Next Article in Journal
The Collapse of Ecosystem Engineer Populations
Next Article in Special Issue
Best Approximation of the Fractional Semi-Derivative Operator by Exponential Series
Previous Article in Journal / Special Issue
Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions
Article Menu

Export Article

Open AccessArticle
Mathematics 2018, 6(1), 8; doi:10.3390/math6010008

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)
View Full-Text   |   Download PDF [280 KB, uploaded 9 January 2018]   |  

Abstract

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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

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

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Mathematics EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top