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

2008 ◽  
Vol 77 (2) ◽  
Author(s):  
Daniel Fulger ◽  
Enrico Scalas ◽  
Guido Germano
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Diffusion Equation ◽  
Random Walks ◽  
Continuous Time ◽  
Fractional Diffusion ◽  
Space Time ◽  
Stochastic Solution ◽  
Continuous Time Random Walks ◽  
Time Fractional Diffusion Equation

scholarly journals A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Gianni Pagnini ◽  
Paolo Paradisi
Keyword(s):  
Diffusion Equation ◽  
Symmetric Space ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Stationary Increments ◽  
Sample Paths ◽  
Stochastic Solution ◽  
Probability Densities ◽  
Time Fractional Diffusion Equation

AbstractThe stochastic solution with Gaussian stationary increments is established for the symmetric space-time fractional diffusion equation when 0 <Numerical simulations are carried out by choosing as Gaussian process the fractional Brownian motion. Sample paths and probability densities functions are shown to be in agreement with the fundamental solution of the symmetric space-time fractional diffusion equation.

scholarly journals Clustered continuous-time random walks: diffusion and relaxation consequences

2012 ◽  
Vol 468 (2142) ◽  
pp. 1615-1628 ◽  
Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler
Keyword(s):  
Random Walks ◽  
Power Law ◽  
Continuous Time ◽  
Waiting Times ◽  
Scaling Limit ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
The Novel ◽  
Fractional Diffusion Equations ◽  
Continuous Time Random Walks

We present a class of continuous-time random walks (CTRWs), in which random jumps are separated by random waiting times. The novel feature of these CTRWs is that the jumps are clustered. This introduces a coupled effect, with longer waiting times separating larger jump clusters. We show that the CTRW scaling limits are time-changed processes. Their densities solve two different fractional diffusion equations, depending on whether the waiting time is coupled to the preceding jump, or the following one. These fractional diffusion equations can be used to model all types of experimentally observed two power-law relaxation patterns. The parameters of the scaling limit process determine the power-law exponents and loss peak frequencies.

NUMERICAL SIMULATIONS FOR SPACE–TIME FRACTIONAL DIFFUSION EQUATIONS

2013 ◽  
Vol 10 (02) ◽  
pp. 1341001 ◽  
Author(s):  
LEEVAN LING ◽  
MASAHIRO YAMAMOTO
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Time Derivative ◽  
Fundamental Solutions ◽  
Fractional Diffusion ◽  
Space Time ◽  
Fractional Diffusion Equation ◽  
Long Time ◽  
Liouville Fractional Derivative ◽  
Time Fractional Diffusion Equation

We consider the solutions of a space–time fractional diffusion equation on the interval [-1, 1]. The equation is obtained from the standard diffusion equation by replacing the second-order space derivative by a Riemann–Liouville fractional derivative of order between one and two, and the first-order time derivative by a Caputo fractional derivative of order between zero and one. As the fundamental solution of this fractional equation is unknown (if exists), an eigenfunction approach is applied to obtain approximate fundamental solutions which are then used to solve the space–time fractional diffusion equation with initial and boundary values. Numerical results are presented to demonstrate the effectiveness of the proposed method in long time simulations.

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