Journal
JOURNAL OF VIBRATION AND CONTROL
Volume 13, Issue 9-10, Pages 1249-1268Publisher
SAGE PUBLICATIONS LTD
DOI: 10.1177/1077546307077468
Keywords
fractional relaxation; fractional calculus; Mittag-Leffler function; complete monotonicity; slowly varying functions
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The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however, we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a positive measure depending on the order-distribution. We consider in some detail two cases of fractional relaxation of distribution order: the double-order and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we give plots of the solutions for moderate and large times.
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