A New Viewpoint to Fourier Analysis in Fractal Space

  • Mengke Liao
  • Xiaojun Yang
  • Qin Yan
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 41)


Fractional analysis is an important method for mathematics and engineering, and fractional differentiation inequalities are great mathematical topic for research. In this paper we point out a new viewpoint to Fourier analysis in fractal space based on the local fractional calculus and propose the local fractional Fourier analysis. Based on the generalized Hilbert space, we obtain the generalization of local fractional Fourier series via the local fractional calculus. An example is given to elucidate the signal process and reliable result.



This work is grateful for the finance supports of the National Natural Science Foundation of China (Grant No. 50904045).


  1. 1.
    R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.MATHCrossRefGoogle Scholar
  2. 2.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific, Singapore, 2009.Google Scholar
  3. 3.
    R.C. Koeller, Applications of Fractional Calculus to the Theory of Viscoelasticity, J. Appl. Mech., 51(2), 299–307 (1984).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    J. Sabatier, O.P. Agrawal, J. A. Tenreiro Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, 2007.MATHCrossRefGoogle Scholar
  5. 5.
    A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 1997.MATHGoogle Scholar
  6. 6.
    A. Carpinter, P. Cornetti, A. Sapora, et al, A fractional calculus approach to nonlocal elasticity, The European Physical Journal, 193(1), 193–204 (2011).Google Scholar
  7. 7.
    N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62, 3135–3145 (2000).CrossRefGoogle Scholar
  8. 8.
    A. Tofight, Probability structure of time fractional Schrödinger equation, Acta Physica Polonica A, 116(2), 111–118 (2009).Google Scholar
  9. 9.
    B.L. Guo, Z.H, Huo, Global well-posedness for the fractional nonlinear schrödinger equation, Comm. Partial Differential Equs., 36(2), 247–255 (2011).MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    O. P. Agrawal, Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, Nonlinear Dyn., 29, 1–4 (2002).CrossRefGoogle Scholar
  11. 11.
    A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys., 35(2) 311–322 (1996).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    H. Jafari, S. Seifi, Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation, Comm. Non. Sci. Num. Siml., 14(5), 2006–2012 (2009).MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Y. Povstenko, Non-axisymmetric solutions to time-fractional diffusion-wave equation in an infinite cylinder, act. Cal. Appl. Anal., 14(3), 418–435 (2011).MathSciNetGoogle Scholar
  14. 14.
    F. Mainardi, G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput., 141(1), 51–62 (2003).MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation, Comput. Math. Appl., 59(5), 1766–1772 (2010).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    F.H, Huang, F. W. Liu, The Space-Time Fractional Diffusion Equation with Caputo Derivatives, J. Appl. Math. Comput., 19(1), 179–190 (2005).MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    K.B, Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.MATHGoogle Scholar
  18. 18.
    K.S, Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons New York, 1993.MATHGoogle Scholar
  19. 19.
    I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.MATHGoogle Scholar
  20. 20.
    S.G, Samko, A.A, Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993.MATHGoogle Scholar
  21. 21.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.MATHGoogle Scholar
  22. 22.
    G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009.MATHCrossRefGoogle Scholar
  23. 23.
    G.A. Anastassiou, Mixed Caputo fractional Landau inequalities, Nonlinear Anal.: T. M. A., 74(16), 5440–5445 (2011).MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted, 2011.Google Scholar
  25. 25.
    K.M. Kolwankar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505–513 (1996).MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    A. Carpinteri, B. Chiaia, P. Cornetti, Static-kinematic duality and the principle of virtual work in the mechanics of fractal media, Comput. Methods Appl. Mech. Eng., 191, 3–19 (2001).MATHCrossRefGoogle Scholar
  27. 27.
    G. Jumarie, On the representation of fractional Brownian motion as an integral with respect to (dt)a, Appl. Math. Lett., 18, 739–748 (2005).MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    G. Jumarie, The Minkowski’s space–time is consistent with differential geometry of fractional order, Phy. Lett. A, 363, 5–11 (2007).MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-differentiable Functions Further Results, Comp. Math. Appl., 51, 1367–1376 (2006).MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions,  Appl. Math. Lett., 22(3), 378–385 (2009).MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    G. Jumarie, Cauchy’s integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order, Appl. Math. Lett., 23, 1444–1450 (2010).MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    G.C. Wu, Adomian decomposition method for non-smooth initial value problems, Math. Comput. Mod., 54, 2104–2108 (2011).MATHCrossRefGoogle Scholar
  33. 33.
    K.M. Kolwankar, A.D. Gangal, Hölder exponents of irregular signals and local fractional derivatives, Pramana J. Phys., 48, 49–68 (1997).Google Scholar
  34. 34.
    K.M. Kolwankar, A.D. Gangal, Local fractional Fokker–Planck equation, Phys. Rev. Lett., 80, 214–217 (1998).MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    X.R. Li, Fractional Calculus, Fractal Geometry, and Stochastic Processes, Ph.D. Thesis, University of Western Ontario, 2003Google Scholar
  36. 36.
    A. Carpinteri, P. Cornetti, K. M. Kolwankar, Calculation of the tensile and flexural strength of disordered materials using fractional calculus, Chaos, Solitons and Fractals, 21(3), 623–632 (2004).MATHCrossRefGoogle Scholar
  37. 37.
    A. Babakhani, V.D. Gejji, On calculus of local fractional derivatives,  J. Math. Anal. Appl., 270, 66–79 (2002).MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    A. Parvate, A. D. Gangal, Calculus on fractal subsets of real line - I: formulation, Fractals, 17(1), 53–81 (2009).MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    F.B. Adda, J. Cresson, About non-differentiable functions, J. Math. Anal. Appl., 263, 721–737 (2001).MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    A. Carpinteri, B. Chiaia, P. Cornetti, A fractal theory for the mechanics of elastic materials, Mater. Sci. Eng. A, 365, 235–240 (2004).CrossRefGoogle Scholar
  41. 41.
    A. Carpinteri, B. Chiaia, P. Cornetti, The elastic problem for fractal media: basic theory and finite element formulation, Comput. Struct., 82, 499–508 (2004).CrossRefGoogle Scholar
  42. 42.
    A. Carpinteri, B. Chiaia, P. Cornetti, On the mechanics of quasi-brittle materials with a fractal microstructure. Eng. Fract. Mech., 70, 2321–2349 (2003).CrossRefGoogle Scholar
  43. 43.
    A. Carpinteri, B. Chiaia, P. Cornetti, A mesoscopic theory of damage and fracture in heterogeneous materials, Theor. Appl. Fract. Mech., 41, 43–50 (2004).CrossRefGoogle Scholar
  44. 44.
    A. Carpinteri, P. Cornetti, A fractional calculus approach to the description of stress and strain localization in fractal media, Chaos, Solitons & Fractals, 13, 85–94 (2002).MATHCrossRefGoogle Scholar
  45. 45.
    A.V. Dyskin, Effective characteristics and stress concentration materials with self-similar microstructure, Int. J. Sol. Struct., 42, 477–502 (2005).MATHCrossRefGoogle Scholar
  46. 46.
    A. Carpinteri, S. Puzzi, A fractal approach to indentation size effect, Eng. Fract. Mech., 73,2110–2122 (2006).CrossRefGoogle Scholar
  47. 47.
    Y. Chen, Y. Yan, K. Zhang, On the local fractional derivative, J. Math. Anal. Appl., 362, 17–33 (2010).MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    X.J Yang, Local Fractional Integral Transforms, Prog. Nonlinear Sci., 4, 1–225 (2011).Google Scholar
  49. 49.
    X.J Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, 2011.Google Scholar
  50. 50.
    X.J Yang, Local Fractional Laplace’s Transform Based on the Local Fractional Calculus, In: Proc. of the CSIE2011, Springer, Wuhan, pp.391–397, 2011.Google Scholar
  51. 51.
    W. P. Zhong, F. Gao, Application of the Yang-Laplace transforms to solution to nonlinear fractional wave equation with fractional derivative, In: Proc. of the 2011 3rd International Conference on Computer Technology and Development, ASME, Chendu, pp.209–213, 2011.Google Scholar
  52. 52.
    X.J. Yang, Z.X. Kang, C.H. Liu, Local fractional Fourier’s transform based on the local fractional calculus, In: Proc. of The 2010 International Conference on Electrical and Control Engineering, IEEE, Wuhan, pp.1242–1245, 2010.CrossRefGoogle Scholar
  53. 53.
    W.P. Zhong, F. Gao, X.M. Shen, Applications of Yang-Fourier transform to local Fractional equations with local fractional derivative and local fractional integral, Adv. Mat. Res, 416, 306–310 (2012).CrossRefGoogle Scholar
  54. 54.
    X.J. Yang, M.K. Liao, J.W. Chen, A novel approach to processing fractal signals using the Yang-Fourier transforms, Procedia Eng., 29, 2950–2954 (2012).CrossRefGoogle Scholar
  55. 55.
    X.J. Yang, Fast Yang-Fourier transforms in fractal space, Adv. Intelligent Trans. Sys., 1(1), 25–28 (2012).Google Scholar
  56. 56.
    G. S. Chen, Generalizations of Hölder’s and some related integral inequalities on fractal space, Reprint, ArXiv:1109.5567v1 [math.CA], 2011.Google Scholar
  57. 57.
    X.J. Yang, A short introduction to Yang-Laplace Transforms in fractal space, Adv. Info. Tech. Management, 1(2), 38–43 (2012).Google Scholar
  58. 58.
    X.J. Yang, The discrete Yang-Fourier transforms in fractal space, Adv. Electrical Eng. Sys., 1(2), 78–81 (2012).Google Scholar
  59. 59.
    A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Water conservancyShihezhi UniversityShihezhiP.R. China
  2. 2.Department of Mathematics and MechanicsChina University of Mining and TechnologyXuzhouP.R. China

Personalised recommendations