Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions

Abstract In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer, Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional derivative operator . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given. Keywords: Riemann–Liouville fractional derivative operatorgeneralized Mittag–Leffler functionHardy-type inequalitiesLaplace transform methodVolterra differintegral equationsfractional differential equationsfractional kinetic equationsLebesgue integrable functionsFox–Wright hypergeometric functions 2000 Mathematics Subject Classification : Primary: 26A3333C2033E12Secondary: 47B3847G10 Acknowledgements The research of the first-named author was supported by a DAAD Post-Doctoral Fellowship during his visit to work with Professor Rudolf Hilfer at the Institute for Computational Physics (ICP) of the University of Stuttgart for 3 months in the academic year 2008–2009.

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