arXiv:0801.4914v1 [math-ph] 31 Jan 2008 FRACALMO PRE-PRINT: www.fracalmo.org Fractional Calculus and Applied Analysis, Vol. 10 No 3 (2007) 269-308 An International Journal for Theory and Applications ISSN 1311-0454 www.diogenes.bg/fcaa/ Time-fractional derivatives in relaxation processes: a tutorial survey Francesco MAINARDI (1) and Rudolf GORENFLO (2) (1) Department of Physics, University of Bologna, and INFN, Via Irnerio 46, I-40126 Bologna, Italy Corresponding Author. E-mail: francesco.mainardi@unibo.it (2) Department of Mathematics and Informatics, Free University Berlin, Arnimallee 3, D-14195 Berlin, Germany E-mail: gorenflo@mi.fu-berlin.de Dedicated to Professor Michele Caputo, Accademico dei Lincei, Rome, on the occasion of his 80-th birthday (May 5, 2007). Abstract The aim of this tutorial survey is to revisit the basic theory of relax- ation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann- Liouville sense and in the Caputo sense. After giving a necessary out- line of the classical theory of linear viscoelasticity, we contrast these two types of fractional derivatives in their ability to take into account initial conditions in the constitutive equations of fractional order. We also provide historical notes on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity. 2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05, Key Words and Phrases: fractional derivatives, relaxation, creep, Mittag- Leffler function, linear viscoelasticity 1