Fractional Differential Equations

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

高等学校計算数学学報 = Numerical mathematics

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

Hilfer–Prabhakar derivatives and some applications

In this paper we consider fractional higher-order stochastic differential equations of the form \[ \left ( \mu + c_\alpha \frac{d^\alpha } {dt^\alpha } \right )^\beta X(t) = \mathcal{E} (t) , \quad \mu >0,\; \beta >0,\; \alpha \in (0,1) \cup \mathbb{N} \] where $\mathcal{E} (t)$ is a Gaussian white noise. We obtain explicitly the covariance functions and the spectral densities of the stochastic processes satisfying the above equations.

/pdf/spectral-densities-related-to-some-fractional-stochastic-3fz1wrej1q.pdf

Spectral densities related to some fractional stochastic differential equations

We consider fractional directional derivatives and establish some connection with stable densities. Solutions to advection equations involving fractional directional derivatives are presented and some properties investigated. In particular we obtain solutions written in terms of Wright functions by exploiting operational rules involving the shift operator. We also consider fractional advection diffusion equations involving fractional powers of the negative Laplace operator and directional derivatives of fractional order and discuss the probabilistic interpretations of solutions.

Wright functions governed by fractional directional derivatives and fractional advection diffusion equations

. We study boundary value problems involving non-local operators for the dynamic boundary conditions. Our analysis includes a detailed description of such operators together with their relations with random times and random functionals. We provide some new characterizations for the boundary behaviour of the Brownian motion based on the interplay between non-local operators and boundary value problems.

Non-local Boundary Value Problems for Brownian motions on the half line

We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities, including an adapted version of the Pascal's rule. We then investigate the associated generating functions, for which we establish a recursive, combinatorial and integral formulation. From this, we derive an asymptotic version of the Binomial Theorem. A combinatorial and asymptotic analysis of some finite sums completes the paper.

Generalized binomials in fractional calculus

In this paper we deal with the generalized Gamma processes and their compositions. For the compositions of two or more than two generalized Gamma processes we give, when possible, the explicit law whereas, in the other cases the representations in terms of Fox's H-functions are given. We also study the connections between iteration and product of random processes by exploiting the properties of the generalized Gamma processes, such a study allows us to obtain some striking result about the compositions of the Cauchy processes or fractional Brownian motions. Furthermore, we find out the partial differential equations governing the generalized Gamma processes and their compositions

Mirko D'Ovidio

Papers

Spectral densities related to some fractional stochastic differential equations

Wright functions governed by fractional directional derivatives and fractional advection diffusion equations

Non-local Boundary Value Problems for Brownian motions on the half line

Generalized binomials in fractional calculus

Generalized Gamma Process: some results about composition and subordination