Fractional Differential Equations

Tables of integrals

Wave propagation and group velocity

Annalen der Physik

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

On the ψ-Hilfer fractional derivative

A review of definitions of fractional derivatives and other operators

We revisit the Mittag-Leffler functions of a real variable t, with one, two and three order-parameters {α,β,γ}, as far as their Laplace transform pairs and complete monotonicity properties are concerned. These functions, subjected to the requirement to be completely monotone for t > 0, are shown to be suitable models for non–Debye relaxation phenomena in dielectrics including as particular cases the classical models referred to as Cole–Cole, Davidson–Cole and Havriliak–Negami. We show 3D plots of the relaxations functions and of the corresponding spectral distributions, keeping fixed one of the three order-parameters.

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

Stability of ψ-Hilfer impulsive fractional differential equations

Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation

E. Capelas de Oliveira

Papers

On the ψ-Hilfer fractional derivative

A review of definitions of fractional derivatives and other operators

Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics

Stability of ψ-Hilfer impulsive fractional differential equations

Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation