Threefold Introduction to Fractional Derivatives

Hilfer, R.

doi:10.1002/9783527622979.ch2

Cited by 164 publications

(122 citation statements)

References 123 publications

(140 reference statements)

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“…As selected reference we mention only Gorenflo and Mainardi [1], [2] and [3], Gorenflo, Vivoli and Mainardi [4], Hilfer [5], Mainardi [11], Meerschaert, Nane and Vellaisamy [12], Metzler and Klafter [13], or Zhang, Meerschaert and Baeumer [16], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On an equation being a fractional differential equation with respect to time and a pseudo-differential equation with respect to space related to Lévy-type processes

Jacob

Chen

2011

fcaa

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It is with great pleasure that we dedicate this paper to Professor Rudolf Gorenflo. In particular, the second named author wants to express his appreciation to an outstanding mathematician who was to him more than 30 years ago an inspiring teacher, and a very humane senior colleague later on.A class of fractional differential equations are investigated. Using estimates for Mittag-Leffler function and a fixed point theorem, we establish the existence and uniqueness of mild solutions of the equations.MSC 2010 : Primary 34A08; Secondary 33E12, 35S05, 44A10, 60B15

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Section: Introductionmentioning

confidence: 99%

On an equation being a fractional differential equation with respect to time and a pseudo-differential equation with respect to space related to Lévy-type processes

Jacob

Chen

2011

fcaa

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“…Remark: We still think that non-local analysis in [20] is a proper tool for modeling anomalous diffusion but the zero extension in (8) does not correspond to the Dirichlet boundary conditions. Instead, in the one-dimensional case, one should apply the extension in [5].…”

Section: Resultsmentioning

confidence: 99%

Models of space-fractional diffusion: A critical review

Izsák

Szekeres

2017

Applied Mathematics Letters

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Space-fractional diffusion problems are investigated from the modeling point of view. It is pointed out that the elementwise power of the Laplacian operator in R n is an inadequate model of fractional diffusion. Also, the approach with fractional calculus using zero extension is not a proper model of homogeneous Dirichlet boundary conditions. At the time, the spectral definition of the fractional Dirichlet Laplacian seems to be in many aspects a proper model of fractional diffusion.

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“…While it is possible to translate the spatial position of a physical system forward and backward in space, it is not possible to translate the temporal position of a physical system backwards in time. Translating an experiment backward in time is not the same as reversing the momenta of all particles in a physical system, as emphasized in [25,26]. These observations combined with Equations (71) and (72) suggest to reformulate the normal irreversibility problem above as:…”

Section: Discussionmentioning

confidence: 99%

“…It suggest a reformulation [25,26] of the much discussed irreversibility problem. The normal problem can be stated as:…”

Section: Discussionmentioning

confidence: 99%

Time Automorphisms on C*-Algebras

Hilfer

2015

Mathematics

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Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C * -algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.

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Threefold Introduction to Fractional Derivatives

Cited by 164 publications

References 123 publications

On an equation being a fractional differential equation with respect to time and a pseudo-differential equation with respect to space related to Lévy-type processes

On an equation being a fractional differential equation with respect to time and a pseudo-differential equation with respect to space related to Lévy-type processes

Models of space-fractional diffusion: A critical review

Time Automorphisms on C*-Algebras

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