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Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. (English) Zbl 1171.26305
The author gives a short background on the definition of the modified Riemann-Liouville derivative for non-differentiable functions and the related fractional Taylor’s series, he displays some formulae involving fractional derivatives. Also some formulae involving integrals with respect to \((dx)^\alpha\) and the Lagrangian technique of constant variation for solving fractional differential equations are presented.

26A33 Fractional derivatives and integrals
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[1] Al-Akaidi, M., Fractal speech processing, (2004), Cambridge University Press · Zbl 1082.94003
[2] Campos, L.MC., On a concept of derivative of complex order with applications to special functions, IMA J. appl. math., 33, 109-133, (1984) · Zbl 0565.30034
[3] Campos, L.M.C., Fractional calculus of analytic and branched functions, () · Zbl 0789.30030
[4] Caputo, M., Linear model of dissipation whose Q is almost frequency dependent II, Geophys. J. R. ast. soc., 13, 529-539, (1967)
[5] Djrbashian, M.M.; Nersesian, A.B., Fractional derivative and the Cauchy problem for differential equations of fractional order, Izv. acad. nauk armjanskoi SSR, 3, 1, 3-29, (1968), (in Russian)
[6] Jumarie, G., Stochastic differential equations with fractional Brownian motion input, Int. J. syst. sc., 24, 6, 1113-1132, (1993) · Zbl 0771.60043
[7] Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to \((d t)^\alpha\), Appl. math. lett., 18, 739-748, (2005) · Zbl 1082.60029
[8] Jumarie, G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion, Appl. math. lett., 18, 817-826, (2005) · Zbl 1075.60068
[9] Jumarie, G., Modified riemann – liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. math. appl., 51, 1367-1376, (2006) · Zbl 1137.65001
[10] Jumarie, G., New stochastic fractional models for malthusian growth, the Poissonian birth prodess and optimal management of populations, Math. comput. modelling, 44, 231-254, (2006) · Zbl 1130.92043
[11] Kober, H., On fractional integrals and derivatives, Quart. J. math. Oxford, 11, 193-215, (1940) · Zbl 0025.18502
[12] Kolwankar, KM.; Gangal, AD., Holder exponents of irregular signals and local fractional derivatives, Pramana J. phys., 48, 49-68, (1997)
[13] Kolwankar, KM.; Gangal, AD., Local fractional Fokker-Planck equation, Phys. rev. lett., 80, 214-217, (1998) · Zbl 0945.82005
[14] Letnikov, A.V., Theory of differentiation of fractional order, Math. sb., 3, 1-7, (1868)
[15] Liouville, J., Sur le calcul des differentielles à indices quelconques, J. ecole polytechnique, 13, 71, (1832), (in french)
[16] Miller, K.S.; Ross, B., ()
[17] Nishimoto, K., Fractional calculus, (1989), Descartes Press Co. Koroyama
[18] Oldham, K.B.; Spanier, J., The fractional calculus. theory and application of differentiation and integration to arbitrary order, (1974), Acadenic Press New York · Zbl 0292.26011
[19] Ortigueira, M.D., Introduction to fractional signal processing. part I: continuous time systems, IEE proc. vision image signal process., I, 62-70, (2000)
[20] Osler, T.J., Taylor’s series generalized for fractional derivatives and applications, SIAM. J. math. anal., 2, 1, 37-47, (1971) · Zbl 0215.12101
[21] Oustaloup, A., (), (in French)
[22] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[23] Ross, B., ()
[24] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives. theory and applications, (1987), Gordon and Breach Science Publishers London · Zbl 0617.26004
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