Time-fractional Diffusion of Distributed Order

Mainardi, Francesco; Mura, A.; Pagnini, Gianni; Gorenflo, Rudolf

doi:10.1177/1077546307087452

Cited by 190 publications

(154 citation statements)

References 39 publications

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“…Soon the distributed order fractional differential equations attracted attention of physicists who recognized that these equations can serve as models for the so-called ultra slow diffusion. In contrast to the slow diffusion, which is characterized by the mean square displacement of the diffusing particles of the power type t α , the mean square displacement in the framework of the ultra slow diffusion is just of logarithmic growth (see e.g., [3]- [4], [15], [17], [21] and the references therein). Another direction of research regarding the distributed order fractional differential equations, which is important for potential applications was to investigate how they preserve the positivity of the initial conditions in time, e.g., to analyze if their fundamental solutions can be interpreted as some probability density functions (see e.g., [3], [6] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Luchko

Yamamoto

2014

Fract Calc Appl Anal

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This article deals with investigation of some important properties of solutions to initial-boundary-value problems for distributed order timefractional diffusion equations in bounded multi-dimensional domains. In particular, we investigate the asymptotic behavior of the solutions as the time variable t → 0 and t → +∞. By the Laplace transform method, we show that the solutions decay logarithmically as t → +∞. As t → 0, the decay rate of the solutions is dominated by the term (t log(1/t)) −1 . Thus the asymptotic behavior of solutions to the initial-boundary-value problem for the distributed order time-fractional diffusion equations is shown to be different compared to the case of the multi-term fractional diffusion equations.MSC 2010 : Primary 35R11; Secondary 35B40, 35S11, 44A10

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Section: Introductionmentioning

confidence: 99%

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Luchko

Yamamoto

2014

Fract Calc Appl Anal

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“…A variety of other solutions to the FFPE have been obtained by Mainardi [129], including the inverse Fourier transform for β = 2, in which case, the solution asymptotically relaxes as the inverse power law t −α/2 .…”

Section: Fractional Phase Space Equationsmentioning

confidence: 99%

A Fractional Probability Calculus View of Allometry

West¹

2014

Systems

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The scaling of respiratory metabolism with body size in animals is considered by many to be a fundamental law of nature. An apparent corollary of this law is the scaling of physiologic time with body size, implying that physiologic time is separate and distinct from clock time. However, these are only two of the many allometry relations that emerge from empirical studies in the physical, social and life sciences. Herein, we present a theory of allometry that provides a foundation for the allometry relation between a network function and the size that is entailed by the hypothesis that the fluctuations in the two measures are described by a scaling of the joint probability density. The dynamics of such networks are described by the fractional calculus, whose scaling solutions entail the empirically observed allometry relations.

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“…Then, interest slowly increased, with several interesting papers being published; of particular note among them are Chechkin et al [5], who applied DOFD to study retarding subdiffusion and accelerating superdiffusion; Lorenzo and Hartley [6], who studied variable-order and distributed-order fractional operators; Chechkin et al [7] applied distributed-order time-fractional operators in the fractional equations; Naber [8] studied distributed-order fractional subdiffusion; Kochubei [9] applied distributed-order operators to the study of ultraslow diffusion; and Mainardi et al [10] applied DOFD to study the diffusion.…”

Section: Introductionmentioning

confidence: 99%

The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials

Caputo

Fabrizio

2017

Fractal Fract

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Abstract:The extension of the fractional order derivative to the distributed order fractional derivative (DOFD) is somewhat simple from a formal point of view, but it does not yet have a simple, obvious analytic form that allows its fast numerical calculation, which is necessary when solving differential equations with DOFD. In this paper, we supply a simple analytic kernel for the Caputo DOFD and the Caputo-Fabrizio DOFD, which may be used for numerical calculation in cases where the weight function is unity. This, in turn, could potentially allow faster solution of differential equations containing DOFD. Utilizing an analytical formulation of simple physical systems with phenomenological equations that include a DOFD, we show the relevant differences between the Caputo DOFD and the Caputo-Fabrizio DOFD. Finally, we propose a model based on DOFD for modeling composed materials that comprise different constituents, and show its compatibility with thermodynamics.

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Time-fractional Diffusion of Distributed Order

Cited by 190 publications

References 39 publications

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

Asymptotic estimates of solutions to initial-boundary-value problems for distributed order time-fractional diffusion equations

A Fractional Probability Calculus View of Allometry

The Kernel of the Distributed Order Fractional Derivatives with an Application to Complex Materials

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