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Fractional variation of Hölderian functions

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Published/Copyright: May 23, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 3

Abstract

The paper demonstrates the basic properties of the local fractional variation operators (termed fractal variation operators). The action of the operators is demonstrated for local characterization of Hölderian functions. In particular, it is established that a class of such functions exhibits singular behavior under the action of fractal variation operators in infinitesimal limit. The link between the limit of the fractal variation of a function and its derivative is demonstrated. The paper presents a number of examples, including the calculation of the fractional variation of Cauchy sequences leading to the Dirac’s delta-function.

Keywords : fractional calculus; non-differentiable functions; singular functions; Hölder classes; pseudodifferential operators

References

[1] F. B. Adda, Geometric interpretation of the fractional derivative. J. Fract. Calc. 11 (1997), 21-51.Search in Google Scholar

[2] F. B. Adda, J. Cresson, About non-differentiable functions. J. Math. Analysis Appl. 263 (2001), 721-737.10.1006/jmaa.2001.7656Search in Google Scholar

[3] F. B. Adda, J. Cresson, Fractional differential equations and the Schrödinger equation. App. Math. Comp. 161 (2005), 324-345.Search in Google Scholar

[4] F. B. Adda, J. Cresson, Corrigendum to “About non-differentiable functions” [J. Math. Anal. Appl. 263 (2001) 721-737]. J. Math. Analysis Appl. 408, No 1 (2013), 409-413; doi: 10.1016/j.jmaa.2013.06.027.10.1016/j.jmaa.2013.06.027Search in Google Scholar

[5] A. Babakhani, V. Daftardar-Gejji, On calculus of local fractional derivatives. J. Math. Analysis Appl. 270, No 1 (2002), 66-79.Search in Google Scholar

[6] R. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheology 27 (1983), 201-210.Search in Google Scholar

[7] R. Bracewell, The Fourier Transform and its Applications, McGraw- Hill Kogakusha, Ltd., Tokyo, 2nd Ed. (1978).Search in Google Scholar

[8] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. R. Ast. Soc. 13, No 5 (1967), 529-539; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3-14.Search in Google Scholar

[9] M. Caputo, F. Mainardi, Linear models of dissipation in anelastic solids. Rivista del Nuovo Cimento 1 (1971), 161-198.10.1007/BF02820620Search in Google Scholar

[10] Y. Chen, Y. Yan, K. Zhang, On the local fractional derivative. J. Math. Anal. Appl. 362, No 1 (2010), 17-33; doi: 10.1016/j.jmaa.2009.08.014.10.1016/j.jmaa.2009.08.014Search in Google Scholar

[11] M. Katz, D. Tall, A Cauchy-Dirac Delta function. Foundations of Science 18, No 1 (2013), 107-123; doi: 10.1007/s10699-012-9289-4.10.1007/s10699-012-9289-4Search in Google Scholar

[12] K. Kolwankar, Local fractional calculus: A review. arXiv:1307.0739v1 (2013).Search in Google Scholar

[13] K. Kolwankar, A. Gangal, Hölder exponents of irregular signals and local fractional derivatives. Pramana J. Phys. 1, No 1 (1997), 49-68.Search in Google Scholar

[14] K. Kolwankar, J. Lévy Véhel, Measuring functions smoothness with local fractional derivatives. Fract. Calc. Appl. Anal. 4, No 3 (2001), 285-301.[15] S. Mallat, W.-L. Hwang, Singularity detection and processing with wavelets. IEEE Trans. Information Theory 38, No 2 (1992), 617-643; doi: 10.1109/18.119727.10.1109/18.119727Search in Google Scholar

[16] L. Nottale, M. Célérier, Emergence of complex and spinor wave functions in scale relativity. i. nature of scale variables. J. Math. Physics 54, No 11 (2013), # 112102; doi: 10.1063/1.4828707.10.1063/1.4828707Search in Google Scholar

[17] K. Oldham, J. Spanier, The Fractional Calculus. Academic Press, London (1970).Search in Google Scholar

[18] B. Ross, The development of fractional calculus 1695-1900. Historia Math. 4 (1977), 75-89.Search in Google Scholar

[19] T. Todorov, A non-standard Delta function. Proc. Am. Math. Soc. 110 (1990), 1143-1144.10.1090/S0002-9939-1990-1037226-6Search in Google Scholar

[20] T. Todorov, Pointwise kernels of Schwartz distributions. Proc. Am. Math. Soc. 114, No 3 (1992), 817-819. Search in Google Scholar

Received: 2014-6-20
Accepted: 2014-12-20
Published Online: 2015-5-23
Published in Print: 2015-6-1

© Diogenes Co., Sofia

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https://doi.org/10.1515/fca-2015-0036
Keywords for this article
fractional calculus; non-differentiable functions; singular functions; Hölder classes; pseudodifferential operators

Articles in the same Issue

  1. Frontmatter
  2. Fcaa Related News, Events And Books (Fcaa-Volume 18-3-2015)
  3. Decay solutions for a class of fractional differential variational inequalities
  4. A biomathematical view on the fractional dynamics of cellulose degradation
  5. The spreading property for a prey-predator reaction-diffusion system with fractional diffusion
  6. Fractional variation of Hölderian functions
  7. Periodic disturbance rejection for fractional-order dynamical systems
  8. Successive approximation: A survey on stable manifold of fractional differential systems
  9. When do fractional differential equations have solutions that are bounded by the Mittag--Leffler function ?
  10. On explicit stability conditions for a linear fractional difference system
  11. Fractional differential inclusions in the Almgren sense
  12. Time-optimal control of fractional-order linear systems
  13. Analytical solutions for the multi-term time-space fractional reaction-diffusion equations on an infinite domain
  14. Nonexistence results for a class of evolution equations in the Heisenberg group
  15. High-order approximation to Caputo derivatives and Caputo-type advection-diffusion equations (II)
  16. Dyadic nonlocal diffusions in metric measure spaces
  17. Fractional derivative anomalous diffusion equation modeling prime number distribution
  18. Time-fractional diffusion equation in the fractional Sobolev spaces
  19. Continuous time random walk models associated with distributed order diffusion equations
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