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We provide a review of theoretical results concerning the Havriliak–Negami (HN) and the Jurlewicz–Weron–Stanislavsky (JWS) dielectric relaxation models. We derive explicit forms of functions characterizing relaxation phenomena in the time domain—the relaxation, response and probability distribution functions. We also explain how to construct and solve relevant evolution equations within these models. These equations are usually solved by using the Schwinger parametrization and the integral transforms. Instead, in this work we replace it by the powerful Efros theorem. That allows one to relate physically admissible solutions to the memory-dependent evolution equations with phenomenologically known spectral functions and, from the other side, with the subordination mechanism emerging from a stochastic analysis of processes underpinning considered relaxation phenomena. Our approach is based on a systematic analysis of the memory-dependent evolution equations. It exploits methods of integral transforms, operational calculus and special functions theory with the completely monotone and Bernstein functions. Merging analytic and stochastic methods enables us to give a complete classification of the standard functions used to describe the large class of the relaxation phenomena and to explain their properties. [ABSTRACT FROM AUTHOR]

We calculate exactly the Laplace transform of the Fréchet distribution in the form γx-(1 + γ)exp (-x-γ), γ > 0, 0 ≤ x < ∞, for arbitrary rational values of the shape parameter γ, i.e. for γ = l/k with l, k = 1, 2, . . . . The method employs the inverse Mellin transform. The closed form expressions are obtained in terms of Meijer G functions and their graphical illustrations are provided. A rescaled Fréchet distribution serves as a kernel of Fréchet integral transform. It turns out that the Fréchet transform of one-sided Lévy law reproduces the Fréchet distribution. [ABSTRACT FROM AUTHOR]

We present a method of generation of exact and explicit forms of one-sided, heavy-tailed Lévy stable probability distributions gα(x), 0 <= x < ∞, 0 < α < 1. We demonstrate that the knowledge of one such a distribution gα(x) suffices to obtain exactly gαp(x), p = 2, 3, ... . Similarly, from known gα(x) and gβ(x), 0 < α, β < 1, we obtain gαβ(x). The method is based on the construction of the integral operator, called Lévy transform, which implements the above operations. For α rational, α = l/k with l < k, we reproduce in this manner many of the recently obtained exact results for gl/k(x). This approach can be also recast as an application of the Efros theorem for generalized Laplace convolutions. It relies solely on efficient definite integration. [ABSTRACT FROM AUTHOR]

The paper is devoted to study the frequency shift in the solution of the generalized telegraph equation with a moving point-wise harmonic source. This equation contains the nonlocality in time derivatives which is expressed by the memory functions η (t) and γ (t) , where η (t) smears the second time-derivative and γ (t) the first one. Moreover, in the Laplace domain we have η ˆ (s) = γ ˆ 2 (s). The generalized telegraph equation with an external source is solved by using the integral decomposition which allows us to write this solution as a product of the solution of the telegrapher equation with harmonic source and f γ ˆ (ξ , t) which is a function of the Laplace transform of memory function γ. Such obtained solution manifests the frequency shift which is illustrated in three examples of the memory functions γ (t) : the localized case, its mixture with power-law, and the power-law case only. We show that only the first two cases have the wave front and the Doppler-like shift. The third example, despite the lack of wave fronts, also manifests the frequency shift. Thus it turns out that the frequency shift occurs regardless of the existence of a wave front, but it is more visible when such a front exists. • The method of integral decomposition is applied to get solutions of the memory dependent evolution equations. • Jugging existence/non-existence of the Doppler shift is separated from the existence/non-existence the wave fronts. • It is shown that the Cattaneo-Vernotte equation may be used to describe the wave properties of the heat transport. [ABSTRACT FROM AUTHOR]