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During the last few decades, differential equations and systems of fractional order (that is arbitrary one, not necessarily integer) begun to play an important role in modeling of various phenomena of physical, engineering, automatization, biological and biomedical, chemical, earth, economics, social relations, etc. nature. The so-called Special Functions of Fractional Calculus (SF of FC) provide an important tool of Fractional Calculus (FC) and Applied Analysis (AA). In particular, they are often used to represent the solutions of fractional differential equations in explicit form. Among the most popular representatives of the SF of FC are: the Mittag-Leffler (ML) function, the Wright generalized hypergeometric function pΨq, the more general Fox H-function, and the Inayat-HussainH-function. The classical Special Functions (called also SF of Mathematical Physics), including the orthogonal polynomials, and the pFq-hypergeometric functions fall in this scheme as examples of the simpler Meijer G-function. In this survey talk, we overview the properties and some applications of an important class of SF of FC, introduced for the first time in our works. For integer m>1 and arbitrary real (or complex, under suitable restrictions) indices ρ1,...,ρm>0 and μ1,...,μm, we define the multi-index (vector-index) Mittag-Leffler functions by: E(1/ρi),(μi)(z) = E)1/ρi),(μi)(m)(z) = K=0∞zk/Γ(μ1+k/ρ1)...Γ(μm+k/ρm) = 1Ψm(1,1)(μ1,1/ρi)1m;z = H1,m+11,1-z|(0,1)(0,1),(1-μi,1/ρi)1m. We propose also a list of examples of SF of FC that are E(1/ρi),(μi)-functions and play important role in pure mathematics and in solving problems from natural, applied and social sciences, and state some open problems. [ABSTRACT FROM AUTHOR]

In this paper, the anomalous diffusion processes are modeled with partial differential equations of the fractional order that are then discussed in details. The anomalous diffusion can be characterized by the property that it no longer follows the Gaussian statistics and in particular one observes a deviation from the linear time dependence of the mean squared displacement. This is the case for many different phenomena including, e.g., the translocation dynamics of a polymer chain through a nanopore, charge carrier transport in amorphous semiconductors, laser cooling in quantum optical systems to mention only few of them. In this paper, we consider the case of the anomalous diffusion that shows a power-low growth of the mean squared displacement in time. Our starting point is a stochastic formulation of the model in terms of the random walk processes. Following this line, a continuous time random walk model in form of a system of the integral equations of convolution type for the corresponding probability density functions is introduced. These so called master equations can be explicitly solved in the Fourier-Laplace domain. The time-fractional differential equation is then derived asymptotically from the master equations for the special classes of the probability density functions with the infinite first moment. For the obtained model equation and its generalizations the initial-boundary-value problems in the bounded domains are discussed. A special focus is on the initial-boundary-value problems for the generalized time-fractional diffusion equation. For this equation, the maximum principle well known for the elliptic and parabolic type PDEs is presented and applied both for the a priori estimates of the solution and for the proof of its uniqueness. Finally, first the existence of the generalized solution and then the existence of the solution under some restrictions are shown. [ABSTRACT FROM AUTHOR]

In this paper, some elements of the theory of the Wright function φ are discussed. The Wright function-along with the Mittag-Leffler function-plays a prominent role in the theory of the partial differential equations of the fractional order that are actively used nowadays for modeling of many phenomena including e.g. the anomalous diffusion processes or in the theory of the complex systems. This function appears there simultaneously as a Green function in the initial-value problems for the model linear equations with the constant coefficients and as a special solution invariant under the groups of the scaling transformations of the fractional differential equations. In this paper, both of these applications are shortly introduced. Whereas the analytical theory of the Wright function is already more or less well developed, its numerical evaluation is still an area of the active research. In this paper, the numerical evaluation of the Wright function is discussed with a focus on the case of the real axis that is very important for applications. In particular, several approaches are presented including the method of series summation, integral representations, and asymptotical expansions. In different parts of the complex plane different numerical techniques are employed. In each case, estimates for accuracy of the computations are provided. [ABSTRACT FROM AUTHOR]