Fractional diffusion equation for fractal-time-continuous-time random walks

1995 ◽  
Vol 6 ◽  
pp. 7-10 ◽  
Author(s):  
P.A. Alemany
Fractals ◽  
2003 ◽  
Vol 11 (supp01) ◽  
pp. 281-289 ◽  
Author(s):  
ENRICO SCALAS ◽  
RUDOLF GORENFLO ◽  
FRANCESCO MAINARDI ◽  
MARCO RABERTO

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.


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.


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