Cantortype cylindricalcoordinate method for differential equations with local fractional derivatives


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1 From the SelectedWorks of XiaoJun Yang April 8, 2013 Cantortype cylindricalcoordinate method for differential equations with local fractional derivatives XiaoJun Yang H. M. Srivastava, H. M. Srivastava J. H. He, J. H. He D. Baleanu, D. Baleanu Available at:
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3 Physics Letters A Contents lists available at SciVerse ScienceDirect Physics Letters A Cantortype cylindricalcoordinate method for differential equations with local fractional derivatives XiaoJun Yang a,b,,h.m.srivastava c, JiHuan He d, Dumitru Baleanu e,f,g a Department of Mathematics and Mechanics, China University of Mining and Technology, Xuhou, Jiangsu, , People s epublic of China b Institute of Applied Mathematics, Qujing Normal University, Qujing , People s epublic of China c Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 34, Canada d National Engineering Laboratory for Modern Silk, College of Textile and Clothing Engineering, Soochow University, 199 enai oad, Suhou , People s epublic of China e Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, Ankara, 06530, Turkey f Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulai University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia g Institute of Space Sciences, MagureleBucharest, omania article info abstract Article history: eceived 1 February 2013 eceived in revised form 5 April 2013 Accepted 8 April 2013 Availableonline9April2013 Communicated by. Wu Keywords: Heatconduction equation Damped wave equation Local fractional derivatives Cantor set In this Letter, we propose to use the Cantortype cylindricalcoordinate method in order to investigate a family of local fractional differential operators on Cantor sets. Some testing examples are given to illustrate the capability of the proposed method for the heatconduction equation on a Cantor set and the damped wave equation in fractal strings. It is seen to be a powerful tool to convert differential equations on Cantor sets from Cantoriancoordinate systems to Cantortype cylindricalcoordinate systems Published by Elsevier B.V. 1. Introduction Everything in the Cartesiancoordinate system is used to be measured with respect to the coordinate axes. However, the Cartesian coordinates are not best suited for every shape. Certain shapes, like circular or spherical ones, cannot even be demonstrated through a function in Cartesiancoordinate system. These shapes are more easily determined in cylindrical or spherical coordinates [1]. They are the equivalent of the origin in the Cartesiancoordinate system. Both classical and fractional differential equations in the coordinate system are switched between Cartesian, cylindrical and spherical coordinates [2,3]. ecently, the Cantoriancoordinate system, which was first construed in [4 6], was set up on fractals. Based on it, the heatconduction equation on Cantor sets without heat generation in fractal media was presented in [4] as follows: * Corresponding author at: Department of Mathematics and Mechanics, China University of Mining and Technology, Xuhou, Jiangsu, , People s epublic of China. Tel.: addresses: X.J. Yang, H.M. Srivastava, J.H. He, D. Baleanu. K 2α 2α T ρ α T α c α t α = 0 1 or K 2α 2α T x 2α + 2α T y 2α + 2α T 2α where 2α = 2α 2α 2α + + x2α y2α 2α ρ α T α c α = 0, 2 tα is the local fractional Laplace operator [4 6], whose local fractional differential operator is denoted as follows [4 11] for other definitions, see also [12 25]: f α x 0 = dα f x dx α α f x f x 0 = lim x x0 x=x0 x x 0 α, 3 where α f x f x 0 = Γ1 + α f x f x0 and f x is satisfied with the following condition [4,15]: f x f x0 τ α x x 0 α, so that see [4 18] /$ see front matter 2013 Published by Elsevier B.V.
4 X.J. Yang et al. / Physics Letters A f x f x0 < ε α 4 with x x 0 <δ,forε,δ>0 and ε,δ. In a similar manner, for a given vector function Ft = F 1 te α 1 + F 2 te α 2 + F 3te α 3, the local fractional vector derivative is defined by see [4] F α t 0 = dα Ft dt α α Ft Ft 0 = lim t t0 t=t0 t t 0 α 5 where e α 1, eα 2 and eα 2 are the directions of the local fractional vector function. The aim of this Letter is to investigate the Cantortype cylindricalcoordinate method within the local fractional vector operator. The layout of the Letter is as follows. In Section 2, weproposeand describe the Cantortype cylindricalcoordinate method. In Section 3, we consider the testing examples. Finally, in Section 4, we present our concluding remarks and observations. 2. Cantortype cylindricalcoordinate method For the following Cantortype cylindrical coordinates [4]: x α = α cos α α, y α = α sin α α, α = α, with > 0,, +, 0<<2π and x 2α + y 2α = 2α,we have the local fractional vector given by r = α cos α α e α 1 + α sin α α e α 2 + α e α 3, 7 so that C α = 1 α r Γ1 + α α = cos α α e α 1 + sin α α e α 2, C α = 1 α r Γ1 + α α = α Γ1 + α sin α α e α 1 + α Γ1 + α cos α α e α 2, C α 3 = 1 α r Γ1 + α α = eα 3. Therefore, we obtain e α = cos α α e α 1 + sin α α e α 2, e α = sin α α e α 1 + cos α α e α 2, 9 e α = eα 3, where C α = eα, Cα = Γ1+α α eα, Cα 3 = eα. Now, by making use of Eq. 9, we can write this last result in matrix form as follows: e α cosα α sin α α 0 e α e α = sin α α cos α α 1 0 e α e α 2, e α 3 which leads to E α i = T α ij Eα j, 11 where e α E α cosα α i, T α sin α α 0 e α ij = sin α α cos α α 0, e α E α 1 j e α Here T α is fractal matrix, which is defined on the generalied Banach space [5,6]. The general basis vectors of two fractal spaces are ij defined, respectively, from the fractal tangent vectors [4], namely, e α E α e α i, E α 1 e α j e α 3 In view of Eqs. 8 and 9, upon differentiating the Cantorian position with respect to the Cantortype cylindrical coordinates implies that e α = 1 α r Γ1 + α α = cos α α e α 1 + sin α α e α 2, e α = 1 α r α α = sin α α e α 1 + cos α α e α 2, 14 e α = 1 α r Γ1 + α α = eα 3. Eq. 14 is orthogonal and normalied everywhere see [5,6]. Hence, we can 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: r,, = r e α + eα + eα 15 where the fractal vector coordinates given by r = r,, e α, r = r,, e α, r = r,, e α 16 are the projections of r on the local fractal basis vectors. The local fractional derivatives with respect to the Cantortype cylindrical coordinates are given by the local fractional differentiation through the Cantorian coordinates as follows: α α x α α y α α α α = x α + y α + α = Γ1 + α cos α α α x α + sin α α α y α = e α α = α, 17 α α x α α y α α α α = x α + y α + α = α sin α α α x α + cos α α α y α = α e α α = α α 18 and α α x α α y α α α α = x α + y α + α = Γ1 + α α α = eα α = α, 19 where α = eα α = α α, α = α e α α = 1 α α α, α α = eα α = α. 20 In light of Eq. 20, the local fractional gradient operator is described as follows: α = e α α + eα α + eα α = e α α α + eα 1 α α α + α eα α. 21
5 1698 X.J. Yang et al. / Physics Letters A A local fractional gradient operator in Cantortype cylindricalcoordinate systems is expressed as follows: α u = e α α α u,, + eα 1 α α α u,, + e α α u,,, 22 α where α u = e α α u,, + eα α u,, + eα α u,,. 23 = e α α r α eα + α r α eα + α r α eα = α r α. 27 By combining Eqs. 25 to 27, Eq. 24 can be reformulated as follows: α r = α r α + 1 α r α α + r α + α r α. 28 We notice that the local fractional curl operator of The local fractional divergence operator of r = e α r + e α r + e α r in Cantortype cylindricalcoordinate systems has the form given by α r = e α α α + eα 1 α α α + α eα α e α r + e α r + e α r = e α α r α + eα 1 α r α α + eα α r α. 24 The first term of Eq. 24 becomes e α α r α α = eα r α eα + α r α eα + α r α eα + r α e α α α e α + r α + r α e α α = e α α r α eα + α r α eα + α r α eα = e α α r α eα + α r α eα + α r α eα = α r α. 25 The second term of Eq. 24 is given by e α 1 α r α α = e α 1 α α r α eα + α r α eα + α r α eα + r α e α α α e α + r α + r α e α α = e α 1 α r α α eα + α r α eα + α r α eα + r e α r e α + 0 = e α 1 α α r α eα + α r α eα + α r α eα + r e α r e α = 1 α r α α + r α. 26 Finally, the third term of Eq. 24 has the following form: e α α r α α = eα r α eα + α r α eα + α r α eα + r α e α α + α e α α r + α e α α r = e α α r α eα + α r α eα + α r α eα r = e α r + e α r + e α r in Cantortype cylindricalcoordinate systems can be computed by α r = e α α α + eα 1 α α α + α eα α e α r + e α r + e α r = e α α r α + eα 1 α r α α + eα α r α. 29 The first term of Eq. 29 is determined by e α α r α α = eα r α eα + α r α eα + α r α eα + r α e α α α e α + r α + r α e α α = e α α r α eα + α r α eα + α r α eα = e α α r α eα + α r α eα + α r α eα = α r α eα α r α eα. 30 The second term of Eq. 29 is represented as follows: e α 1 α α r α = eα α r α eα + α r α eα + α r α e α + r α + r = e α 1 α α e α α α eα + r α e α α α r α eα + α r α eα + α r α eα + r e α r e α + 0 = e α 1 α r α α + eα 1 α r α α + eα r α. 31 The third term of Eq. 29 is expressed in the form: e α α r α α = eα r α eα + α r α eα + α r α eα + r α e α α α e α + r α + r α e α α = e α α r α eα + α r α eα + α r α eα = e α α r α α eα r α. 32 Substituting Eqs. 30 to 32 into Eq. 29, it follows that
6 α α r r = α eα α r α eα + e α 1 α r α α + eα 1 α r α α + e α r α + e α α r α α eα r α 1 α r = α α α r α e α α + r α α r α e α α r + α + r α 1 α r α α e α. 33 Consequently, the form of the local fractional Laplace operator is given by 2α = e α α α + eα 1 α α α + α eα α e α α α + eα 1 α α α + eα = e α α α e α α α + eα + e α 1 α α α e α α α + eα + e α α α e α α α + eα α α 1 α α α + eα X.J. Yang et al. / Physics Letters A α α 1 α α α + eα 1 α α α + eα α α The first term of Eq. 34 is rewritten as follows: e α α α e α α α + eα 1 α α α + α eα α = e α e α 2α 2α + α 1 α eα α α α + e α 2α α α α α. 34 = 2α. 35 2α The second term of Eq. 34 is represented by e α 1 α α α e α α α + eα 1 α α α + α eα α = e α 1 α e α 2α α α + α α e α α α + 1 α α e α α α α + e α α 1 α α α α + e α 2α α α = e α 1 α e α 2α α α + α eα α e α α α α + e α α 1 α α α α + e α 2α α α = 1 α 1 α α α α α + 1 α α α = 1 2α 2α 2α + 1 α α α. 36 The third term of Eq. 34 is computed by means of the following formula: e α α α e α α α + eα = e α e α 2α α α + eα α + α eα α α 1 α α α α 1 α α + e α 2α 2α = 2α. 37 2α As a result of our computations, the expression for the local fractional Laplace operator assumes the following form: 2α = 2α 2α + 1 2α 2α 2α + 1 α α α + 2α. 38 2α 3. Testing examples In order to demonstrate the effectiveness of the proposed method, we have chosen several differential equations on Cantor sets. Example 1. Let us consider the heatconduction equation on Cantor sets without heat generation in fractal media, namely, K 2α 2α T x, y,, t x 2α + 2α T x, y,, t y 2α + 2α T x, y,, t 2α ρ α T x, y,, t α c α t α = By using Eq. 38, wefindfromeq.39 that K 2α 2α T,,Z, t 2α + 1 2α T,,Z, t 2α 2α + 1 α T,,Z, t α α + 2α T,,Z, t 2α ρ α T,,Z, t α c α t α = 0, 40 which is the form of the heatconduction equation on Cantor sets in Cantortype cylindricalcoordinate systems. Example 2. Consider the damped wave equation in fractal strings as given below: 2α ux, y,, t t 2α α ux, y,, t t α 2α ux, y,, t x 2α + 2α ux, y,, t y 2α + 2α ux, y,, t 2α Applying Eq. 38 to Eq. 41, weget 2α u,,z, t t 2α α u,,z, t t α 2α u,,z, t 2α + 1 2α u,,z, t 2α 2α + 1 α u,,z, t α α + 2α u,,z, t 2α = = 0, 42 which is the form of the damped wave equation in fractal strings in Cantortype cylindricalcoordinate systems. 4. Concluding remarks and observations In our present investigation, we have proposed and developed a new Cantortype cylindricalcoordinate method. The equivalent forms of differential equations on Cantor sets are then investigated within the proposed method. We notice that this method is different from the fractional complextransform method on the fractional differential operator see, for details, [8,26 28]. The former is an equivalent form of differential equations on Cantor sets converting from Cantoriancoordinate systems to Cantortype cylindricalcoordinate systems, while the latter is one form
7 1700 X.J. Yang et al. / Physics Letters A from Cantoriancoordinate systems to Cartesiancoordinate systems, because it is always most convenient to view curvilinear coordinate systems through the eyes of particular global Cartesiancoordinate systems in a flat Euclidean space. However, the Cantortype cylindricalcoordinate method is used to view fractal curvilinear coordinate systems through the eyes of particular global Cantoriancoordinate systems in a flat fractal space. Several examples for the heatconduction equation on a Cantor set and the damped wave equation in fractal strings were tested by applying the Cantortype cylindricalcoordinate method. It is similar to the method based upon conversion from the Cartesiancoordinate systems to the cylindricalcoordinate systems. Acknowledgements The work is supported by PAPD A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and Project for Six Kinds of Top Talents in Jiangsu Province, China Grant No. ZBZZ035. eferences [1] W.M. Boothby, An Introduction to Differential Manifolds and iemannian Geometry, Academic Press, New York, [2] Z.Z. Ganji, D.D. Ganji, A.D. Ganji, M. ostamian, Numer. Methods Partial Differential Equations [3] S. Momani, Z. Odibat, Appl. Math. Comput [4] X.J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, [5] X.J. Yang, Progr. Nonlinear Sci [6] X.J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, [7] X.J. Yang, D. Baleanu, Therm. Sci. 2012, TSCI Y, in press. [8] M.S. Hu, D. Baleanu, X.J. Yang, Math. Probl. Engrg [9] M.S. Hu,.P. Agarwal, X.J. Yang, Abstr. Appl. Anal [10] G.A. Anastassiou, O. Duman, Advances in Applied Mathematics and Approximation Theory, Springer, New York, [11] W.P. Zhong, F. Gao, X.M. Shen, Adv. Mat. es [12] K.M. Kolwankar, A.D. Gangal, Phys. ev. Lett [13] D. Baleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control, Springer, New York, [14] J.A.T. Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. Numer. Simul [15] Y. Chen, Y. Yan, K.W. Zhang, J. Math. Anal. Appl [16] W. Chen, Chaos Solitons Fractals [17] F. Ben Adda, J. Cresson, J. Math. Anal. Appl [18] G. Jumarie, Comput. Math. Appl [19] A. Carpinteri, P. Cornetti, K.M. Kolwankar, Chaos Solitons Fractals [20] A. Carpinteri, B. Chiaia, P. Cornetti, Comput. Methods Appl. Mech. Engrg [21] A. Carpinteri, B. Chiaia, P. Cornetti, Engrg. Fract. Mech [22] A. Babakhani, V. DaftardarGejji, J. Math. Anal. Appl [23] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, New York, [24] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, [25] I. Podlubny, Fractional Differential Equations, Academic Press, New York, [26] J.H. He, S.K. Elagan, Z.B. Li, Phys. Lett. A [27] Z.B. Li, J.H. He, Math. Comput. Appl [28] J.H. He, Abstr. Appl. Anal