The book is devoted to the mathematical foundations of nonextensive statistical mechanics. This is the first book containing the systematic presentation of the mathematical theory and concepts related to nonextensive statistical mechanics, a current generalization of Boltzmann-Gibbs statistical mechanics introduced in 1988 by one of the authors and based on a nonadditive entropic functional extending the usual Boltzmann-Gibbs-von Neumann-Shannon entropy. Main mathematical tools like the q-exponential function, q-Gaussian distribution, q-Fourier transform, q-central limit theorems, and other related objects are discussed rigorously with detailed mathematical rational. The book also contains recent results obtained in this direction and challenging open problems. Each chapter is accompanied with additional useful notes including the history of development and related bibliographies for further reading.

The chapter discusses fractional generalizations of the well-known Duhamel principle. The Duhamel principle, introduced nearly 200 years ago, reduces the Cauchy problem for the inhomogeneous differential equation to the Cauchy problem for a corresponding homogeneous differential equation. Unlike the classical case, fractional versions of the Duhamel principle require a fractional derivative of the inhomogeneous term in the initial condition of the reduced equation. We present generalizations of the Duhamel principle to wide classes of single time-fractional and distributed order pseudo-differential equations, both containing Caputo-Djrbashian and Riemann-Liouville derivatives. The abstract case also presented to capture initialboundary value problems for equations given on bounded domains. Note that the fractional Duhamel principle for equations containing Caputo-Djrbashian derivatives and Riemann-Liouville derivatives significantly differ. A number of applications of the fractional Duhamel principle have been appeared recent years. Here, we discuss some of these applications, as well.

The book is devoted to the fundamental relationship between three objects: a stochastic process, stochastic differential equations driven by that process and their associated Fokker–Planck–Kolmogorov equations. This book discusses wide fractional generalizations of this fundamental triple relationship, where the driving process represents a time-changed stochastic process; the Fokker–Planck–Kolmogorov equation involves time-fractional order derivatives and spatial pseudo-differential operators; and the associated stochastic differential equation describes the stochastic behavior of the solution process. It contains recent results obtained in this direction. This book is important since the latest developments in the field, including the role of driving processes and their scaling limits, the forms of corresponding stochastic differential equations, and associated FPK equations, are systematically presented. Examples and important applications to various scientific, engineering, and economics problems make the book attractive for all interested researchers, educators, and graduate students.

The book systematically presents the theories of pseudo-differential operators with symbols singular in dual variables, fractional order derivatives, distributed and variable order fractional derivatives, random walk approximants, and applications of these theories to various initial and multi-point boundary value problems for pseudo-differential equations. Fractional Fokker-Planck-Kolmogorov equations associated with a large class of stochastic processes are presented. Fractional Duhamel's principle is developed and presented in detail. A complex version of the theory of pseudo-differential operators with meromorphic symbols based on the recently introduced complex Fourier transform is developed and applied for initial and boundary value problems for systems of complex differential and pseudo-differential equations. Each chapter contains Additonal Notes on historical facts, relevant results, open problems, and references for further reading.