On Mittag-Leffler-type functions in fractional evolution processes

Mainardi, Francesco; Gorenflo, Rudolf

doi:10.1016/s0377-0427(00)00294-6

Cited by 474 publications

(281 citation statements)

References 23 publications

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“…2, 3, 4, 5, 6, 7, 8, 9, 10, applications and enhancements of these techniques were presented. The relevance of fractional calculus in the phenomenological description of anomalous diffusion has been discussed within applications of statistical mechanics in physics, chemistry and biology [11,12,13,14,15,16,17] as well as finance [18,19,20,21,22]; even human travel and the spreading of epidemics were modeled with fractional diffusion [23]. A direct Monte Carlo approach to fractional Fokker-Planck dynamics through the underlying CTRW requires random numbers drawn from the Mittag-Leffler distribution.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

2008

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We present a numerical method for the Monte Carlo simulation of uncoupled continuous-time random walks with a Lévy α-stable distribution of jumps in space and a Mittag-Leffler distribution of waiting times, and apply it to the stochastic solution of the Cauchy problem for a partial differential equation with fractional derivatives both in space and in time. The one-parameter MittagLeffler function is the natural survival probability leading to time-fractional diffusion equations. Transformation methods for Mittag-Leffler random variables were found later than the well-known transformation method by Chambers, Mallows, and Stuck for Lévy α-stable random variables and so far have not received as much attention; nor have they been used together with the latter in spite of their mathematical relationship due to the geometric stability of the Mittag-Leffler distribution. Combining the two methods, we obtain an accurate approximation of space-and time-fractional diffusion processes almost as easy and fast to compute as for standard diffusion processes.

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Section: Introductionmentioning

confidence: 99%

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

2008

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“…The Mittag-Le er function Eα(z) with α > 0 is defined by the following series representation, valid in the whole complex plane [24,25]:…”

Section: Definitionmentioning

confidence: 99%

Numerical solution of two dimensional time fractional-order biological population model

Prakash

Kumar

2016

Open Physics

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Abstract:In this work, we provide an approximate solution of a parabolic fractional degenerate problem emerging in a spatial diffusion of biological population model using a fractional variational iteration method (FVIM). Four test illustrations are used to show the proficiency and accuracy of the projected scheme. Comparisons between exact solutions and numerical solutions are presented for different values of fractional order α.

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“…Somewhat less restrictive integral definitions have also been given in the literature by making use of Mittag-Leffler-type functions [35]. For any real number β > 0, denote by I β f ( x ) the Riemann-Liouville integral

I_{t_{1}, t_{2}}^{β} u false(x false) = \frac{1}{normalΓ false(β false)} \int_{t_{1}}^{t_{2}} u false(t false) {false(x - t false)}^{β - 1} d t .

…”

Section: Theory and Algorithmmentioning

confidence: 99%

“…Some interesting properties and history of Riemann-Liouville fractional derivatives have been discussed in Ref [35]. …”

Section: Theory and Algorithmmentioning

confidence: 99%

High-order fractional partial differential equation transform for molecular surface construction

Chen

Wei

2012

Computational and Mathematical Biophysics

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Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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On Mittag-Leffler-type functions in fractional evolution processes

Cited by 474 publications

References 23 publications

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

Numerical solution of two dimensional time fractional-order biological population model

High-order fractional partial differential equation transform for molecular surface construction

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