Abstract and Applied Analysis

Volume 2014 (2014), Article ID 484323, 7 pages

http://dx.doi.org/10.1155/2014/484323

Research Article

Local Fractional Variational Iteration Method for Local Fractional Poisson Equations in Two Independent Variables

Li Chen,¹ Yang Zhao,² Hossein Jafari,3,4 J. A. Tenreiro Machado,⁵ and Xiao-Jun Yang⁶

¹School of Mathematics and Statistics, Zhengzhou Normal University, Zhengzhou 450044, China

²Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China

³Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran

⁴Department of Mathematical Sciences, University of South Africa, Pretoria 7945, South Africa

⁵Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Rua Dr. Antonio Bernardino de Almeida 431, 4200-072 Porto, Portugal

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

Received 13 March 2014; Accepted 25 March 2014; Published 10 April 2014

Academic Editor: Jordan Hristov

Copyright © 2014 Li Chen 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 Poisson equations in two independent variables that appear in mathematical physics involving the local fractional derivatives are investigated in this paper. The approximate solutions with the nondifferentiable functions are obtained by using the local fractional variational iteration method.

1. Introduction

As it is known the Poisson equation plays an important role in mathematical physics [1, 2]; that is, it describes the electrodynamics and intersecting interface (see, e.g., [3, 4] and the cited references therein). The solution of this equation was discussed by using different methods [5–9]. We notice that recently fractional Poisson equations based on fractional derivatives were analyzed in [10] and the existence and approximations of its solutions can be found in [11]. The Legendre wavelet method was used to find the fractional Poisson equation with Dirichlet boundary conditions [12]. In [13], the Dirichlet problem for the fractional Poisson’s equation with Caputo derivatives was reported. Furthermore, the fractional Poisson equation based on the shifted Grünwald estimate was obtained in [14].

The variational iteration method structured in [15–17] was applied to deal with the following type of equations: Helmholtz [18], Burger’s and coupled Burger’s [19], Klein-Gordon [20], KdV [21], the oscillation [22], Schrodinger [23], reaction-diffusion [24], diffusion equation [25], Bernoulli equation [26], and others. The extended variational iteration method, called the fractional variational iteration method, was developed and applied to handle some fractional differential equations within the modified Riemann-Liouville derivative [27–31]. More recently, the local fractional variational iteration method, initiated in [32], was used to find the nondifferentiable solutions for the heat-conduction [32], Laplace [33], damped and dissipative wave [34], Helmholtz [35] and Fokker-Planck [36] equations, the wave equation on Cantor sets [37], and the fractal heat transfer in silk cocoon hierarchy [38] with local fractional derivative.

We mention that developing a numerical algorithm for local fractional differential equations on Cantor set is not straightforward. Thus, in this paper, we deal with the local fractional Poisson equation in two independent variables, namely,

where the nondifferentiable functions and are adopted the local fractional differential operators and denotes the fractal dimension, subject to the initial and boundary conditions

We recall that the local fractional Laplace equation presented in [33] is a special case of the local fractional Poisson equation with source term . Taking all the above thinks into account, the aim of this paper is to find the nondifferentiable solutions for (1) with different conditions by utilizing the local fractional variational iteration algorithm.

The paper has the following organization. In Section 2 the concepts of local fractional complex derivatives and integrals are briefly reviewed. In Section 3 the local fractional variational iteration method is recalled. In Section 4 the nondifferentiable solutions for local fractional Poisson equations are presented. Finally, Section 5 outlines the main conclusions.

2. A Brief Review of the Local Fractional Calculus

Definition 1 (see [32–38]). Let the function , if it satisfies the condition

where , for , , and .

Definition 2 (see [32–38]). Let . The local fractional derivative of of order is defined as

where

The formulas of local fractional derivatives of special functions [37] used in the paper are as follows:

where is a local fractional continuous function, is a constant, and is a set of positive integers.

Definition 3 (see [32–38]). Let . The local fractional integral of of order in the interval is defined as

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

The formulas of local fractional integrals of special functions used in the paper are presented as follows [37]:

where is a local fractional continuous function, is a constant, and is a set of positive integers.

3. Analysis of the Method

The local fractional variational iteration method structured in [32] was applied to deal with the local fractional differential equations arising in mathematical physics (see, e.g., [33–38]). In this section, we introduce the idea of the local fractional variational iteration method.

Let us consider the local fractional operator equation in the form

where and are linear and nonlinear local fractional operators, respectively, and is the source term within the nondifferentiable function.

Local fractional variational iteration algorithm reads as

where is a fractal Lagrange multiplier and .

Therefore, a local fractional correction functional was structured as follows:

where is considered as a restricted local fractional variation and is a fractal Lagrange multiplier. That is, [27, 30].

After the fractal Lagrangian multiplier is determined, for , the successive approximations of the solution can be readily given by using any selective local fractional function . Consequently, we obtain the solution in the following form:

The local fractional variational method was compared with the fractional series expansion and decomposition technologies.

If , then we have the local fractional variational iteration formula [32–34, 36, 37] as follows:

The above formula plays an important role in dealing with the -order local fractional differential equation with either linearity or nonlinearity.

4. The Nondifferentiable Solutions for Local Fractional Poisson Equations

In this section we investigate the nondifferentiable solutions for the local fractional Poisson equations in two independent variables with different initial-boundary conditions.

Example 1. We analyze the local fractional Poisson equation in the following form:

subject to the initial and boundary conditions, namely,

In view of (17) and (18), we take the initial value given by

From (13), the local fractional iteration procedure is given by

Making use of (19) and (20), we get the first approximation as follows:

The second approximation can be written as

The third approximation reads as

The fourth approximation is as follows:

and so on.

Finally, by direct calculations we obtain

Hence, we report the nondifferentiable solution of (14)

and its graph is shown in Figure 1.

Figure 1: Plot of the nondifferentiable solution of (14) with the parameter .

Example 2. Next we discuss the local fractional Poisson equations as

with the initial and boundary conditions given as follows:

In view of (13), the local fractional iteration procedure becomes

where the initial value is given by

Making use of (29) and (30), the first approximation reads as follows:

The expression of the second approximation is as follows:

The third approximation becomes

The fourth approximation is given by

Therefore, we get the nondifferentiable solution of (27)

and the corresponding graph is depicted in Figure 2.

Figure 2: Plot of the nondifferentiable solution of (27) with the parameter .

Example 3. The next particular case is the local fractional Poisson equations as follows:

subject to the initial and boundary conditions

We start with the initial value as follows:

The local fractional iteration procedure leads us to

In view of (38) and (39), we obtain the following successive approximations:

and so on.

Thus, the nondifferentiable solution of (36) has the form

and its graph is shown in Figure 3.

Figure 3: Plot of the nondifferentiable solution of (36) with the parameter .

5. Conclusions

The local fractional operators started to be deeply investigated during the last few years. One of the major problems is to find new methods and techniques to solve some given important local fractional partial differential equations on Cantor set. In this line of thought we consider that three local fractional Poisson equations with differential initial and boundary values were solved by using the local fractional variational iteration method. The graphs of the nondifferentiable solutions were also obtained.

Conflict of Interests

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

Acknowledgment

This work was supported by the Natural Science Foundation of Henan Province, China.

References

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1998.

H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, New York, NY, USA, 2005.

D. J. Griffiths and R. College, Introduction to Electrodynamics, Prentice Hall, Upper Saddle River, NJ, USA, 1999.

R. B. Kellogg, “On the Poisson equation with intersecting interfaces,” Applicable Analysis, vol. 4, no. 2, pp. 101–129, 1974.

R. W. Hockney, “A fast direct solution of Poisson's equation using Fourier analysis,” Journal of the Association for Computing Machinery, vol. 12, no. 1, pp. 95–113, 1965.

P. Neittaanmäki and J. Saranen, “On finite element approximation of the gradient for solution of Poisson equation,” Numerische Mathematik, vol. 37, no. 3, pp. 333–337, 1981.

F. Civan and C. M. Sliepcevich, “Solution of the Poisson equation by differential quadrature,” International Journal for Numerical Methods in Engineering, vol. 19, no. 5, pp. 711–724, 1983.

W. Dörfler, “A convergent adaptive algorithm for Poisson's equation,” SIAM Journal on Numerical Analysis, vol. 33, no. 3, pp. 1106–1124, 1996.

Z. Cai and S. Kim, “A finite element method using singular functions for the Poisson equation: corner singularities,” SIAM Journal on Numerical Analysis, vol. 39, no. 1, pp. 286–299, 2002.

Y. Derriennic and M. Lin, “Fractional Poisson equations and ergodic theorems for fractional coboundaries,” Israel Journal of Mathematics, vol. 123, no. 1, pp. 93–130, 2001.

M. Sanz-Solé and I. Torrecilla, “A fractional poisson equation: existence, regularity and approximations of the solution,” Stochastics and Dynamics, vol. 9, no. 4, pp. 519–548, 2009.

M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini, and F. Fereidouni, “Two-dimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions,” Engineering Analysis with Boundary Elements, vol. 37, no. 11, pp. 1331–1338, 2013.

V. D. Beibalaev and R. P. Meilanov, “The Dirihlet problem for the fractional Poisson’s equation with Caputo derivatives: a finite difference approximation and a numerical solution,” Thermal Science, vol. 16, no. 2, pp. 385–394, 2012.

B. Abdollah and V. Sohrab, “Fractional finite difference method for solving the fractional Poisson equation based on the shifted Grünwald estimate,” Walailak Journal of Science and Technology, vol. 15, no. 2, pp. 427–435, 2013.

J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.

J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.

J. H. He, G. C. Wu, and F. Austin, “The variational iteration method which should be followed,” Nonlinear Science Letters A, vol. 1, no. 1, pp. 1–30, 2010.

S. Momani and S. Abuasad, “Application of He's variational iteration method to Helmholtz equation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006.

M. A. Abdou and A. A. Soliman, “Variational iteration method for solving Burger's and coupled Burger's equations,” Journal of Computational and Applied Mathematics, vol. 181, no. 2, pp. 245–251, 2005.

S. Abbasbandy, “Numerical solution of non-linear Klein-Gordon equations by variational iteration method,” International Journal for Numerical Methods in Engineering, vol. 70, no. 7, pp. 876–881, 2007.

S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalized KdV equations using He's polynomials,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 2, pp. 227–234, 2009.

V. Marinca, N. Herişanu, and C. Bota, “Application of the variational iteration method to some nonlinear one-dimensional oscillations,” Meccanica, vol. 43, no. 1, pp. 75–79, 2008.

A.-M. Wazwaz, “A study on linear and nonlinear Schrodinger equations by the variational iteration method,” Chaos, Solitons & Fractals, vol. 37, no. 4, pp. 1136–1142, 2008.

M. Dehghan and F. Shakeri, “Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem,” Journal of Computational and Applied Mathematics, vol. 214, no. 2, pp. 435–446, 2008.

S. Das, “Approximate solution of fractional diffusion equation-revisited,” International Review of Chemical Engineering, vol. 4, no. 5, pp. 501–504, 2012.

J. Hristov, “An exercise with the He’s variation iteration method to a fractional Bernoulli equation arising in a transient conduction with a non-linear boundary heat flux,” International Review of Chemical Engineering, vol. 4, no. 5, pp. 489–497, 2012.

G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A: General, Atomic and Solid State Physics, vol. 374, no. 25, pp. 2506–2509, 2010.

J.-H. He, “A short remark on fractional variational iteration method,” Physics Letters A: General, Atomic and Solid State Physics, vol. 375, no. 38, pp. 3362–3364, 2011.

N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,” Journal of King Saud University: Science, vol. 23, no. 4, pp. 413–417, 2011.

S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters A: General, Atomic and Solid State Physics, vol. 375, no. 3, pp. 309–313, 2011.

E. A. Hussain and Z. M. Alwan, “Comparison of variational and fractional variational iteration methods in solving time-fractional differential equations,” Journal of Advance in Mathematics, vol. 6, no. 1, pp. 849–858, 2014.

X. J. 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.

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.

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. 1, article 86, pp. 1–11, 2013.

A. M. Yang, Z. S. Chen, H. M. Srivastava, and X. J. Yang, “Application of the local fractional series expansion method and the variational iteration method to the Helmholtz equation involving local fractional derivative operators,” Abstract and Applied Analysis, vol. 2013, Article ID 259125, 6 pages, 2013.

X. J. Yang and D. Baleanu, “Local fractional variational iteration method for Fokker-Planck equation on a Cantor set,” Acta Universitaria, vol. 23, no. 2, pp. 3–8, 2013.

D. Baleanu, J. A. T. Machado, C. Cattani, M. C. Baleanu, and X.-J. Yang, “Local fractional variational iteration and decomposition methods for wave equation on Cantor sets within local fractional operators,” Abstract and Applied Analysis, vol. 2014, Article ID 535048, 6 pages, 2014.

J. H. He and F. J. Liu, “Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy,” Nonlinear Science Letters A, vol. 4, no. 1, pp. 15–20, 2013.