Preprint No.
A-02-03
A.V. Chechkin, Rudolf Gorenflo, I.M. Sokolov
Retarding sub- and accelerating
super-diffusion governed by distributed order fractional diffusion
equations
Abstract:
We propose diffusion-like equations with time and space
fractional derivatives of distributed order for the kinetic description of
anomalous diffusion and relaxation phenonema, whose diffusion exponent
varies with time and which, correspondingly, cannot be viewed as a
self-affine random process possessing a unique Hurst exponent. We
prove
the positivity of the solutions of the proposed equations and establish
the relation to the Continuous Time Random Walk theory. We show that the
distributed-order time-fractional diffusion equation describes the
sub-diffusion random process which is subordinated to the Wiener process
and whose diffusion exponent decreases in time (retarding sub-diffusion)
leading to superslow diffusion, for which the mean-square displacement
grows logarithmically in time. We also demonstrate that the
distributed-order space-fractional diffusion equation describes
super-diffusion phenomena when the diffusion exponent increases in time
(accelerating super-diffusion).
Keywords: anomalous diffusion, integro-differential equations,
fractional calculus.
Mathematics Subject Classification (MSC2000):
26A33, 33E12, 45E10, 45K05, 60J60
Language: ENG
Available: Pr-A-02-03.ps
Contact: Rudolf Gorenflo, Freie Universität Berlin, Fachbereich Mathematik und Informatik, Arnimallee 2-6, D-14195 Berlin, Germany (gorenflo@math.fu-berlin.de)
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