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Mathematics 2018, 6(1), 8;

A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients

Department of Physics and Astronomy, University of Bologna, and the National Institute of Nuclear Physics (INFN), Via Irnerio, 46, I-40126 Bologna, Italy
Received: 14 December 2017 / Revised: 3 January 2018 / Accepted: 5 January 2018 / Published: 9 January 2018
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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In this note, we show how an initial value problem for a relaxation process governed by a differential equation of a non-integer order with a constant coefficient may be equivalent to that of a differential equation of the first order with a varying coefficient. This equivalence is shown for the simple fractional relaxation equation that points out the relevance of the Mittag–Leffler function in fractional calculus. This simple argument may lead to the equivalence of more general processes governed by evolution equations of fractional order with constant coefficients to processes governed by differential equations of integer order but with varying coefficients. Our main motivation is to solicit the researchers to extend this approach to other areas of applied science in order to have a deeper knowledge of certain phenomena, both deterministic and stochastic ones, investigated nowadays with the techniques of the fractional calculus. View Full-Text
Keywords: Caputo fractional derivatives; Mittag–Leffler functions; anomalous relaxation Caputo fractional derivatives; Mittag–Leffler functions; anomalous relaxation

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Mainardi, F. A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients. Mathematics 2018, 6, 8.

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