Time-fractional derivatives in relaxation processes: a tutorial survey. (English) Zbl 1157.26304

Summary: The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Riemann-Liouville sense arid in the Caputo sense. After giving a necessary outline 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 oil the origin’s of the Caputo derivative and on the use of fractional calculus in viscoelasticity.


26A33 Fractional derivatives and integrals
33E12 Mittag-Leffler functions and generalizations
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
44A10 Laplace transform
45K05 Integro-partial differential equations
74D05 Linear constitutive equations for materials with memory


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