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On Multi-Dimensional Random Walk Models Approximating Symmetric Space-Fractional Diffusion Processes

Umarov, SabirGorenflo, Rudolf — 2005

Fractional Calculus and Applied Analysis

Mathematics Subject Classification: 26A33, 47B06, 47G30, 60G50, 60G52, 60G60. In this paper the multi-dimensional analog of the Gillis-Weiss random walk model is studied. The convergence of this random walk to a fractional diffusion process governed by a symmetric operator defined as a hypersingular integral or the inverse of the Riesz potential in the sense of distributions is proved. * Supported by German Academic Exchange Service (DAAD).

Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey

Mainardi, FrancescoGorenflo, Rudolf — 2007

Fractional Calculus and Applied Analysis

2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10, 45K05, 74D05, The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classica theory of linear viscoelasticity, we contrast these two types of fractiona derivatives in their ability...

Renewal Processes of Mittag-Leffler and Wright Type

Mainardi, FrancescoGorenflo, RudolfVivoli, Alessandro — 2005

Fractional Calculus and Applied Analysis

2000 MSC: 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05. After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider...

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