Laplace transform and the Mittag-Leffler function

CAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL E NÍVEL SUPERIORThe exponential function is solution of a linear differential equation with constant coefficients, and the Mittag-Leffler function is solution of a fractional linear differential equation with constant coefficients. Using infinite series and Laplace transform, we introduce the Mittag-Leffler function as a generalization of the exponential function. Particular cases are recovered. © 2013 Taylor & Francis.The exponential function is solution of a linear differential equation with constant coefficients, and the Mittag-Leffler function is solution of a fractional linear differential equation with constant coefficients. Using infinite series and Laplace transform, we introduce the Mittag-Leffler function as a generalization of the exponential function. Particular cases are recovered454595604CAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL E NÍVEL SUPERIORCAPES - COORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL E NÍVEL SUPERIORsem informaçãoAblowitz, M.J., Fokas, A.S., (1999) Complex variables: Introduction and applications (Cambridge texts in applied mathematics), , Cambridge (UK),: Cambridge University PressSneddon, I.N., (1995) Fourier transforms, , New York (NY),: Dover Publications IncCoddington, E.A., (1989) An introduction to ordinary differential equations, , New York (NY),: Dover Publications IncKevorkian, J., (1990) Partial differential equations, analytical solution techniques, , Belmont (CA),: Wadsworth & Brooks/Cole Advanced Book & SoftwareCamargo, R.F., Chiacchio, A.O., Oliveira, E.C., One-sided and two-sided Green's function (2013) Bound Value Probl, 2013, p. 45Miller, K.S., Ross, B., (1993) An introduction to the fractional calculus and fractional differential equations, , New York (NY),: WileyMittag-Leffler, G.M., A generalization of the Laplace-Abel integral (1903) C R Acad Sci Paris, 137, pp. 537-539Prabhakar, T.R., A singular integral equation with generalized Mittag-Leffler function in the kernel (1971) Yokohama Math J, 19, pp. 7-15Mainardi, F., (2010) Fractional calculus and waves in linear viscoelasticity, , London,: Imperial College PressCapelas de Oliveira, E., (2012) Special functions and applications, , 2nd, São Paulo,: Livraria Editora da Física, PortugueseMathai, A.M., Haubold, H.J., (2008) Special functions for applied scientists, , New York (NY),: Springer ScienceOliveira, E.C., Rodrigues Jr., W.A., (2006) Analytical functions and applications, , São Paulo,: Livraria Editora da Física, PortugueseMittag-Leffler, G.M., On the new function Eα(x) (1903) C R Acad Sci Paris, 137, pp. 554-558Humbert, P., Agarwal, R.P., On the Mittag-Leffler function and some of its generalizations (1953) Bull Sci Math Ser, 77, pp. 180-185Shukla, A.K., Prajapati, J.C., On a generalization of Mittag-Leffler function and its properties (2007) J Math Anal Appl, 336, pp. 797-811Gehlot, K.S., The generalized k-Mittag-Leffler function (2012) Int J Contemp Math Sci, 7, pp. 2213-2219Silva Costa, F., (2011) Fox's H function and applications, , Campinas: Unicamp, Portugues

A review of definitions of fractional derivatives and other operators

Given the increasing number of proposals and definitions of operators in the scope of fractional calculus, it is important to introduce a systematic classification. Nonetheless, many of the definitions that emerged in the literature can not be considered as fractional derivatives. We analyze a list of expressions to have a general overview of the concept of fractional (integrals) derivatives. Moreover, some formulae that do not involve the term fractional, are also included due to their particular interest in the area388195208sem informaçãosem informaçã