scholarly journals Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach

2011 ◽  
Vol 4 (4) ◽  
pp. 873-900 ◽  
Author(s):  
Naoufel Ben Abdallah ◽  
◽  
Antoine Mellet ◽  
Marjolaine Puel ◽  
◽  
...  
Keyword(s):  
Kinetic Equations ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Hilbert Expansion

scholarly journals Fractional Diffusion Limit for Collisional Kinetic Equations

2010 ◽  
Vol 199 (2) ◽  
pp. 493-525 ◽  
Author(s):  
Antoine Mellet ◽  
Stéphane Mischler ◽  
Clément Mouhot
Keyword(s):  
Kinetic Equations ◽  
Fractional Diffusion ◽  
Diffusion Limit

scholarly journals Generalized fractional Poisson process and related stochastic dynamics

2020 ◽  
Vol 23 (3) ◽  
pp. 656-693 ◽  
Author(s):  
Thomas M. Michelitsch ◽  
Alejandro P. Riascos
Keyword(s):  
Poisson Process ◽  
Continuous Time ◽  
Counting Process ◽  
Stochastic Dynamics ◽  
Waiting Times ◽  
Fractional Diffusion ◽  
Diffusion Limit ◽  
Fractional Operators ◽  
Fractional Poisson Process ◽  
Erlang Process

AbstractWe survey the ‘generalized fractional Poisson process’ (GFPP). The GFPP is a renewal process generalizing Laskin’s fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prabhakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the ‘well-scaled’ diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems.

Diffusion limit of a Boltzmann–Poisson system with nonlinear equilibrium state

2019 ◽  
Vol 16 (01) ◽  
pp. 131-156
Author(s):  
Lanoir Addala ◽  
Mohamed Lazhar Tayeb
Keyword(s):  
Nonlinear Diffusion ◽  
Collision Operator ◽  
Nonlinear Diffusion Equation ◽  
One Dimension ◽  
Diffusion Limit ◽  
Poisson System ◽  
Nonlinear Relaxation ◽  
Contraction Property ◽  
Hilbert Expansion ◽  
Nonlinear Equilibrium

The diffusion approximation for a Boltzmann–Poisson system is studied. Nonlinear relaxation type collision operator is considered. A relative entropy is used to prove useful [Formula: see text]-estimates for the weak solutions of the scaled Boltzmann equation (coupled to Poisson) and to prove the convergence of the solution toward the solution of a nonlinear diffusion equation coupled to Poisson. In one dimension, a hybrid Hilbert expansion and the contraction property of the operator allow to exhibit a convergence rate.

CONTINUOUS TIME RANDOM WALK AND TIME FRACTIONAL DIFFUSION: A NUMERICAL COMPARISON BETWEEN THE FUNDAMENTAL SOLUTIONS

2005 ◽  
Vol 05 (02) ◽  
pp. L291-L297 ◽  
Author(s):  
FRANCESCO MAINARDI ◽  
ALESSANDRO VIVOLI ◽  
RUDOLF GORENFLO
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Continuous Time Random Walk ◽  
Diffusion Limit ◽  
Numerical Comparison ◽  
Time Fractional Diffusion Equation ◽  
Quality Of Approximation

We consider the basic models for anomalous transport provided by the integral equation for continuous time random walk (CTRW) and by the time fractional diffusion equation to which the previous equation is known to reduce in the diffusion limit. We compare the corresponding fundamental solutions of these equations, in order to investigate numerically the increasing quality of approximation with advancing time.

scholarly journals Multiscale numerical schemes for kinetic equations in the anomalous diffusion limit

2015 ◽  
Vol 353 (8) ◽  
pp. 755-760 ◽  
Author(s):  
Nicolas Crouseilles ◽  
Hélène Hivert ◽  
Mohammed Lemou
Keyword(s):  
Anomalous Diffusion ◽  
Kinetic Equations ◽  
Diffusion Limit ◽  
Numerical Schemes

scholarly journals Anomalous diffusion and transport in heterogeneous systems separated by a membrane

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160502 ◽  
Author(s):  
E. K. Lenzi ◽  
H. V. Ribeiro ◽  
A. A. Tateishi ◽  
R. S. Zola ◽  
L. R. Evangelista
Keyword(s):  
Anomalous Diffusion ◽  
Heterogeneous System ◽  
Particle Distribution ◽  
Kinetic Equations ◽  
Heterogeneous Systems ◽  
Particle Dynamics ◽  
Fractional Diffusion ◽  
Semipermeable Membrane ◽  
Fractional Diffusion Equations ◽  
Rich Variety

Diffusion of particles in a heterogeneous system separated by a semipermeable membrane is investigated. The particle dynamics is governed by fractional diffusion equations in the bulk and by kinetic equations on the membrane, which characterizes an interface between two different media. The kinetic equations are solved by incorporating memory effects to account for anomalous diffusion and, consequently, non-Debye relaxations. A rich variety of behaviours for the particle distribution at the interface and in the bulk may be found, depending on the choice of characteristic times in the boundary conditions and on the fractional index of the modelling equations.

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