Skip to main content
SpringerLink
Log in

A New Viewpoint to Fourier Analysis in Fractal Space

  • Conference paper
  • First Online:
Advances in Applied Mathematics and Approximation Theory

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 41))

  • 1490 Accesses

Abstract

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 is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.

    Book  MATH  Google Scholar 

  2. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific, Singapore, 2009.

    Google Scholar 

  3. R.C. Koeller, Applications of Fractional Calculus to the Theory of Viscoelasticity, J. Appl. Mech., 51(2), 299–307 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  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.

    Book  MATH  Google Scholar 

  5. A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer, New York, 1997.

    MATH  Google Scholar 

  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. N. Laskin, Fractional quantum mechanics, Phys. Rev. E, 62, 3135–3145 (2000).

    Article  Google Scholar 

  8. A. Tofight, Probability structure of time fractional Schrödinger equation, Acta Physica Polonica A, 116(2), 111–118 (2009).

    Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  10. O. P. Agrawal, Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, Nonlinear Dyn., 29, 1–4 (2002).

    Article  Google Scholar 

  11. A. M. A. El-Sayed, Fractional-order diffusion-wave equation, Int. J. Theor. Phys., 35(2) 311–322 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  14. F. Mainardi, G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput., 141(1), 51–62 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  17. K.B, Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.

    MATH  Google Scholar 

  18. K.S, Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons New York, 1993.

    MATH  Google Scholar 

  19. I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.

    MATH  Google Scholar 

  20. S.G, Samko, A.A, Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam, 1993.

    MATH  Google Scholar 

  21. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

    MATH  Google Scholar 

  22. G.A. Anastassiou, Fractional Differentiation Inequalities, Research Monograph, Springer, New York, 2009.

    Book  MATH  Google Scholar 

  23. G.A. Anastassiou, Mixed Caputo fractional Landau inequalities, Nonlinear Anal.: T. M. A., 74(16), 5440–5445 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  24. G.A. Anastassiou, Univariate right fractional Ostrowski inequalities, CUBO, accepted, 2011.

    Google Scholar 

  25. K.M. Kolwankar, A.D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos, 6, 505–513 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Jumarie, The Minkowski’s space–time is consistent with differential geometry of fractional order, Phy. Lett. A, 363, 5–11 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  29. G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor Series of Non-differentiable Functions Further Results, Comp. Math. Appl., 51, 1367–1376 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  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).

    Article  MathSciNet  MATH  Google Scholar 

  32. G.C. Wu, Adomian decomposition method for non-smooth initial value problems, Math. Comput. Mod., 54, 2104–2108 (2011).

    Article  MATH  Google Scholar 

  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. K.M. Kolwankar, A.D. Gangal, Local fractional Fokker–Planck equation, Phys. Rev. Lett., 80, 214–217 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  35. X.R. Li, Fractional Calculus, Fractal Geometry, and Stochastic Processes, Ph.D. Thesis, University of Western Ontario, 2003

    Google Scholar 

  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).

    Article  MATH  Google Scholar 

  37. A. Babakhani, V.D. Gejji, On calculus of local fractional derivatives,  J. Math. Anal. Appl., 270, 66–79 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Parvate, A. D. Gangal, Calculus on fractal subsets of real line - I: formulation, Fractals, 17(1), 53–81 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  39. F.B. Adda, J. Cresson, About non-differentiable functions, J. Math. Anal. Appl., 263, 721–737 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  40. A. Carpinteri, B. Chiaia, P. Cornetti, A fractal theory for the mechanics of elastic materials, Mater. Sci. Eng. A, 365, 235–240 (2004).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  MATH  Google Scholar 

  45. A.V. Dyskin, Effective characteristics and stress concentration materials with self-similar microstructure, Int. J. Sol. Struct., 42, 477–502 (2005).

    Article  MATH  Google Scholar 

  46. A. Carpinteri, S. Puzzi, A fractal approach to indentation size effect, Eng. Fract. Mech., 73,2110–2122 (2006).

    Article  Google Scholar 

  47. Y. Chen, Y. Yan, K. Zhang, On the local fractional derivative, J. Math. Anal. Appl., 362, 17–33 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  48. X.J Yang, Local Fractional Integral Transforms, Prog. Nonlinear Sci., 4, 1–225 (2011).

    Google Scholar 

  49. X.J Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, 2011.

    Google Scholar 

  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. 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. 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.

    Chapter  Google Scholar 

  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).

    Article  Google Scholar 

  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).

    Article  Google Scholar 

  55. X.J. Yang, Fast Yang-Fourier transforms in fractal space, Adv. Intelligent Trans. Sys., 1(1), 25–28 (2012).

    Google Scholar 

  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. 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. X.J. Yang, The discrete Yang-Fourier transforms in fractal space, Adv. Electrical Eng. Sys., 1(2), 78–81 (2012).

    Google Scholar 

  59. A. Vretblad, Fourier Analysis and Its Applications, Springer-Verlag, New York, 2003.

    MATH  Google Scholar 

Download references

Acknowledgement

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

Author information

Authors and Affiliations

  1. College of Water conservancy, Shihezhi University, Shihezhi, 832003, P.R. China

    Mengke Liao & Qin Yan

  2. Department of Mathematics and Mechanics, China University of Mining and Technology, Xuzhou, 221008, P.R. China

    Xiaojun Yang

Authors
  1. Mengke Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaojun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Qin Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xiaojun Yang .

Editor information

Editors and Affiliations

  1. , Department of Mathematical Sciences, The University of Memphis, Memphis, 38152, Tennessee, USA

    George A. Anastassiou

  2. , Department of Mathematics, TOBB Economics and Technology University, Sögütözü, Ankara, 06530, Turkey

    Oktay Duman

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this paper

Cite this paper

Liao, M., Yang, X., Yan, Q. (2013). A New Viewpoint to Fourier Analysis in Fractal Space. In: Anastassiou, G., Duman, O. (eds) Advances in Applied Mathematics and Approximation Theory. Springer Proceedings in Mathematics & Statistics, vol 41. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6393-1_26

Download citation

Publish with us

Policies and ethics

Search

Navigation