Helmholtz and Diffusion Equations Associated with Local Fractional Derivative Operators Involving the Cantorian and Cantor-Type Cylindrical Coordinates

Abstract

The main object of this paper is to investigate the Helmholtz and diffusion equations on the Cantor sets involving local fractional derivative operators. The Cantor-type cylindrical-coordinate method is applied to handle the corresponding local fractional differential equations. Two illustrative examples for the Helmholtz and diffusion equations on the Cantor sets are shown by making use of the Cantorian and Cantor-type cylindrical coordinates.

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Ya-Juan Hao 1 and H. M. Srivastava 2 and Hossein Jafari 3 and Xiao-Jun Yang 4

Recommended by J. A. Tenreiro Machado

1, College of Science, Yanshan University, Qinhuangdao 066004, China
2, Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, V8W 3R4, Canada
3, Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar 47415-416, Iran
4, Department of Mathematics and Mechanics, China University of Mining and Technology, Jiangsu, Xuzhou 221008, China

Received 9 June 2013; Accepted 7 July 2013

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

In the Euclidean space, we observe several interesting physical phenomena by using the differential equations in the different styles of planar, cylindrical, and spherical geometries. There are many models for the anisotropic perfectly matched layers [1], the plasma source ion implantation [2], fractional paradigm and intermediate zones in electromagnetism [3, 4], fusion [5], reflectionless sponge layers [6], time-fractional heat conduction [7], singular boundary value problems [8], and so on (see also the references cited in each of these works).

The Helmholtz equation was applied to deal with problems in such fields as electromagnetic radiation, seismology, transmission, and acoustics. Kreß and Roach [9] discussed the transmission problems for the Helmholtz equation. Kleinman and Roach [10] studied the boundary integral equations for the three-dimensional Helmholtz equation. Karageorghis [11] presented the eigenvalues of the Helmholtz equation. Heikkola et al. [12] considered the parallel fictitious domain method for the three-dimensional Helmholtz equation. Fu and Mura [13] suggested the volume integrals of the inhomogeneous Helmholtz equation. Samuel and Thomas [14] proposed the fractional Helmholtz equation.

Diffusion theory has become increasingly interesting and potentially useful in solids [15, 16]. Some applications of physics, such as superconducting alloys [17], lattice theory [18], and light diffusion in turbid material [19], were considered. Fractional calculus theory (see [20-28]) was applied to model the diffusion problems in engineering, and fractional diffusion equation was discussed (see, e.g., [29-36]).

Recently, the local fractional calculus theory was applied to process the nondifferentiable phenomena in fractal domain (see [37-48] and the references cited therein). There are some local fractional models, such as the local fractional Fokker-Planck equation [37], the local fractional stress-strain relations [38], the local fractional heat conduction equation [45], wave equations on the Cantor sets [47], and the local fractional Laplace equation [48].

The main aim of this paper is present in the mathematical structure of the Helmholtz and diffusion equations within local fractional derivative and to propose their forms in the Cantor-type cylindrical coordinates by using the Cantor-type cylindrical-coordinate method [46].

Our present investigation is structured as follows. In Section 2, the Helmholtz equation on the Cantor sets with local fractional derivative is investigated. The diffusion equation on the Cantor sets based upon the local fractional vector calculus is structured in Section 3. The Helmholtz and diffusion equations in the Cantor-type cylindrical coordinates are studied in Section 4. Finally, the conclusions are presented in Section 5.

2. The Helmholtz Equation on the Cantor Sets

In order to derive the Helmholtz equation on the Cantor sets, if the local fractional derivative is defined through [43-46] [figure omitted; refer to PDF] with [figure omitted; refer to PDF] then the wave equation on the Cantor sets was suggested in [44] by [figure omitted; refer to PDF] where the local fractional Laplace operator is given by [43, 44, 48] [figure omitted; refer to PDF] where 1/^a2α is a constant and u(r,t) is satisfied with local fractional continuous conditions (see [47]).

Using separation of variables in nondifferentiable functions, which begins by assuming that the fractal wave function u(r,t) may be separable, namely, [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] In the three-dimensional Cantorian coordinate system, by following (7), we have [figure omitted; refer to PDF] where the operator is a local fractional derivative operator.

For the two-dimensional Cantorian coordinate system, the local fractional homogeneous Helmholtz equation is given by [figure omitted; refer to PDF] For a fractal dimension α=1 , (9) becomes [figure omitted; refer to PDF] which is the classical Helmholtz equation [10].

In view of (9), the inhomogeneous Helmholtz equation reads as follows: [figure omitted; refer to PDF] where f(x,y,z) is a local fractional continuous function.

In the two-dimensional Cantorian coordinate system, following (12), the local fractional inhomogeneous Helmholtz equation can be suggested by [figure omitted; refer to PDF] where f(x,y) is a local fractional continuous function.

We notice that the fractional Helmholtz equation was applied to deal with the differentiable wave equations in [14]. However, the Helmholtz equation with local fractional derivative arises in physical problems in such areas as, for example, fractal electromagnetic radiation, seismology, and acoustics, because their wave functions are the local fractional continuous functions (nondifferentiable functions). So, the Helmholtz equation on the Cantor sets can be used to describe the fractal electromagnetic radiation, the fractal seismology, the fractal acoustics, and so on.

3. Diffusion Equation on the Cantor Sets

In this section, we derive the diffusion equation on the Cantor sets with local fractional vector calculus [44].

Let us recall Fick's law within the local fractional derivative, which was presented as [figure omitted; refer to PDF] where [straight phi](r,t) and J(r,t) are local fractional continuous functions.

It is noticed that the flux of the diffusing material in any part of the fractal system is proportional to the local fractional density gradient. If the diffusion coefficient D([straight phi])=D is constant, the local fractional Fick law was suggested as [44] [figure omitted; refer to PDF] which was expressed as [44] [figure omitted; refer to PDF] where the local fractional vector integral is defined as [44] [figure omitted; refer to PDF] with N elements of area with a unit normal local fractional vector _nP , Δ^SP(β) [arrow right]0 as N[arrow right]∞ for β=2α , and [straight phi](r,t) is the density of the diffusing material in local fractional field.

The conservation of mass within local fractional vector operator was presented as [44] [figure omitted; refer to PDF] where local fractional volume integral is given by [44] [figure omitted; refer to PDF] with N elements of volume Δ^VP(γ) [arrow right]0 as N[arrow right]∞ for γ=(3/2)β=3α .

Following (18), and by using the divergence theorem of local fractional field [44], we have [figure omitted; refer to PDF] where J(r,t) is the flux of the diffusing material in local fractional field.

Submitting (14) into (20), we obtain [figure omitted; refer to PDF] which is the so-called diffusion equation on the Cantor sets. This result differs from the fractional diffusion equation [29-36].

For the diffusion coefficient D([straight phi])=D , (21) becomes [figure omitted; refer to PDF] In the three-dimensional Cantorian coordinate system, following (22), we have [figure omitted; refer to PDF] In the two-dimensional Cantorian coordinate system, we get [figure omitted; refer to PDF] In the one-dimensional Cantorian coordinate system, we obtain [48] [figure omitted; refer to PDF] We notice that when fractal dimension α is equal to 1, we get the classical diffusion equation [15, 16]. However, the diffusion equation on the Cantor sets with local fractional derivative is derived from local fractional field, whose quantities are local fractional continuous functions.

4. The Cantor-Type Cylindrical-Coordinate Method to the Helmholtz and Diffusion Equations on the Cantor Sets

Let us consider the Cantor-type cylindrical coordinates, which read as follows: [figure omitted; refer to PDF] with R∈(0,+∞) , z∈(-∞,+∞) , θ∈(0,π] , and ^x2α +^y2α =^R2α .

We now have a local fractional vector given by [figure omitted; refer to PDF] such that [46] [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Submitting (29) into (9) and (12), it yields [figure omitted; refer to PDF] which is the Helmholtz equation in the Cantor-type cylindrical coordinates.

In the like manner, from (23), we get [figure omitted; refer to PDF] which is the diffusion equation in the Cantor-type cylindrical coordinates.

5. Concluding Remarks and Observations

In the present work, we have derived the Helmholtz and diffusion equations on the Cantor sets in the Cantorian coordinates, which are based upon the local fractional derivative operators. By applying the Cantor-type cylindrical-coordinate method, we have also investigated the Helmholtz and diffusion equations on the Cantor sets in the Cantor-type cylindrical coordinates. Furthermore, we have presented two illustrative examples for the corresponding fractional Helmholtz and diffusion equations on the Cantor sets by using the Cantorian and Cantor-type cylindrical coordinates.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 11102181) and in part by Natural Science Foundation of Hebei Province (no. A2012203117).

References

[1] F. L. Teixeira, W. C. Chew, "Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates," IEEE Microwave and Guided Wave Letters , vol. 7, no. 11, pp. 371-373, 1997.

[2] J. T. Scheuer, M. Shamim, J. R. Conrad, "Model of plasma source ion implantation in planar, cylindrical, and spherical geometries," Journal of Applied Physics , vol. 67, no. 3, pp. 1241-1245, 1990.

[3] N. Engheta, "On fractional paradigm and intermediate zones in electromagnetism. I. Planar observation," Microwave and Optical Technology Letters , vol. 22, no. 4, pp. 236-241, 1999.

[4] N. Engheta, "On fractional paradigm and intermediate zones in electromagnetism. II. Cylindrical and spherical observations," Microwave and Optical Technology Letters , vol. 23, no. 2, pp. 100-103, 1999.

[5] E. M. Schetselaar, "Fusion by the IHS transform: should we use cylindrical or spherical coordinates?," International Journal of Remote Sensing , vol. 19, no. 4, pp. 759-765, 1998.

[6] P. G. Petropoulos, "Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates," SIAM Journal on Applied Mathematics , vol. 60, no. 3, pp. 1037-1058, 2000.

[7] X. Jiang, M. Xu, "The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems," Physica A , vol. 389, no. 17, pp. 3368-3374, 2010.

[8] R. Buckmire, "Investigations of nonstandard, Mickens-type, finite-difference schemes for singular boundary value problems in cylindrical or spherical coordinates," Numerical Methods for Partial Differential Equations , vol. 19, no. 3, pp. 380-398, 2003.

[9] R. Kreß, G. F. Roach, "Transmission problems for the Helmholtz equation," Journal of Mathematical Physics , vol. 19, no. 6, pp. 1433-1437, 1978.

[10] R. E. Kleinman, G. F. Roach, "Boundary integral equations for the three-dimensional Helmholtz equation," SIAM Review , vol. 16, pp. 214-236, 1974.

[11] A. Karageorghis, "The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation," Applied Mathematics Letters , vol. 14, no. 7, pp. 837-842, 2001.

[12] E. Heikkola, T. Rossi, J. Toivanen, "A parallel fictitious domain method for the three-dimensional Helmholtz equation," SIAM Journal on Scientific Computing , vol. 24, no. 5, pp. 1567-1588, 2003.

[13] L. S. Fu, T. Mura, "Volume integrals of ellipsoids associated with the inhomogeneous Helmholtz equation," Wave Motion , vol. 4, no. 2, pp. 141-149, 1982.

[14] M. S. Samuel, A. Thomas, "On fractional Helmholtz equations," Fractional Calculus & Applied Analysis , vol. 13, no. 3, pp. 295-308, 2010.

[15] P. G. Shewmon Diffusion in Solids , McGraw-Hill, New York, NY, USA, 1963.

[16] G. R. Richter, "An inverse problem for the steady state diffusion equation," SIAM Journal on Applied Mathematics , vol. 41, no. 2, pp. 210-221, 1981.

[17] K. D. Usadel, "Generalized diffusion equation for superconducting alloys," Physical Review Letters , vol. 25, no. 8, pp. 507-509, 1970.

[18] D. Wolf-Gladrow, "A lattice Boltzmann equation for diffusion," Journal of Statistical Physics , vol. 79, no. 5-6, pp. 1023-1032, 1995.

[19] A. Ishimaru, "Diffusion of light in turbid material," Applied Optics , vol. 28, no. 12, pp. 2210-2215, 1989.

[20] I. Podlubny Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications , vol. 198, of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.

[21] R. Hilfer Applications of Fractional Calculus in Physics , World Scientific Publishing, River Edge, NJ, USA, 2000.

[22] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo Theory and applications of fractional differential equations , vol. 204, of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.

[23] J. Sabatier, O. P. Agrawal, J. A. T. Machado Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering , Springer, Dordrecht, The Netherlands, 2007.

[24] F. Mainardi Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , Imperial College Press, London, UK, 2010.

[25] G. M. Zaslavsky, "Chaos, fractional kinetics, and anomalous transport," Physics Reports , vol. 371, no. 6, pp. 461-580, 2002.

[26] D. Baleanu, J. A. T. Machado, A. C. J. Luo Fractional Dynamics and Control , Springer, New York, NY, USA, 2012.

[27] J. Klafter, S. C. Lim, R. Metzler Fractional Dynamics in Physics: Recent Advances , World Scientific Publishing, Singapore, 2012.

[28] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo Fractional Calculus Models and Numerical Methods , vol. 3, of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Singapore, 2012.

[29] W. Wyss, "The fractional diffusion equation," Journal of Mathematical Physics , vol. 27, no. 11, pp. 2782-2785, 1986.

[30] A. S. Chaves, "A fractional diffusion equation to describe Lévy flights," Physics Letters A , vol. 239, no. 1-2, pp. 13-16, 1998.

[31] I. M. Sokolov, A. V. Chechkin, J. Klafter, "Fractional diffusion equation for a power-law-truncated Lévy process," Physica A , vol. 336, no. 3-4, pp. 245-251, 2004.

[32] A. V. Chechkin, R. Gorenflo, I. M. Sokolov, "Fractional diffusion in inhomogeneous media," Journal of Physics A , vol. 38, no. 42, pp. L679-L684, 2005.

[33] F. Mainardi, G. Pagnini, "The Wright functions as solutions of the time-fractional diffusion equation," Applied Mathematics and Computation , vol. 141, no. 1, pp. 51-62, 2003.

[34] C. Tadjeran, M. M. Meerschaert, H.-P. Scheffler, "A second-order accurate numerical approximation for the fractional diffusion equation," Journal of Computational Physics , vol. 213, no. 1, pp. 205-213, 2006.

[35] J. Hristov, "Approximate solutions to fractional subdiffusion equations," European Physical Journal , vol. 193, no. 1, pp. 229-243, 2011.

[36] J. Hristov, "A short-distance integral-balance solution to a strong subdiffusion equation: a Weak Power-Law Profile," International Review of Chemical Engineering , vol. 2, no. 5, pp. 555-563, 2010.

[37] K. M. Kolwankar, A. D. Gangal, "Local fractional Fokker-Planck equation," Physical Review Letters , vol. 80, no. 2, pp. 214-217, 1998.

[38] A. Carpinteri, B. Chiaia, P. Cornetti, "Static-kinematic duality and the principle of virtual work in the mechanics of fractal media," Computer Methods in Applied Mechanics and Engineering , vol. 191, no. 1-2, pp. 3-19, 2001.

[39] F. Ben Adda, J. Cresson, "About non-differentiable functions," Journal of Mathematical Analysis and Applications , vol. 263, no. 2, pp. 721-737, 2001.

[40] A. Babakhani, V. Daftardar-Gejji, "On calculus of local fractional derivatives," Journal of Mathematical Analysis and Applications , vol. 270, no. 1, pp. 66-79, 2002.

[41] G. Jumarie, "Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions," Applied Mathematics Letters , vol. 22, no. 3, pp. 378-385, 2009.

[42] W. Chen, H. Sun, X. Zhang, D. Korosak, "Anomalous diffusion modeling by fractal and fractional derivatives," Computers & Mathematics with Applications , vol. 59, no. 5, pp. 1754-1758, 2010.

[43] X. J. Yang Local Fractional Functional Analysis and Its Applications , Asian Academic Publisher, Hong Kong, China, 2011.

[44] X. J. Yang Advanced Local Fractional Calculus and Its Applications , World Science Publisher, New York, NY, USA, 2012.

[45] X. J. Yang, D. Baleanu, "Fractal heat conduction problem solved by local fractional variation iteration method," Thermal Science , vol. 17, no. 2, pp. 625-628, 2013.

[46] X. J. Yang, H. M. Srivastava, J. H. He, D. Baleanu, "Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives," Physics Letters A , vol. 377, no. 28-30, pp. 1696-1700, 2013.

[47] M.-S. Hu, R. P. Agarwal, X.-J. Yang, "Local fractional Fourier series with application to wave equation in fractal vibrating string," Abstract and Applied Analysis , vol. 2012, 2012.

[48] Y. J. Yang, D. Baleanu, X. J. Yang, "A local fractional variational iteration method for Laplace equation within local fractional operators," Abstract and Applied Analysis , vol. 2013, 2013.