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On calculus of local fractional derivatives. (English) Zbl 1005.26002

K. M. Kolwankar and A. D. Gangal [“Fractional differentiability of nowhere differentiable functions and dimensions”, Chaos 6, No. 4, 505-513 (1996; Zbl 1055.26504)] introduced a certain “localization” \(d^\alpha f(x)\) of the Riemann-Liouville fractional derivative to study the local behaviour of nowhere differentiable functions (see reviewer’s remark on this notion in the review in Zbl to the paper by F. Ben Adda and J. Cresson [“About non-differentiable functions”, J. Math Anal. Appl. 263, No. 2, 721-737 (2001; Zbl 0995.26006)]). The authors study some properties of this “localization”, such as the Leibniz rule for the product, the chain rule and others.

MSC:

26A33 Fractional derivatives and integrals
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References:

[1] Oldham, K.; Spanier, J., The Fractional Calculus (1970), Academic Press: Academic Press London · Zbl 0428.26004
[2] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press London · Zbl 0918.34010
[3] Srivastava, H. M.; Owa, S., Univalent Functions, Fractional Calculus and Their Applications (1989), Halsted Press/Wiley: Halsted Press/Wiley New York · Zbl 0683.00012
[4] Kolwankar, K. M.; Gangal, A. D., Fractional differentiability of nowhere differentiable functions and dimension, Chaos, 6, 505-513 (1996) · Zbl 1055.26504
[5] Kolwankar, K. M.; Gangal, A. D., Hölder exponents of irregular signals and local fractional derivatives, Pramana J. Phys., 48, 49-68 (1997)
[6] Kolwankar, K. M.; Gangal, A. D., Local fractional Fokker-Planck equation, Phys. Rev. Lett., 48, 49-52 (1997)
[7] Risken, H., The Fokker-Planck Equation (1984), Springer-Verlag: Springer-Verlag Berlin · Zbl 0546.60084
[8] Kolwankar, K. M.; Gangal, A. D., Local fractional derivatives and fractal functions of several variables, (Proceedings of the Conference on Fractals in Engineering, Archanon (1997))
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