Time-fractional derivatives in relaxation processes: a tutorial survey

By Francesco Mainardi, Rudolf Gorenflo. 2008

220 citations recorded by the source corpus.

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 Rieamann-Liouville sense and 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 on the origins of the Caputo derivative and on the use of fractional calculus in viscoelasticity.

Subfields: Control and Systems Engineering, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9957b9e145c9a4584598d710e69717f92e87a509

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Citations in the corpus

Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse (2013, background): “Nowadays it is worth of being referred to as the Queen Function of Fractional Calculus [10].”
Evaluation of Fractional Integrals and Derivatives of Elementary Functions: Overview and Tutorial (2019, background): “For a more comprehensive introduction to fractional integrals and fractional derivatives we refer the reader to any of the available textbooks [3,5,9–12] or review papers [13,14].”
Fractional-order uniaxial visco-elasto-plastic models for structural analysis (2016, background): “skin, bone, respiratory tissue, tendons [45–49]), as well as other relaxation phenomena in chemistry, electronics, magnetic systems [50,44,51] and social sciences (e.”
Numerical Solution of Fractional Differential Equations: A Survey and a Software Tutorial (2018, background): “For a more comprehensive introduction to this subject, the reader is referred to any of the available textbooks [1–5] or review papers [6,7] and, in particular, to the book by Diethelm [8] by which this introductory section is mainly inspired.”
Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions (2010, background): “Several authors [16, 24] called the general operators in (1.”
The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey (2010, background): “[1], [21], [42], [40], so that it could be referred as the Queen function of fractional calculus(4).”
Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat-transfer problems (2016, background): “, Riemann-Liouville and Caputo FD, were considered to describe the anomalous relaxation [2, 5, 8-10].”
Visco-elastic behavior through fractional calculus: An easier method for best fitting experimental results (2011, background): “At the beginning of the twentieth century, Nutting (1921), Gemant (1936) observed that, for visco-elastic materials such as rubber, bitumen, polymers, concrete etc.”
Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics (2011, result): “In fact, as pointed out by Gorenflo and Mainardi [5, 19], functions of Mittag-Leffler type enter as solutions of many problems dealt with fractional calculus so that they like to refer to the Mittag-Leffler function to as the Queen function of Fractional Calculus, in contrast with its role of a Cind”
A practical guide to Prabhakar fractional calculus (2020, background): “Indeed, in light of the striking relevance of this function for the mathematical modeling of several phenomena, in 2007 the ML function was named the Queen function of fractional calculus [99] by F.”
Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator (2012, unknown): “Several authors (see [10,11]) called (1.”