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

Authors: Yang-Yang Li; Yang Zhao; Gong-Nan Xie; Dumitru Baleanu; Xiao-Jun Yang; Kai Zhao
Journal: Advances in Mathematical Physics
 (2014)

---

## 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.


---

## Body

## 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. Section2 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 functionu along a fractal line lα was defined as [14, 20]
(1)∫l(α)u(xP,yP,zP)·dl(α)=lim⁡N→∞∑P=1Nu(xP,yP,zP)·ΔlP(α),
where the quantity ΔlP(α) is elements of line, |ΔlP(α)|→0 as N→∞, and α∈(0,1].The local fractional surface integral was defined as [14, 20–23]
(2)∬u(rP)·dS(β)=lim⁡N→∞∑P=1Nu(rP)·nPΔSP(β),
where the quantity ΔSP(β) is elements of surface, the quantity nP is N elements of area with a unit normal local fractional vector, and ΔSP(β)→0 as N→∞ for β=2α.The local fractional volume integral of the functionu was defined as [14, 20–23]
(3)∭u(rP)dV(γ)=lim⁡N→∞∑P=1Nu(rP)ΔVP(γ),
where the quantity ΔVP(γ) is the elements of volume, ΔVP(γ)→0 as N→∞, and γ=3α.The local fractional Stokes’ theorem of the fractal field states that [13, 20]
(4)∮l(α)u·dl(α)=∬S(β)(∇α×u)·dS(β).
The electric Gauss law for the fractal electric field was suggested as [20]
(5)∯S(β)D·dS(β)=∭V(γ)ρdV(γ),
which leads to
(6)∇α·D=ρ,
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]
(7)∮l(α)H·dl(α)=∬S(β)(Ja+∂αD∂tα)·dS(β),
which leads to
(8)∇α×H=Ja+∂αD∂tα,
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]
(9)∮l(α)E·dl(α)+∂α∂tα∬S(β)B·dS(β)=0,
which leads to
(10)∇α×E=-∂αB∂tα,
where the constitutive relationships in fractal electric field are
(11)D=εfE
with the fractal dielectric permittivity εf and the fractal dielectric field E.The magnetic Gauss law for the fractal magnetic field was written as [20]
(12)∯S(β)B·dS(β)=0,
which leads to
(13)∇α·B=0,
where the constitutive relationships in fractal magnetic field are
(14)H=μfB
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
(15)∇α·(εfE)=ρ,
so that
(16)εf(∇α·E)=ρ,
where ρ denotes the free charges density in fractal homogeneous medium, εf 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(17)∇α·E=ρεf.
If the electrostatics in fractal domain is described by the expression
(18)∇α×E=-∂αB∂tα=0,
then the fractal electric field within the local fractional gradient is
(19)E=-∇αψ,
where the quantity ψ is a nondifferentiable term and E is the fractal dielectric field.In view of (16) and (19), we obtain
(20)εf[∇α·(-∇αψ)]=ρ,
which leads to
(21)-εf[∇α·∇αψ]=ρ,
where the quantity ψ is a nondifferentiable term and εf denotes the fractal dielectric permittivity.From (21) we arrive at
(22)∇α·∇αψ=-ρεf.
Let us define the local fractional operator
(23)∇α·∇α=∇2α.
Making use of (22) and (23), we have
(24)∇2αψ=-ρεf.
In the Cantorian coordinates, from (24), the local fractional Poisson equation arising in electrostatics in fractal domain can be written as
(25)∂2α∂x2αψ(x,y,z)+∂2α∂y2αψ(x,y,z)+∂2α∂z2αψ(x,y,z)=-ρ(x,y,z)εf,
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]
(26)∇2α=∂2α∂x2α+∂2α∂y2α+∂2α∂z2α.
In the Cantorian coordinates, from (25), the local fractional Laplace equation arising in electrostatics in fractal domain is
(27)∂2α∂x2αψ(x,y,z)+∂2α∂y2αψ(x,y,z)+∂2α∂z2αψ(x,y,z)=0,
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
(28)∂2α∂x2αψ(x,y)+∂2α∂y2αψ(x,y)=-ρ(x,y)εf,
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
(29)∂2α∂x2αψ(x,y)+∂2α∂y2αψ(x,y)=0,
where the quantity ψ(x,y) is a nondifferentiable function.For the boundary conditions on the fractal potential(30)ψ(0)=0,ψ(b)=8,
we have the local fractional Laplace’s equation
(31)∂2α∂z2αψ(z)=0,
which leads to the nondifferentiable solution given y(32)ψ(z)=8(zb)α
and its graph is shown in Figure 1.Figure 1
Plot of (31) with parameters α=ln⁡2/ln⁡3 and b=1.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]
(33)xα=Rαcos⁡αθα,yα=Rαsinαθα,zα=zα
with R∈(0,+∞), z∈(-∞,+∞), θ∈(0,π], and x2α+y2α=R2α.From (33) we have [25]
(34)∇αϕ(R,θ,z)=eRα∂α∂Rαϕ+eθα1Rα∂α∂θαϕ+ezα∂α∂zαϕ,(35)∇2αϕ(R,θ,z)=∂2α∂R2αϕ+1R2α∂2α∂θ2αϕ+1Rα∂α∂Rαϕ+∂2α∂z2αϕ,
where
(36)eRα=cos⁡αθαe1α+sinαθαe2α,eθα=-sinαθαe1α+cos⁡αθαe2α,ezα=e3α,
and the local fractional vector suggested by
(37)r=Rαcos⁡αθαe1α+Rαsinαθαe2α+zαe3α=rReRα+rθeθα+rzezα.
In view of (35), the local fractional Poisson equation in the Cantor-type cylindrical coordinates is written as
(38)∂2α∂R2αψ(R,θ,z)+1R2α∂2α∂θ2αψ(R,θ,z)+1Rα∂α∂Rαψ(R,θ,z)+∂2α∂z2αψ(R,θ,z)=-1εfρ(R,θ,z),
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
(39)∂2α∂R2αψ(R,θ,z)+1R2α∂2α∂θ2αψ(R,θ,z)+1Rα∂α∂Rαψ(R,θ,z)+∂2α∂z2αψ(R,θ,z)=0,
where the quantity ψ(R,θ,z) is a nondifferentiable function.We now consider the Cantor-type circle coordinates given by [14]
(40)xα=Rαcos⁡αθα,yα=Rαsinαθα
with R∈(0,+∞), θ∈(0,2π], and x2α+y2α=R2α.Making use of (37), we obtain
(41)∇αϕ(R,θ)=eRα∂α∂Rαϕ+eθα1Rα∂α∂θαϕ,(42)∇2αϕ(R,θ)=∂2α∂R2αϕ+1R2α∂2α∂θ2αϕ+1Rα∂α∂Rαϕ,
where [14]
(43)eRα=cos⁡αθαe1α+sinαθαe2α,eθα=-sinαθαe1α+cos⁡αθαe2α,
and the local fractional vector is suggested by [14]
(44)r=Rαcos⁡αθαe1α+Rαsinαθαe2α=rReRα+rθeθα.
From (28) and (42) the local fractional Poisson equation in fractal domain with two variables can be written as
(45)∂2α∂R2αψ(R,θ)+1R2α∂2α∂θ2αψ(R,θ)+1Rα∂α∂Rαψ(R,θ)=-1εfρ(R,θ),
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
(46)∂2α∂R2αψ(R,θ)+1R2α∂2α∂θ2αψ(R,θ)+1Rα∂α∂Rαψ(R,θ)=0,
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.



---

*Source: 590574-2014-04-16.xml*