Acknowledgments

Fractional derivatives were invented by Leibniz soon after their integer-order cousins. Recently, they have found practical appli-cations in many areas of science and engineering. Fractional dif-fusion equations replace the integer derivatives in the traditional diffusion equation with their fractional counterparts. These mod-els for anomalous diffusion govern limits of continuous time ran-dom walks, where a random waiting time separates random parti-cle jumps. A power law probability distribution for particle jumps leads to fractional derivatives in space. Power law waiting times correspond to time-fractional derivatives. Particle traces are ran-dom fractals, whose dimension relates to the orders of the frac-tional derivatives. Parameter estimation requires novel statisti-cal techniques, since power law data contains many outliers, and these tail values are the most important feature of the data. Nu-merical methods apply nonlocal variants of standard Euler finit