Local Fractional Poisson and Laplace Equations with Applications to Electrostatics in Fractal Domain

Abstract

From the local fractional calculus viewpoint, Poisson and Laplace equations were presented in this paper. Their applications to the electrostatics in fractal media are discussed and their local forms in the Cantor-type cylindrical coordinates are also obtained.

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Yang-Yang Li 1,2 and Yang Zhao 3 and Gong-Nan Xie 4 and Dumitru Baleanu 5,6,7 and Xiao-Jun Yang 8 and Kai Zhao 1

Academic Editor:Carlo Cattani

1, Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences, Changchun 130102, China
2, University of Chinese Academy of Sciences, Beijing 100049, China
3, Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China
4, School of Mechanical Engineering, Northwestern Polytechnical University, Xi'an, Shaanxi 710048, China
5, Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
6, Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
7, Institute of Space Sciences, Magurele, 077125 Bucharest, Romania
8, Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu 221008, China

Received 19 March 2014; Accepted 1 April 2014; 16 April 2014

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.

1. Introduction

Poisson and Laplace equations had successfully played an important role in electrodynamics [1-3]. Mathematically, they are two-order partial differential equations and exist in the spaces of different kinds [4, 5]. Their solutions were studied by different. There are approximate and numerical methods for them, such as the finite difference method [6], the finite element method [7], the random walk method [8], the quadrilateral quadrature element [9], and the complex polynomial method [10].

Since Mandelbrot [10] described the fractals, the fractional calculus [11-13] and local fractional calculus [14-16] were applied to the real world problem based on them. For example, Engheta discussed the fractional-order electromagnetic theory [17]. Tarasov studied the fractal distribution of charges [18]. Calcagni et al. suggested the electric charge in multiscale space and times [19]. In [20], the local fractional approach for Maxwell's equations was considered. Local fractional calculus [20-25] has been successfully applied to describe dynamical systems with the nondifferentiable functions. For example, the Maxwell theory on Cantor sets was studied in [20]. The Heisenberg uncertainty relation was discussed by using the local fractional Fourier analysis [21]. The system of Navier-Stokes equations arising in fractal flows was reported in [22]. The local fractional nonhomogeneous heat equations arising in fractal heat flow were presented in [23]. The fractal forest gap within the local fractional derivative was investigated in [24].

In the present paper, the local fractional Poisson and Laplace equations within the nondifferentiable functions arising in electrostatics in fractal domain and in the Cantor-type cylindrical coordinates [25] based upon the local fractional Maxwell equations [20] will be derived from the fractional vector calculus.

The outline of the paper is depicted below. Section 2 introduces the local fractional Maxwell equations. Section 3 discusses the local fractional Poisson and Laplace equations arising in electrostatics in fractal media. In Section 4, the local fractional Poisson and Laplace equations in the Cantor-type cylindrical coordinates are presented. Finally, Section 5 is devoted to the conclusions.

2. Local Fractional Maxwell's Equations

In this section, the local fractional Maxwell's equations are introduced and the concepts of the local fractional vector calculus are reviewed. We first introduce the local fractional vector calculus and its theorems [14, 20-23].

The local fractional line integral of the function u along a fractal line ^lα was defined as [14, 20] [figure omitted; refer to PDF] where the quantity Δ^lP(α) is elements of line, |Δ^lP(α) |[arrow right]0 as N[arrow right]∞ , and α∈(0,1] .

The local fractional surface integral was defined as [14, 20-23] [figure omitted; refer to PDF] where the quantity Δ^SP(β) is elements of surface, the quantity _nP is N elements of area with a unit normal local fractional vector, and Δ^SP(β) [arrow right]0 as N[arrow right]∞ for β=2α .

The local fractional volume integral of the function u was defined as [14, 20-23] [figure omitted; refer to PDF] where the quantity Δ^VP(γ) is the elements of volume, Δ^VP(γ) [arrow right]0 as N[arrow right]∞ , and γ=3α .

The local fractional Stokes' theorem of the fractal field states that [13, 20] [figure omitted; refer to PDF] The electric Gauss law for the fractal electric field was suggested as [20] [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where the quantity ρ denotes the free charges density and the quantity D is the fractal electric displacement.

The Ampere law in the fractal magnetic field was presented as [20] [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where the quantity H is the fractal magnetic field strength and the quantity _Ja denotes the fractal conductive current.

The Faraday law in the fractal electric field reads as [20] [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where the constitutive relationships in fractal electric field are [figure omitted; refer to PDF] with the fractal dielectric permittivity _{[straight epsilon]f} and the fractal dielectric field E .

The magnetic Gauss law for the fractal magnetic field was written as [20] [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where the constitutive relationships in fractal magnetic field are [figure omitted; refer to PDF] with the fractal magnetic permeability _μf and the fractal magnetic field B .

3. Local Fractional Poisson and Laplace Equations in Fractal Media

In this section, we derive the local fractional Poisson and Laplace equations arising in electrostatics in fractal media.

In view of (11), from (6) we have [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] where ρ denotes the free charges density in fractal homogeneous medium, _{[straight epsilon]f} denotes the fractal dielectric permittivity, and E denotes the fractal dielectric field.

Hence, the local fractional differential form of Gauss's law in local fractional divergence operator reads as [figure omitted; refer to PDF] If the electrostatics in fractal domain is described by the expression [figure omitted; refer to PDF] then the fractal electric field within the local fractional gradient is [figure omitted; refer to PDF] where the quantity ψ is a nondifferentiable term and E is the fractal dielectric field.

In view of (16) and (19), we obtain [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] where the quantity ψ is a nondifferentiable term and _{[straight epsilon]f} denotes the fractal dielectric permittivity.

From (21) we arrive at [figure omitted; refer to PDF] Let us define the local fractional operator [figure omitted; refer to PDF] Making use of (22) and (23), we have [figure omitted; refer to PDF] In the Cantorian coordinates, from (24), the local fractional Poisson equation arising in electrostatics in fractal domain can be written as [figure omitted; refer to PDF] where both ψ(x,y,z) and ρ(x,y,z) are nondifferentiable functions; the local fractional operator ^∇2α in the Cantorian coordinates was written as [14] [figure omitted; refer to PDF] In the Cantorian coordinates, from (25), the local fractional Laplace equation arising in electrostatics in fractal domain is [figure omitted; refer to PDF] where the quantity ψ(x,y,z) is a nondifferentiable function.

From (25) the local fractional Laplace equation arising in electrostatics in fractal domain with two variables can be written as [figure omitted; refer to PDF] where both ψ(x,y) and ρ(x,y) are nondifferentiable functions.

From (27) the local fractional Laplace equation arising in electrostatics in fractal domain with two variables can be written as [figure omitted; refer to PDF] where the quantity ψ(x,y) is a nondifferentiable function.

For the boundary conditions on the fractal potential [figure omitted; refer to PDF] we have the local fractional Laplace's equation [figure omitted; refer to PDF] which leads to the nondifferentiable solution given y [figure omitted; refer to PDF] and its graph is shown in Figure 1.

Figure 1: Plot of (31) with parameters α=ln...2/ln...3 and b=1 .

[figure omitted; refer to PDF]

We notice that the local fractional Poisson's equation shows the potential behavior in the fractal regions with nondifferentiable functions where there is the free charge, while local fractional Laplace's equation governs the nondifferentiable potential behavior in fractal regions where there is no free charge.

4. Local Fractional Poisson and Laplace Equations in the Cantor-Type Cylindrical Coordinates

In this section, the local fractional Poisson and Laplace equations in the Cantor-type cylindrical coordinates are considered. We first start with the Cantor-type cylindrical coordinate method.

We now consider the Cantor-type cylindrical coordinates given by [14, 25] [figure omitted; refer to PDF] with R∈(0,+∞) , z∈(-∞,+∞) , θ∈(0,π] , and ^x2α +^y2α =^R2α .

From (33) we have [25] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and the local fractional vector suggested by [figure omitted; refer to PDF] In view of (35), the local fractional Poisson equation in the Cantor-type cylindrical coordinates is written as [figure omitted; refer to PDF] where both ψ(R,θ,z) and ρ(R,θ,z) are nondifferentiable functions.

From (35), the local fractional Laplace equation in the Cantor-type cylindrical coordinates is [figure omitted; refer to PDF] where the quantity ψ(R,θ,z) is a nondifferentiable function.

We now consider the Cantor-type circle coordinates given by [14] [figure omitted; refer to PDF] with R∈(0,+∞) , θ∈(0,2π] , and ^x2α +^y2α =^R2α .

Making use of (37), we obtain [figure omitted; refer to PDF] where [14] [figure omitted; refer to PDF] and the local fractional vector is suggested by [14] [figure omitted; refer to PDF] From (28) and (42) the local fractional Poisson equation in fractal domain with two variables can be written as [figure omitted; refer to PDF] where both ψ(R,θ) and ρ(R,θ) are nondifferentiable functions.

From (29) and (42) the local fractional Laplace equation in fractal domain with two variables can be written as [figure omitted; refer to PDF] where the quantity ψ(R,θ) is a nondifferentiable function.

5. Conclusions

In this work we derived the local fractional Poisson and Laplace equations arising in electrostatics in fractal domain from local fractional vector calculus. The local fractional Poisson and Laplace equations in the Cantor-type cylindrical coordinates were also discussed. The nondifferentiable solution for local fractional Laplace equation was also given.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 41371345 and no. 41201335).

Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

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