The two forms of fractional relaxation of distributed order

Details The two forms of fractional relaxation of distributed order The two forms of fractional relaxation of distributed order

2007-01-07

arxiv.org/abs/cond-mat/0701131v2

Journal of Vibration and Control, Vol. 13, No. 9-10, pp.1249-1268 (2007)

Physics

Condensed Matter

Statistical Mechanics

18 pages, 4 figures. International Symposium on Mathematical Methods in Engineering, (MME06), Ankara, Turkey, April 27-29, 200

Scientific paper

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 with some detail two cases of fractional relaxation of distributed 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 exhibit plots of the solutions for moderate and large times.

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