Mathematics Colloquium: Fractional Calculus, Anomalous Diffusion, and Probability
4:10 p.m. Murrow 55
Dr. Mark M. Meerschaert
Fractional derivatives were invented by Leibniz soon after their integer-order cousins. Recently, they have found practical applications in many areas of science and engineering. Fractional diffusion equations replace the integer derivatives in the traditional diffusion equation with their fractional counterparts. These models for anomalous diffusion govern limits of continuous time random walks, where a random waiting time separates random particle 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 random fractals, whose dimension relates to the orders of the fractional derivatives. Parameter estimation requires novel statistical techniques, since power law data contains many outliers, and these tail values are the most important feature of the data. Numerical methods apply nonlocal variants of standard Euler finite difference algorithms. Since fractional derivatives are nonlocal operators, boundary value problems are more delicate. This talk will review the basic mathematical and statistical ideas, and some recent applications to hydrology, biology, and medicine.