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On the Riemann-Liouville fractional calculus and some recent applications. (English) Zbl 0868.26004
Summary: When B. Mandelbrot discussed the problem of fractional Brownian motion in his classic book “The fractal geometry of nature” (1982; Zbl 0504.28001), he already pointed out some strong relations to the Riemann-Liouville fractional integral 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 propagation are generalized to fractional order differential equations. The basic idea behind all that is that the order of differentiation need not be an integer but a fractional number (i.e., ${d}^{q}/d{t}^{q}$ with $0). Applications to slow relaxation processes in complex systems like polymers or even biological tissue and to self-similar protein dynamics will be discussed. In addition, we investigate a fractional diffusion equation and we present the corresponding probability density function for the location of a random walker on a fractal object. Fox functions play a dominant part.

##### MSC:
 26A33 Fractional derivatives and integrals (real functions) 28A80 Fractals 35R60 PDEs with randomness, stochastic PDE