisher Site | Google Scholar | MathSciNet - Y. J. Yang, D. Baleanu, and X. J. Yang, “A local fractional variational iteration method for Laplace equation within local fractional operators,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 202650, 6 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - X. Yang, D. Baleanu, and W. Zhong, “Approximate solutions for diffusion equations on Cantor space-time,”
*Proceedings of the Romanian Academy A*, vol. 14, no. 2, pp. 127–133, 2013. View at: Google Scholar | MathSciNet - X. J. Yang,
*Advanced Local Fractional Calculus and Its Applications*, World Science, New York, NY, USA, 2012. - A. Liangprom and K. Nonlaopon, “On the convolution equation related to the diamond Klein-Gordon operator,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 908491, 16 pages, 2011. View at: Publisher Site | Google Scholar | MathSciNet - C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at: Google Scholar - X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,”
*Boundary Value Problems*, no. 1, pp. 131–146, 2013. View at: Google Scholar - A. Yang, X. Yang, and Z. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 351057, 5 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - M. Hu, R. P. Agarwal, and X.-J. Yang, “Local fractional Fourier series with application to wave equation in fractal vibrating string,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 567401, 15 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - S. Q. Wang, Y. J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 176395, 7 pages, 2014. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

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.