Partial fractional differential equations and some of their applications

Abstract

Our paper is devoted to investigation of partial differential equations of fractional order. We give a historical survey of results in this field basically concerning differential equations with Riemann–Liouville and Caputo partial fractional derivatives. We pay a special attention to application of the method of Fourier, Laplace and Mellin integral transforms to study partial fractional differential equations. We present recent results on explicit solutions of Cauchy-type and Cauchy problems for model homogeneous partial differential equations with Riemann–Liouville and Caputo partial fractional derivatives generalizing the classical heat and wave equations. Explicit solutions of the above problems are given in terms of the Mittag–Leffler function, and of the so-called H-function and its special cases such as the Wright and generalized Wright functions.

We discuss applications of partial fractional differential equations to the modelling of anomalous phenomena in nature and in the theory of complex systems. Special interest has been paid to the anomalous diffusion processes such as super-slow diffusion (or sub-diffusion) and super-fast diffusion (or super-diffusion) processes. We present new fractional derivative model, which allows us to have strong control of both the sub- and super-diffusion processes which means the control of the temporal behavior of the speed of spreading via analysis of the second space moment or moments of some other order.