We review results on the spherically symmetric, asymptotically flat Einstein-Vlasov system. We focus on a recent result where we found explicit conditions on the initial data which guarantee the formation of a black hole in the evolution. Among these data there are data such that the corresponding solutions exist globally in Schwarzschild coordinates. We put these results into a more general co...

In the framework of the one-dimensional Fractal Time Random Walk (FTRW) relaxation model, we rigorously show that the frequency domain response takes, in both nonbiased and biased walks, the only possible Cole-Cole form. The underlying reason for this is the speciic form of the relaxation function (the survival probability of a relaxing system) determined in this model by the Mittag-Leeer distr...

The present paper aims at the study and derivation of Saigo generalized fractional integral operator involving product of Hfunction of one variable and general class of polynomials. On account of the most general nature of the operator, H-function and general class of polynomials occurring in the main result, a large number of known and new results involving Rieman-Liouville, ErdélyiKober Fract...

*Correspondence: [email protected] 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK Full list of author information is available at the end of the article Abstract We establish analogues of the mean value theorem and Taylor’s theorem for fractional differential operators defined using a Mittag–Leffler kernel. We formulate a new model for the fract...

This paper deals with a new mathematical model of the hamstring muscle group. The proposed viscoelastic model includes fractional derivatives of stretching force and elongation as well as restrictions on the coefficients that follow from the second law of thermodynamics. On the basis of experimental data four coefficients of the model have been calculated by numerical procedure. Obtained result...

It has been shown that anomalous relaxation in dielectrics can be described in terms of equations with fractional derivatives. The solutions of the resulting equation with fractional derivatives are expressed by the Mittag –Leffler function and the Fox function. The conditions of a change from the Debye relaxation to “slow” (anomalous) relaxation with a power time dependence have been examined ...

In this paper we propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Then we provide the fractional comparison principle. Third, we extend the application of Riemann-Liouville fractional systems by using Caputo fractional systems. Finally, an illustrative example is provided as a proof of concept. keywords Fractional order dynamic system, Nona...

Random variables with Mittag-Leffler distribution can take values either in the set of non-negative integers or in the positive real line. There can be of two different types, one (type-1) heavy-tailed with index α ∈ (0, 1), the other (type-2) possessing all its moments. We investigate various stochastic processes where they play a key role, among which: the discrete space/time Neveu branching ...

In this article, we survey the asymptotic stability analysis of fractional differential systems with the Prabhakar fractional derivatives. We present the stability regions for these types of fractional differential systems. A brief comparison with the stability aspects of fractional differential systems in the sense of Riemann-Liouville fractional derivatives is also given.