Fractional diffusion equation on fractals; Self-similar stationary solutions in a force field derived from a logarithmic potential
type: Item
instance of: scholarly article
title: Fractional diffusion equation on fractals; Self-similar stationary solutions in a force field derived from a logarithmic potential
zbMATH Open document ID: 0810.35171
DOI: 10.1016/0960-0779(94)90143-0
author: M. O. Vlad
published in: Chaos, Solitons and Fractals
publication date: 1995-04-10T00:00:00Z
review text: Consider the fractional diffusion equation in the form \[ {{\partial^{1/2} C} \over {\partial^{1/2} t}}= D^{1/2} \Biggl( {{\partial C} \over r} + {{n-1} \over r} C\Biggr) \qquad \text{as } r\to\infty, \tag{\(*\)} \] where \[ {{d^ a C} \over {dt^ 2}}= {1\over {\Gamma(1- a)}} {d\over {dt}} \int_ 0^ t {{C(r,\tau)} \over {(t- \tau)^ a}} d\tau, \qquad \Gamma(x)= \int_ 0^ \infty y^{x-1} e^{-y} dy, \] is the fractional time derivative of \(C(r,t)\) of order \(a\in (0,1]\). The author generalizes the fractional diffusion equation \((*)\) for Euclidean diffusion in force fields. The heuristic argument of Giona and Roman is used for generalizing the fractional diffusion equation in force fields to the fractal case. In this paper the author solves the problem of fluctuating fractal concentration profiles by using the theory of random point processes [see \textit{N. G. Van Kampen}, Stochastic processes in physics and chemistry (1981; Zbl 0511.60038); \textit{D. L. Snyder}, Random point processes (1975; Zbl 0385.60052); \textit{D. R. Cox} and \textit{V. Isham}, Point processes (1980; Zbl 0441.60053); \textit{S. K. Srinivasan}, Stochastic point processes and their applications (1974; Zbl 0287.60058)].
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 35Q58
zbMATH DE Number: 562531
zbMATH Keywords: fractional diffusion equation
zbMATH Keywords: fractal concentration profiles
zbMATH Keywords: random point processes
reviewed by: Stanisław Wedrychowicz
MaRDI profile type: Publication
full work available at URL: 0960-0779(94)90143-0
OpenAlex ID: W1997279961
cites work: Fractional diffusion equation on fractals: one-dimensional case and asymptotic behaviour
cites work: Fractional diffusion equation on fractals: three-dimensional case and scattering function
cites work: Q3864730
cites work: Q4775246
description: scientific article; zbMATH DE number 562531