Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions. (English) Zbl 1171.26305

Jumarie, Guy

Appl. Math. Lett. 22, No. 3, 378-385 (2009).

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.

Reviewer: Tej Singh Nahar (Bhilwara)

MSC:

26A33

Fractional derivatives and integrals

Keywords:

fractional calculus; modified Riemann-Liouville derivative; fractional Taylor’s series; Mittag-Leffler function

References:

[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, (Kalia, R. N., Recent Advances in Fractional Calculus (1993), Global Publishing Company) · 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., (An Introduction to the Fractional Calculus and Fractional Differential Equations (1933), Wiley: Wiley New York)
[17]	Nishimoto, K., Fractional Calculus (1989), Descartes Press Co.: 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: 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., (La derivation non entiere: Theorie, synthese et applications (1995), Editions Hermes: Editions Hermes Paris), (in French) · Zbl 0864.93004
[22]	Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[23]	Ross, B., (Fractional Calculus and its Applications. Fractional Calculus and its Applications, Lecture Notes in Mathematics, vol. 457 (1974), Springer: Springer Berlin)
[24]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and derivatives. Theory and Applications (1987), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers London · Zbl 0617.26004