Cantor-type spherical-coordinate Method for Differential Equations within Local Fractional Derivatives

Mohamad Rafi Segi Rahmat, Dumitru Baleanu*, and Xiao-Jun Yang Cantor-type spherical-coordinate Method for Differential Equations within Local Fractional Derivatives Abstract: In this article, we utilize the Cantor-type spherical coordinate method to investigate a family of local fractional differential operators on Cantor sets. Some examples are discussed to show the capability of this method for the damped wave, Helmholtz and heat conduction equations defined on Cantor sets. We show that it is a powerful tool to convert differential equations on Cantor sets from Cantoriancoordinate systems to Cantor-type spherical-coordinate systems. Keywords: Cantor sets; Local fractional derivatives; Local fractional dynamic equation; Cantor-type Spherical-coordinate Introduction In Euclidean space we observe several interesting physical phenomena by using differential equations in the styles of planar, cylindrical, and spherical geometries. There are many anysotropic models in cylindrical and spherical coordinates Teixeira and Chew, 997; Scheuer et al., 990; Engheta, 990; Engheta, 999; Schetselaar, 998; Petropoulos, 2000. Fractional calculus has a long history and there exist many applications Kilbas et al., 2006; Podlubny, 999; Oldham and Spanier, 970; Srivastava and Owa, 989. However, fractional derivatives do not reflect the local geometric behaviors for specific functions. Therefore, attempts have been made recently to define a local version of the fractional derivative and there have been successively applied to model the nondifferentiability of phenomena in fractal physical media Adda and Cresson, 200; Kolwankar and Gangal, 996; Kolwankar and Gangal, 997; Kolwankar and Gangal, 998; Babakhani and Daftardar-Gejji, 2002; Chen et al., 200; Yang, 20a, b; Yang, 202; Yang and Baleanu, 203; Yang et al., 203; Yan et al., 204; Zhao et al., 203; Yang et al., 204; Srivastava et al., 204. *Corresponding Author: Dumitru Baleanu: Department of Mathematics, Çankaya University, Öğretmenler Cad. 4 06530, Balgat, Ankara,Turkey and Institute of Space Sciences, Magurele, Bucharest, Romania, E-mail: dumitru@cankaya.edu.tr Mohamad Rafi Segi Rahmat: School of Applied Mathematics, The University of Nottingham Malaysia Campus, Jalan Broga, 43500 Semenyih, Selangor D.E, Malaysia Xiao-Jun Yang: Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, Jiangsu, 22008, China 205 Mohamad Rafi Segi Rahmat et al. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. Download Date 9/8/8 7:32 PM

232 Mohamad Rafi Segi Rahmat et al. We recall that the Cantorian-coordinate system, firstly described in Yang, 20a, b; Yang, 202, was set up on fractals. Based on this system, 3-dimensional fractal dynamical equations such as the heat conduction equation without heat generation in fractal media Yang, 202, the damped wave equation in fractal strings Yang, 202, Maxwell s equations on Cantor sets Zhao et al., 203, Helmholtz and diffusion equations on Cantor sets Zhao et al., 203 are presented. In the Refs. Yang et al., 203; Hao et al., 203, the authors proposed to use Cantor-type cylindrical-coordinate method to investigate a family of local fractional differential equation. It is a powerful tool to convert the differential equations on Cantor sets from a Cantorian-coordinate system to Cantor-type cylindrical-coordinate system. We recall that many physical situations have spherical symmetry, e.g. the gravitational field of a single body and the electric field of a point charge exhibit spherical symmetry. The aim of this chapter is to structure and investigate the Cantor-type sphericalcoordinate method within framework of the local fractional vector operator. The layout of this chapter is given below. In Section 2, we propose and describe the Cantortype spherical-coordinate method. In Section 3, we investigate equivalent forms of certain differential equations on a Cantor set with the proposed method. 2 Mathematical Tools In this section, we briefly present some of the concepts of the local derivative of a scalar function and a vector function on Cantor set. Let f x be a local fractional continuous function on the interval a, b or f x C a, b if for x a, b, x x 0 < δ, ϵ, δ > 0, the following condition f x f x 0 < ϵ 2 is valid see Yang, 20 a,b. The local fractional derivative of f x C a, b of order 0 < in the interval [a, b] is defined as see Yang, 20a,b; Yang, 202 where f x 0 = d f x 0 dx = f x f x 0 x x 0, 3 f x f x 0 = Γ f x f x 0 with gamma function Γ, and the condition defined by Eq. holds. The local fractional partial differential operator of order 0 < is given by see Yang, 20a, b t ux 0, t = ux 0, t ux 0, t 0 t t 0 4 Download Date 9/8/8 7:32 PM

Cantor-type spherical-coordinate Method for Differential Equations 233 where ux 0, t ux 0, t 0 = Γ ux 0, t ux 0, t 0. In a similar manner, for a given vector function Ft = F te F 2 te 2 F 3 te 3, the local derivative is defined by see Yang,202 F t 0 = d Ft 0 dt = Ft Ft 0 t t 0, 5 where e, e 2, e 3 are the directions of the local fractional vector function. The following product rule is valid see Yang, 202 D x [f xgx] = D x f xgx f xd x gx. 6 3 Cantor-type Spherical-coordinate Method We recall that the Cantor-type spherical-coordinates are defined by Yang,20b; Yang,202: x = sin ϕ cos θ, y = sin ϕ sin θ, 7 z = cos ϕ, with > 0, 0 < ϕ < π, 0 θ 2π and x 2 y 2 z 2 = 2. For these Cantor type spherical coordinates, we have the local fractional vector given by r = sin ϕ cos θ e sin ϕ sin θ e 2 cos ϕ e 3, 8 so that C = r Γ = sin ϕ cos θ e sin ϕ sin θ e2 cos ϕ e3, C ϕ = r Γ ϕ = Γ cos ϕ cos θ e Γ cos ϕ sin θ e2 C θ = Γ r θ = Γ sin ϕ sin θ e Γ sin ϕ cos θ e2. Γ sin ϕ e3, Therefore, we obtain e = sin ϕ cos θ e sin ϕ sin θ e2 cos ϕ e3, e ϕ = cos ϕ cos θ e cos ϕ sin θ e2 sin ϕ e3, 0 e θ = sin θ e cos θ e2, 9 Download Date 9/8/8 7:32 PM

234 Mohamad Rafi Segi Rahmat et al. where C = e, C ϕ = Γ e ϕ, C θ = Γ sin ϕ e θ. Now, by making use of Eq. 0, we can write this result in matrix form as follows: e e ϕ e θ = sin ϕ cos θ sin ϕ sin θ cos ϕ cos ϕ cos θ cos ϕ sin θ sin ϕ sin θ cos θ 0 e e2 e3 In view of Eq. 9 and 0, upon differentiating the Cantorian position vector with respect to the Cantor-type spherical coordinates it follows that e = r Γ = sin ϕ cos θ e sin ϕ sin θ e2 cos ϕ e3, r ϕ = cos ϕ cos θ e cos ϕ sin θ e2 sin ϕ e3, e ϕ = 2 e θ = r Γ sin ϕ θ = sin θ e cos θ e2. Eq. 2 is locally orthogonal and normalized everywhere see Yang, 20a, b. Hence, it is possible to define a local fractal basis with an orientation, which is derived from one fractal space to another fractal space. Based on this, a local fractional vector field can be defined as follows: where the fractal vector coordinates are given by r, ϕ, θ = r e, e ϕ, e θ 3 r = r, ϕ, θ e, r ϕ = r, ϕ, θ e ϕ, r θ = r, ϕ, θ e θ 4 are the projections of r on the local fractal basis vectors. The variations of orthogonal vectors with respect to the Cantor-type spherical coordinates are given as follows: e = 0; e ϕ = 0 ; e θ = 0 ; e ϕ = e ϕ ; e θ = sin ϕ e θ ; e ϕ ϕ = e ; e θ ϕ = 0 ; e ϕ θ = cos ϕ e θ e θ θ = sin ϕ e cos ϕ e ϕ The local fractional derivatives with respect to the Cantor-type spherical coordinates are given by the local fractional differentiation through the Cantoriancoordinates as follows: x y z = x y z = Γ sin ϕ cos θ x sin ϕ sin θ y cos ϕ z = e =, 6 5 Download Date 9/8/8 7:32 PM

Cantor-type spherical-coordinate Method for Differential Equations 235 ϕ = x ϕ x = cos ϕ cos θ y z ϕ y ϕ z x cos ϕ sin θ y sin ϕ z = e ϕ = ϕ, 7 where θ = x θ x y θ y z θ z = Γ sin ϕ sin θ x cos θ y = sin ϕ e θ = sin ϕ θ, 8 =, ϕ = ϕ, θ = sin ϕ θ. 9 In light of Eq. 9, the local fractional gradient operator is described as follows: = e e ϕ ϕ e θ θ = e e ϕ ϕ e θ sin ϕ θ 20 A local fractional gradient operator in Cantor-type spherical coordinate system is expressed as follows: = e, ϕ, θ e ϕ ϕ, ϕ, θ e θ sin ϕ, ϕ, θ, 2 θ where = e, ϕ, θ e ϕ ϕ, ϕ, θ e θ θ, ϕ, θ. 22 The local fractional divergence operator of r = r e r ϕ e ϕ r θ e θ in Cantor-type spherical-coordinate systems is given by r = e e ϕ ϕ e θ sin ϕ θ r e r ϕ e ϕ r θ e θ = e r e ϕ r ϕ e θ sin ϕ r θ 23 Download Date 9/8/8 7:32 PM

236 Mohamad Rafi Segi Rahmat et al. With the help of the partial derivatives obtained in Eq. 5 and the product rule Eq. 6, the first term of Eq. 23 becomes e r = r e e r ϕ e ϕ r θ θ e θ 0 0 0 = r. 24 The second term of Eq. 23 is given by e ϕ We notice that r ϕ = r e ϕ ϕ e r ϕ ϕ e ϕ r θ ϕ e θ r e ϕ r ϕ e = r ϕ ϕ r. 25 r θ = r θ e r ϕ θ e ϕ r θ θ e θ r θ sin ϕ e r θ cos ϕ e ϕ r sin ϕ r ϕ cos ϕ e θ. Hence, the third term of Eq. 23 has the following form: e θ r sin ϕ θ = r θ sin ϕ θ r sin ϕ r ϕ cos ϕ = r θ sin ϕ θ r r ϕ cos ϕ sin ϕ 26 By combining Eqs. 24 to 26, Eq. 23 can be written as follows: It follows that r = r = r r = r sin ϕ r ϕ ϕ r r θ sin ϕ θ r ϕ ϕ r ϕ cos ϕ sin ϕ ϕ r ϕ sin ϕ r r ϕ cos ϕ sin ϕ r θ sin ϕ θ 2 r. 27 r θ sin ϕ θ 2 r. 28 Now, we derive the local fractional curl operator of r = r e r ϕ e ϕ r θ e θ in Cantor-type spherical-coordinate systems. Download Date 9/8/8 7:32 PM

Cantor-type spherical-coordinate Method for Differential Equations 237 r = e e ϕ ϕ e θ sin ϕ θ r e r ϕ e ϕ r θ e θ = e r e ϕ r ϕ e θ sin ϕ r θ 29 With the help of the partial derivatives obtained in Eq. 5 and the product rule Eq. 6, the first term of Eq. 29 becomes e r = r e e r ϕ e ϕ r θ θ e θ 0 0 0 = r ϕ e θ r θ θ e ϕ. 30 The second term of Eq. 29 is given by e ϕ r ϕ = e ϕ We notice that r ϕ e r ϕ ϕ e ϕ r θ ϕ e θ r e ϕ r ϕ e = r ϕ e θ r θ ϕ e r ϕ e θ. 3 r θ = r θ e r ϕ θ e ϕ r θ θ e θ r θ sin ϕ e r θ cos ϕ e ϕ r sin ϕ r ϕ cos ϕ e θ. Hence, the third term of Eq. 29 has the following form: e θ sin ϕ r θ = sin ϕ = r θ e ϕ r ϕ θ e r θ sin ϕ e ϕ r θ cos ϕ e r r ϕ sin ϕ θ e ϕ sin ϕ θ e r θ e ϕ r θ cos ϕ sin ϕ e. 32 Substituting Eqs. 30 to 32 into Eq. 29, we obtain r ϕ r = e θ r θ θ e ϕ r sin ϕ θ e ϕ r = θ ϕ r ϕ sin ϕ θ = r sin ϕ θ sin ϕ r ϕ e θ r θ ϕ e r ϕ e θ r θ θ sin ϕ r ϕ θ e r θ r θ ϕ r θ sin ϕ r θ ϕ e ϕ r θ cos ϕ sin ϕ e e r θ cos ϕ sin ϕ e r ϕ ϕ e r ϕ r ϕ e θ Download Date 9/8/8 7:32 PM

238 Mohamad Rafi Segi Rahmat et al. r sin ϕ θ r θ θ r θ e r ϕ ϕ r ϕ r ϕ e θ. 33 Consequently, the local fractional Laplace operator in Cantor-type sphericalcoordinates is given by 2 = e e ϕ ϕ e θ sin ϕ θ e e ϕ ϕ e θ sin ϕ θ = e e e ϕ e θ ϕ sin ϕ e ϕ ϕ e θ sin ϕ θ e e ϕ ϕ e θ sin ϕ θ e e ϕ ϕ e θ sin ϕ θ θ The first term of Eq. 34 is written as follows:. 34 e = e = 2 2 e e ϕ ϕ e θ sin ϕ θ 2 e ϕ ϕ e θ e 2 sin ϕ θ 35 The second term of Eq. 34 becomes e ϕ ϕ e e ϕ ϕ e θ sin ϕ θ = e ϕ e 2 ϕ e ϕ e ϕ ϕ ϕ e ϕ e θ ϕ sin ϕ θ = ϕ ϕ = 2 2 ϕ 2. 36 The third term of Eq. 34 is represented by e θ sin ϕ θ e e ϕ ϕ e θ sin ϕ θ Download Date 9/8/8 7:32 PM

Cantor-type spherical-coordinate Method for Differential Equations 239 = e θ = sin ϕ e 2 θ e θ e ϕ θ ϕ e ϕ θ e θ sin ϕ θ θ e θ θ sin ϕ θ sin ϕ cos ϕ ϕ θ sin ϕ θ cos ϕ 2 sin ϕ ϕ sin ϕ θ sin ϕ θ sin ϕ = θ = cos ϕ 2 sin ϕ ϕ 2 2 sin 2. 37 ϕ θ2 As a result, the expansion for the local fractional Laplace operator assumes the following form: 2 = 2 2 2 2 ϕ 2 2 cos ϕ 2 sin ϕ ϕ 2 2 sin 2 ϕ θ 2 = 2 2 2 sin ϕ ϕ sin ϕ ϕ 2 2 sin 2 ϕ 2 θ 2. 38 4 Examples Below we consider several differential equations on Cantor sets. Example 30. Let us consider the heat-conduction equation on Cantor sets without heat generation in fractal media see Yang et al., 203, namely where K 2 2 Tx, y, z, t c Tx, y, z, t t = 0, 39 2 = 2 x 2 2 2 y2 z 2 is the local Laplace operator Yang,20b; Yang,202. By using Eq. 38, Eq. 39 is transformed into [ K 2 2 T, ϕ, θ, t 2 2 sin ϕ ϕ sin ϕ T, ϕ, θ, t ϕ 2 T, ϕ, θ, t 2 T, ϕ, θ, t 2 sin 2 ϕ θ 2 ] c T t = 0 40 which is the form of the heat-conduction equation on Cantor sets in Cantor-type spherical coordinate system. Download Date 9/8/8 7:32 PM

240 Mohamad Rafi Segi Rahmat et al. Example 3. Consider the damped wave equation in fractal strings as given below: 2 ux, y, z, t t 2 ux, y, z, t t 2 ux, y, z, t = 0. 4 Applying Eq. 38 into Eq. 39, we get 2 u, ϕ, θ, t t 2 u, ϕ, θ, t t [ 2 u, ϕ, θ, t 2 2 sin ϕ ϕ sin ϕ u, ϕ, θ, t ϕ 2 u, ϕ, θ, t 2 ] u, ϕ, θ, t 2 sin 2 ϕ θ 2 = 0 42 which is the form of the damped wave equation in fractal strings in cantor-type sphericalcoordinate system. Example 32. Let us consider the inhomogeneous Helmholtz equation on Cantor sets Hao et al., 203, namely 2 Mx, y, z x 2 2 Mx, y, z y 2 2 Mx, y, z z 2 ω 2 Mx, y, z = f x, y, z, 43 where f x, y, z is a local fractional continuous function. By using Eq. 34 into Eq. 35, we find that 2 M, ϕ, θ 2 2 sin ϕ ϕ sin ϕ M, ϕ, θ ϕ 2 M, ϕ, θ 2 sin 2 ϕ 2 M, ϕ, θ θ 2 ω 2 M, ϕ, θ = f, ϕ, θ, 44 which is the form of the Helmholtz equation on Cantor sets in Cantor-type spherical coordinates system. 5 Conclusions In this chapter, we discussed a new Cantor-type spherical coordinate method and the equivalent forms of differential equations on Cantor sets were investigated. Several examples of differential equations on Cantor sets, e.g. heat-conduction equation without heat generation on fractal media, damped wave equation in fractal strings, Helmholtz equation on Cantor sets were tested by applying the Cantor-type spherical coordinate method. Download Date 9/8/8 7:32 PM

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