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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