Random Walk Models for Time-Fractional Diffusion

Mainardi, Francesco; Gorenflo, Rudolf; Moretti, Daniele; Paradisi, Paolo

doi:10.1142/9789812777720_0016

Cited by 5 publications

(8 citation statements)

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“…There are many alternative definitions that extend spatial diffusion operators to fractional order-α pseudo-differential operators. One of the oldest and conceptually simplest extensions is the isotropic equation of Feller [61] (see also [62]). This is extended here to a fractional-order nonlinear reaction–diffusion equation ∂tθ=−false[−∇2false]α/2∫0θDfalse(θfalse¯false) dθfalse¯+Rfalse(θfalse),using the Riesz functional calculus to define a power of the unbounded positive definite self-adjoint subtracted Laplacian operator −∇2.…”

Section: Beyond Standard Diffusionmentioning

confidence: 99%

“…[63]), including some truncated Lévy distributions with α<1. By integrating Green’s function given in [62], in the case 0<α<1, the cumulative distribution function is 1−2∑n=1normal∞Γfalse(nα+1false)n!nπαsin⁡(nπα2)|xfalse|−nαfalse(Dtfalse)n.For example, 30 specimens of Australian snapper (C. Auratus) were acoustically tagged in the Cabbage Tree Island no-take area at Port Stephens NSW [64]. From the 22 that had signals recorded for 1 year, the median home range of 95% occupancy was only 0.33 km2.…”

Section: Beyond Standard Diffusionmentioning

confidence: 99%

“…There are many alternative definitions that extend spatial diffusion operators to fractional order-α pseudodifferential operators. One of the oldest and conceptually simplest extensions is the isotropic equation of Feller [61] (see also [62]). This is extended here to a fractional-order nonlinear reaction-diffusion equation…”

Section: (D) Nonlinear Diffusion Fractional-order In Spacementioning

confidence: 99%

“…[63]), including some truncated Lévy distributions with α < 1. By integrating Green's function given in [62], in the case 0 < α < 1, the cumulative distribution function is…”

Section: (D) Nonlinear Diffusion Fractional-order In Spacementioning

confidence: 99%

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Conditionally integrable PDEs, non-classical symmetries and applications

2023

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Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

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Section: Beyond Standard Diffusionmentioning

confidence: 99%

Section: Beyond Standard Diffusionmentioning

confidence: 99%

Section: (D) Nonlinear Diffusion Fractional-order In Spacementioning

confidence: 99%

“…[63]), including some truncated Lévy distributions with α < 1. By integrating Green's function given in [62], in the case 0 < α < 1, the cumulative distribution function is…”

Section: (D) Nonlinear Diffusion Fractional-order In Spacementioning

confidence: 99%

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Conditionally integrable PDEs, non-classical symmetries and applications

2023

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“…Fractional calculus enables one to generalize standard diffusion equation by replacing the space and time indices with non-integers. The fundamental solutions of these generalized diffusion equations are intimately connected to some non-Gaussian probability distributions [1] which occur in many fields ranging from finance to physics. This has led to the growing interest in fractional calculus to enlarge the domain of stochastic models by non-Gaussian distributions.…”

Section: Introductionmentioning

confidence: 99%

Komunyakaa and the Kolkata Book Fair: Entries from a Diary of a Journey to India

Datta¹,

Komunyakaa²,

Sen³

2005

Callaloo

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In this work a Feynman-Kac path integral method based on Lévy measure has been proposed for solving the Cauchy problems associated with the space-time fractional Schrödinger equations arising in interacting systems in fractional quantum mechanics. The Continuous Time Random Walk(CTRW) model is used to simulate the underlying Lévy process-a generalized Wiener process. Since we are interested to capture the lowest energy state of the quantum systems, we use Pareto distributions as opposed to Mittag-Leffler random variables, which are more suitable for finite time. Adopting the CTRW model 1 we have been able to simulate the space-time fractional diffusion process with comparable simplicity and covergence rate as in the case of standard diffusion processes. We hope this paves an elegant way to solve space-time diffusion equations numerically through Fractional Feynman-Kac path integral technique as an alternative to fractional calculus.

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