Stochastic solution to partial differential equations of fractional orders

V. V. Uchaikin ; V. V. Saenko

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 6 (2003) no. 2, pp. 197-203

Résumé

Partial differential equations containing the fractional derivatives $\partial^{\beta}f/\partial t^{\beta}(0\beta\leq 1)$ and $(-\Delta_m)^{\alpha/2}(0\alpha2)$. are considered. These equations generalize the ordinary diffusion equation to an anomalous one and can be solved by $m$-dimensional isotropic random walk with delay. In contrast to the ordinary case, a free path distribution should have a heavy tail of the inverse power type with the exponent $\alpha$, and the delay time distribution should have a similar tail with the exponent $\beta$.