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 the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models.

https://www.mdpi.com/2504-3110/4/3/44/pdf

Fractional SIS Epidemic Models

In this paper various types of compositions involving independent fractional Brownian motions \(B^{j}_{H_{j}}(t)\), t>0, j=1,2, are examined.

Composition of Processes and Related Partial Differential Equations

We consider an ensemble of Ornstein-Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time- dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized gray Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

Centre-of-mass like superposition of Ornstein-Uhlenbeck processes: a pathway to non-autonomous stochastic differential equations and to fractional diffusion

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere represented by a time-changed spherical Brownian motion of order \nu \in (0,1] with which some anisotrophies can be captured in Cosmology. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order \nu. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotopic random field is indexed. The time dependent random fields we present in this work is therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.

Time dependent random fields on spherical non-homogeneous surfaces

We relate the convergence of time-changed processes driven by fractional equations to the convergence of corresponding Dirichlet forms. The fractional equations we dealt with are obtained by considering a general fractional operator in time.

Mirko D'Ovidio

Papers

Fractional SIS Epidemic Models

Composition of Processes and Related Partial Differential Equations

Centre-of-mass like superposition of Ornstein-Uhlenbeck processes: a pathway to non-autonomous stochastic differential equations and to fractional diffusion

Time dependent random fields on spherical non-homogeneous surfaces

Fractional equations via convergence of forms