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 work, we construct compositions of vector processes of the form , t > 0, , β ∈ (0, 1], , whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S2βn whose random time is represented by the inverse , t > 0, of the superposition of independent positively skewed stable processes, , t > 0, (H2ν1, Hν2, independent stable subordinators). As special cases for n = 1, and β = 1, we examine the telegraph process T at Brownian time B ([14]) and establish the equality in distribution , t > 0. Furthermore the iterated Brownian motion ([2]) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes, we present their counterparts as Brownian motion at delayed stable-distributed time.

/pdf/time-changed-processes-governed-by-space-time-fractional-zeqyic6c1l.pdf

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

From Sturm–Liouville problems to fractional and anomalous diffusions

https://www.dss.uniroma1.it/it/system/files/pubblicazioni/15-DOVIDIO-RT-12-2011.pdf

On the fractional counterpart of the higher-order equations

Some fractional and anomalous diffusions are driven by equations involving fractional derivatives in both time and space. Such diffusions are processes with randomly varying times. In representing the solutions to those diffusions, the explicit laws of certain stable processes turn out to be fundamental. This paper directs one's efforts towards the explicit representation of solutions to fractional and anomalous diffusions related to Sturm-Liouville problems of fractional order associated to fractional power function spaces. Furthermore, we study a new version of the Bochner's subordination rule and we establish some connections between subordination and space-fractional operator

From Sturm-Liouville problems to fractional and anomalous diffusions

We present the stochastic solution to a generalized fractional partial differential equation (fPDE) involving a regularized operator related to the so-called Prabhakar operator and admitting as spe...

Mirko D'Ovidio

Papers

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

From Sturm–Liouville problems to fractional and anomalous diffusions

On the fractional counterpart of the higher-order equations

From Sturm-Liouville problems to fractional and anomalous diffusions

Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions