We study a fractional time derivative generalization of a previous Natale-Salusti model about nonlinear temperature and pressure waves, propagating in fluid-saturated porous rocks. Their analytic solutions, i.e., solitary shock waves characterized by a sharp front, are here generalized, introducing a formalism that allows memory mechanisms. In realistic wave propagation in porous media we must take into account spatial or temporal variability of permeability, diffusivity, and other coefficients due to the system ``history.'' Such a rock fracturing or fine particulate migration could affect the rock and its pores. We therefore take into account these phenomena by introducing a fractional time derivative to simulate a memory-conserving formalism. We also discuss this generalized model in relation to the theory of dynamic permeability and tortuosity in fluid-saturated porous media. In such a realistic model we obtain exact solutions of Burgers' equation with time fractional derivatives in the inviscid case.

From the point of view of the general theory of the hyper-Bessel operators, we consider a particular operator that is suitable to generalize the standard process of relaxation by taking into account both memory effects of power law type and time variability of the characteristic coefficient. According to our analysis, the solutions are still expressed in terms of functions of the Mittag-Leffler type as in case of fractional relaxation with constant coefficient but exhibit a further stretching in the time argument due to the presence of Erdélyi-Kober fractional integrals in our operator. We present solutions, both singular and regular in the time origin, that are locally integrable and completely monotone functions in order to be consistent with the physical phenomena described by non-negative relaxation spectral distributions.

In this paper, we show several connections between special functions arising from generalized Conway-Maxwell-Poisson (COM-Poisson) type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.

In this paper we consider space-time fractional telegraph equations, where the time derivatives are intended in the sense of Hilfer and Hadamard while the space-fractional derivatives are meant in the sense of Riesz-Feller. We provide the Fourier transforms of the solutions of some Cauchy problems for these fractional equations. Probabilistic interpretations of some specific cases are also provided.

This paper is concerned with the fractionalized diffusion equations governing the law of the fractional Brownian motion B_H(t). We obtain solutions of these equations which are probability laws extending that of B_H(t). Our analysis is based on McBride fractional operators generalizing the hyper-Bessel operators L and converting their fractional power L^α into Erdélyi–Kober fractional integrals. We study also probabilistic properties of the random variables whose distributions satisfy space-time fractional equations involving Caputo and Riesz fractional derivatives. Some results emerging from the analysis of fractional equations with time-varying coefficients have the form of distributions of time-changed random variables.