Thursday 6.2.2020, 14:15h in Kuščerjev seminar, Jadranska 19.

This talk covers the mathematical foundation of the Mittag-Leffler functions, fractional integrals and derivatives, and many recent novel definitions of generalized operators which appear to have many applications nowadays. We pay special attention to the analysis of the complete monotonicity of the Mittag-Leffler function which is a very important prerequisite for its application in modeling different anomalous dynamics processes. We will give a number of definitions and useful properties of different fractional operators, starting with those named as Riemann-Liouville fractional derivative and integral, Caputo fractional derivative, composite (or so-called Hilfer) derivative, and generalized integral operators which contain generalized Mittag-Leffler functions in the kernel. We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals, called Hilfer-Prabhakar derivatives. Many useful properties and relations in fractional calculus that are used in modeling anomalous diffusion and non-exponential relaxation will be presented. The Cauchy-type problems of fractional differential equations and their solutions, existence and uniqueness theorems, and different methods for solving fractional differential equations (integral transform method, operational method), will be considered in this talk. Fractional PDEs are a useful tool for the modeling of many anomalous phenomena in nature. We’ll consider fractional diffusion equations and their connections with the continuous time random walk (ctrw) theory. Furthermore, we show some applications of Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the difference–differential equations governing the dynamics of generalized renewal stochastic processes.

* Zivorad Tomovski, Ss. Cyril and Methodius University, Skopje