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Discrete and continuous random walk models for space-time fractional diffusion. (English) Zbl 1125.76067
In recent years a number of evolution equations have been proposed that can describe phenomena of anomalous diffusion where the variance of density may be infinite or no longer proportional to the first power of time. In particular, these evolution equations include generalized diffusion equations containing fractional derivatives, and integral equations for random walks subordinated to a renewal processes.
Here the authors introduce an integral equation for so-called continuous-time (CTRW) random walk which differs from the usual models in that the steps of the walker occur at random time generated by a renewal process. The authors show how the integral equation for CTRW reduces to fractional diffusion equations under the assumption that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of fractional derivatives. Illustrating examples, numerical results and plot of simulation are given.
For related papers see: A. V. Chechkin, R. Gorenflo, I. M. Sokolov and V. Yu. Gonchar, Fract. Calc. Appl. Anal. 6, No. 3, 259–279 (2003; Zbl 1089.60046); E. W. Montroll and G. H. Weiss, J. Math. Phys. 6, 167–181 (1965).

MSC:
76R50 Diffusion
76M35 Stochastic analysis applied to problems in fluid mechanics
60J60 Diffusion processes
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