scholarly journals Space–time fractional diffusion equations and asymptotic behaviors of a coupled continuous time random walk model

2013 ◽  
Vol 392 (23) ◽  
pp. 5801-5807 ◽  
Author(s):  
Long Shi ◽  
Zuguo Yu ◽  
Zhi Mao ◽  
Aiguo Xiao ◽  
Hailan Huang
Author(s):  
Sabir Umarov

AbstractIn this paper continuous time and discrete random walk models approximating diffusion processes associated with time-fractional and spacedistributed order differential equations are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change process is the inverse to a Levy’s stable subordinator with the stability index β ∈ (0, 1). In the paper the convergence of modeled continuous time and discrete random walks to time-changed processes associated with distributed order fractional diffusion equations are proved using an analytic method.


2021 ◽  
Vol 34 (4) ◽  
Author(s):  
M. Muge Karaman ◽  
Jiaxuan Zhang ◽  
Karen L. Xie ◽  
Wenzhen Zhu ◽  
Xiaohong Joe Zhou

Author(s):  
Karina Weron ◽  
Aleksander Stanislavsky ◽  
Agnieszka Jurlewicz ◽  
Mark M. Meerschaert ◽  
Hans-Peter Scheffler

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.


2005 ◽  
Vol 05 (02) ◽  
pp. L291-L297 ◽  
Author(s):  
FRANCESCO MAINARDI ◽  
ALESSANDRO VIVOLI ◽  
RUDOLF GORENFLO

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.


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