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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 316978, 6 pages
http://dx.doi.org/10.1155/2013/316978
Research Article

Mappings for Special Functions on Cantor Sets and Special Integral Transforms via Local Fractional Operators

1Electronic and Information Technology Department, Jiangmen Polytechnic, Jiangmen 529090, China
2College of Instrumentation & Electrical Engineering, Jilin University, Changchun 130061, China
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
4Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey
5Institute of Space Sciences, Magurele, 077125 Bucharest, Romania
6Mihail Sadoveanu Theoretical High School, District 2, Street Popa Lazar No. 8, 021586 Bucharest, Romania
7Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou Campus, Xuzhou, Jiangsu 221008, China

Received 27 August 2013; Accepted 24 September 2013

Academic Editor: Ali H. Bhrawy

Copyright © 2013 Yang Zhao et al. 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.

Abstract

The mappings for some special functions on Cantor sets are investigated. Meanwhile, we apply the local fractional Fourier series, Fourier transforms, and Laplace transforms to solve three local fractional differential equations, and the corresponding nondifferentiable solutions were presented.

1. Introduction

Special functions [1] play an important role in mathematical analysis, function analysis physics, and so on. We recall here some very well examples, the Gamma function [2], hypergeometric function [3], Bessel functions [4], Whittaker function [5], G-function [6], q-special functions [7], Fox’s H-functions [8], Mittag-Leffler function [9], and Wright’s function [10].

The Mittag-Leffler function had successfully been applied to solve the practical problems [1115]. For example, the Mittag-Leffler-type functions in fractional evolution processes were suggested [15]. Solutions for fractional reaction-diffusion equations via Mittag-Leffler-type functions were discussed [16]. The Mittag-Leffler stability of fractional order nonlinear dynamic systems was presented [17]. Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics were proposed [18]. In [19], the anomalous relaxation via the Mittag-Leffler functions was reported. The continuous-time finance based on the Mittag-Leffler function was given [20]. In [21], the fractional radial diffusion in a cylinder based on the Mittag-Leffler function was investigated. In [22], the Mittag-Leffler stability theorem for fractional nonlinear systems with delay was considered. The stochastic linear Volterra equations of convolution type based on the Mittag-Leffler function were suggested in [23].

Recently, based on the Mittag-Leffler functions on Cantor sets via the fractal measure, the special integral transforms based on the local fractional calculus theory were suggested in [24]. In this work, some applications for the local fractional calculus theory are studied in [2436]. The main aim of this paper is to investigate the mappings for special functions on Cantor sets and some applications of special integral transforms to nondifferentiable problems.

The paper is organized as follows. In Section 2, the mappings for special functions on Cantor sets are investigated. In Section 3, the special integral transforms within local fractional calculus and some applications to nondifferentiable problems are presented. Finally, in Section 4, the conclusions are presented.

2. Mappings for Special Functions on Cantor Sets

In order to give the mappings for special functions on Cantor sets, we first recall some basic definitions about the fractal measure theory [25].

Let Lebesgue-Cantor staircase function be defined as [25] where is a cantor set, is the -dimensional Hausdorff measure, is local fractional integral operator [2431], and is a Gamma function.

Following (1), we obtain which is a Lebesgue-Cantor staircase function. For its graph, please see [28].

In this way, we define some real-valued functions on Cantor sets as follows [2426].

The Cantor staircase function is defined as [25] and its graph is shown in Figure 1.

316978.fig.001
Figure 1: Graph of for .

The Mittag-Leffler functions on Cantor sets are given by [24, 25] and we draw the corresponding graph in Figure 2.

316978.fig.002
Figure 2: Graph of for .

The sine on Cantor sets is defined by [24, 25] and its corresponding graph is depicted in Figure 3.

316978.fig.003
Figure 3: Graph of for .

The cosine on Cantor sets is [24, 25] with graph in Figure 4.

316978.fig.004
Figure 4: Graph of for .

Hyperbolic sine on Cantor sets is defined by [24, 25] and we draw its graphs as shown in Figure 5.

316978.fig.005
Figure 5: Graph of for .

Hyperbolic cosine on Cantor sets is defined as [24, 25] and its graph is shown in Figure 6.

316978.fig.006
Figure 6: Graph of for .

Following (4)–(8), we have where is a fractal unit of an imaginary number [24, 2632].

If for and , satisfies the condition [2426] for we write it as follows:

3. Special Integral Transforms within Local Fractional Calculus

In this section, we introduce the conceptions of special integral transforms within the local fractional calculus concluding the local fractional Fourier series and Fourier and Laplace transforms. After that, we present three illustrative examples.

3.1. Definitions of Special Integral Transforms within Local Fractional Calculus

We here present briefly some results used in the rest of the paper.

Let . Local fractional trigonometric Fourier series of is given by [24, 2628] The local fractional Fourier coefficients read as We notice that the above results are obtained from Pythagorean theorem in the generalized Hilbert space [24, 2628].

Let . The local fractional Fourier transform of is suggested by [24, 2932] The inverse formula is expressed as follows [24, 2932]: Let . The local fractional Laplace transform of is defined as [24, 32, 33] The inverse formula local fractional Laplace transform of is derived as [24, 32, 33] where is local fractional continuous, , and .

For more details of special integral transforms via local fractional calculus, see [24, 32, 33] and the references therein.

3.2. Applications of Local Fractional Fourier Series and Fourier and Laplace Transforms to the Differential Equation on Cantor Sets

We now present the powerful tool of the methods presented above in three illustrative examples.

Example 1. Let us begin with the local fractional differential equation on Cantor set in the following form: where and are constants and the nondifferentiable function is periodic of period so that it can be expanded in a local fractional Fourier series as follows: Here, we give a particular solution in the following form:
Following (20), we have Submitting (20)-(21) into (18), we obtain Hence, we get Therefore, we can calculate In view of (24), we give the solution of (18) as follows:

Example 2. We now consider the following differential equation on Cantor sets: subject to the initial value condition where is constant and is the local fractional continuous function so that its local fractional Fourier transform exists.
Application of local fractional Fourier transform gives so that From (29), we have Therefore, taking the inverse formula of local fractional Fourier transform, we have

Example 3. Let us find the solution to the differential equation on Cantor sets subject to the initial value condition where is the local fractional continuous function so that its local fractional Laplace transform exists.
Taking the local fractional Laplace transform, from (32), we have so that When the local fractional convolution of two functions is given by [24] and the local fractional Laplace transform of is [24] the inverse formula of the local fractional Laplace transform together with the local fractional convolution theorem gives the solution

4. Conclusions

In this work, we investigated the mappings for special functions on Cantor sets and special integral transforms via local fractional calculus, namely, the local fractional Fourier series, Fourier transforms, and Laplace transforms, respectively. These transformations were applied successfully to solve three local fractional differential equations, and the nondifferentiable solutions were reported.

References

  1. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, UK, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  2. Á. Elbert and A. Laforgia, “On some properties of the gamma function,” Proceedings of the American Mathematical Society, vol. 128, no. 9, pp. 2667–2673, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. N. Virchenko, S. L. Kalla, and A. Al-Zamel, “Some results on a generalized hypergeometric function,” Integral Transforms and Special Functions, vol. 12, no. 1, pp. 89–100, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Choi and P. Agarwal, “Certain unified integrals associated with Bessel functions,” Boundary Value Problems, vol. 2013, no. 1, pp. 1–9, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  5. P. J. McNamara, “Metaplectic Whittaker functions and crystal bases,” Duke Mathematical Journal, vol. 156, no. 1, pp. 1–31, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. E. W. Barnes, “The theory of the G-function,” Quarterly Journal of Mathematics, vol. 31, pp. 264–314, 1899.
  7. R. Floreanini and L. Vinet, “Uq( sl (2)) and q-special functions,” Contemporary Mathematics, vol. 160, p. 85, 1994.
  8. H. M. Srivastava, K. C. Gupta, and S. P. Goyal, The H-Functions of One and Two Variables with Applications, South Asian Publishers, New Delhi, India, 1982. View at MathSciNet
  9. R. Gorenflo, A. A. Kilbas, and S. V. Rogosin, “On the generalized Mittag-Leffler type functions,” Integral Transforms and Special Functions, vol. 7, no. 3-4, pp. 215–224, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. K. Saxena and M. Saigo, “Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function,” Fractional Calculus & Applied Analysis, vol. 8, no. 2, pp. 141–154, 2005. View at Zentralblatt MATH · View at MathSciNet
  11. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Singapore, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. View at MathSciNet
  13. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.
  14. S. Hu, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  15. F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. R. K. Saxena, A. M. Mathai, and H. J. Haubold, “Fractional reaction-diffusion equations,” Astrophysics and Space Science, vol. 305, no. 3, pp. 289–296, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. E. C. de Oliveira, F. Mainardi, and J. Vaz Jr., “Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics,” The European Physical Journal Special Topics, vol. 193, no. 1, pp. 161–171, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. D. S. F. Crothers, D. Holland, Y. P. Kalmykov, and W. T. Coffey, “The role of Mittag-Leffler functions in anomalous relaxation,” Journal of Molecular Liquids, vol. 114, no. 1-3, pp. 27–34, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuous-time finance,” Physica A, vol. 284, no. 1–4, pp. 376–384, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. N. N. Achar and J. W. Hanneken, “Fractional radial diffusion in a cylinder,” Journal of Molecular Liquids, vol. 114, no. 1–3, pp. 147–151, 2004. View at Publisher · View at Google Scholar · View at Scopus
  22. S. J. Sadati, D. Baleanu, A. Ranjbar, R. Ghaderi, and T. Abdeljawad, “Mittag-Leffler stability theorem for fractional nonlinear systems with delay,” Abstract and Applied Analysis, vol. 2010, Article ID 108651, 7 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. S. W. J. Welch, R. A. L. Rorrer, and R. G. Duren Jr., “Application of time-based fractional calculus methods to viscoelastic creep and stress relaxation of materials,” Mechanics Time-Dependent Materials, vol. 3, no. 3, pp. 279–303, 1999. View at Publisher · View at Google Scholar · View at Scopus
  24. X. J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic Publisher, Hong Kong, China, 2011.
  25. X. J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science, New York, NY, USA, 2012.
  26. M.-S. Hu, R. P. Agarwal, and X.-J. Yang, “Local fractional Fourier series with application to wave equation in fractal vibrating string,” Abstract and Applied Analysis, vol. 2012, Article ID 567401, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Y. Zhang, A. Yang, and X.-J. Yang, “1-D heat conduction in a fractal medium: a solution by the local fractional Fourier series method,” Thermal Science, vol. 17, no. 3, pp. 953–956, 2013. View at Publisher · View at Google Scholar
  28. Y.-J. Yang, D. Baleanu, and X.-J. Yang, “Analysis of fractal wave equations by local fractional Fourier series method,” Advances in Mathematical Physics, vol. 2013, Article ID 632309, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  29. X. J. Yang, D. Baleanu, and J. A. T. Machado, “Mathematical aspects of Heisenberg uncertainty principle within local fractional Fourier analysis,” Boundary Value Problems, vol. 2013, no. 1, pp. 131–146, 2013. View at Publisher · View at Google Scholar
  30. A. M. Yang, Y. Z. Zhang, and Y. Long, “The Yang-Fourier transforms to heat-conduction in a semi-infinite fractal bar,” Thermal Science, vol. 17, no. 3, pp. 707–7713, 2013. View at Publisher · View at Google Scholar
  31. F. Gao, W. P. Zhong, and X. M. Shen, “Applications of Yang-Fourier transform to local fractional equations with local fractional derivative and local fractional integral,” Advanced Materials Research, vol. 461, pp. 306–310, 2012. View at Publisher · View at Google Scholar · View at Scopus
  32. J.-H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  33. C. F. Liu, S. S. Kong, and S. J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,” Thermal Science, vol. 17, no. 3, pp. 715–721, 2013. View at Publisher · View at Google Scholar
  34. G. S. Chen, “Generalizations of hölder’s and some related integral inequalities on fractal space,” Journal of Function Spaces and Applications, vol. 2013, Article ID 198405, 9 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  35. A. Carpinteri, B. Chiaia, and P. Cornetti, “The elastic problem for fractal media: basic theory and finite element formulation,” Computers and Structures, vol. 82, no. 6, pp. 499–508, 2004. View at Publisher · View at Google Scholar · View at Scopus
  36. A. K. Golmankhaneh, V. Fazlollahi, and D. Baleanu, “Newtonian mechanics on fractals subset of real-line,” Romania Reports in Physics, vol. 65, pp. 84–93, 2013.