ON THE RIEMANN-LIOUVILLE FRACTIONAL CALCULUS AND SOME RECENT APPLICATIONS

When Benoit Mandelbrot discussed the problem of fractional Brownian mo tion in his classic book The Fractal Geometry of Nature, he already po inted out some strong relations to the Riemann-Liouville fractional in tegral and differential calculus. Over the last decade several papers have appeared in which integer-order, standard differential equations modeling processes of relaxation, oscillation, diffusion and wave prop agation are generalized to fractional order differential equations. Th e basic idea behind all that is that the order of differentiation need not be an integer but a fractional number (i.e. d(q)/dt(q) with 0 < q < 1). Applications to slow relaxation processes in complex systems li ke polymers or even biological tissue and to selfsimilar protein dynam ics will be discussed. In addition, we investigate a fractional diffus ion equation and we present the corresponding probability density func tion for the location of a random walker on a fractal object. Fox-func tions play a dominant part.