Prof. Dr. Rudolf Gorenflo
Institut für Mathematik I, FU Berlin
Difference schemes of random walk type for space-fractional diffusion
Abstract: In recent years in physics and in other fields, e.g. stochastic economical processes, pseudo-differential evolutionary equations modelling processes of anomalous diffusion are becoming more and more popular. A mathematical model proposed in 1952 by William Feller consists of replacing in the common diffusion equation the second-order spatial differential operator by a fractional power of it into which also non-symmetric behaviour can be built in. The resulting process is of Markov-type, mathematically a semigroup. Discrete or semi-discrete models can be obtained by discretization in space and in time, or only in space, respectively. By taking care in devicing these they can be interpreted not only as difference schemes for approximating the solution of initial value problems but also as random walk models for simulating particle paths by the Monte Carlo technique. A report is given on several possible accesses to find consistent models, accesses based either on the properties of stable probability distributions or on different possible representations of the spatial pseudodifferential operators. A few graphs of numerical case studies are shown. The results have been obtained in collaboration with Francesco Mainardi in Bologna under support of Centro Nazionale di Ricerche, Roma.
| Zeit: | Dienstag, 13. November 1998, 14.15 Uhr |
| Ort: | FU Berlin, Institut für Mathematik II, Arnimallee 2-6, 14195 Berlin, im EG Raum 032 |