General Fractional Calculus, Evolution Equations, and Renewal Processes
TLDR
In this paper, the authors developed a fractional calculus and theory of diffusion equations associated with operators in the time variable, where k is a nonnegative locally integrable function, and the solution of the Cauchy problem for the relaxation equation was proved (under some conditions upon k) continuous on [0, ∞) and completely monotone.Abstract:
We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form \({(\mathbb D_{(k)} u)(t)=\frac{d}{dt} \int \nolimits_0^tk(t-\tau )u(\tau )\,d\tau-k(t)u(0)}\) where k is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation \({\mathbb D_{(k)} u=-\lambda u}\), λ > 0, proved to be (under some conditions upon k) continuous on [0, ∞) and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process N(E(t)) as a renewal process. Here N(t) is the Poisson process of intensity λ, E(t) is an inverse subordinator.read more
Citations
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Book ChapterDOI
Multi-index Mittag-Leffler Functions
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Generalized fractional derivatives generated by a class of local proportional derivatives
TL;DR: In this paper, the Laplace transform of the generalized proportional fractional derivatives and integrals is calculated and used to solve Cauchy linear fractional type problems, which is a special case of the proportional derivative controller.
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On fractional calculus with general analytic kernels
TL;DR: This work provides a fundamental connection with classical fractional calculus by writing these general fractional operators in terms of the original Riemann–Liouville fractional integral operator.
Journal ArticleDOI
On Fractional Operators and Their Classifications
Dumitru Baleanu,Arran Fernandez +1 more
TL;DR: Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695 as discussed by the authors, and it has become a thriving field of research not only in mathematics but also in other parts of science such as physics, biology, and engineering.
Journal ArticleDOI
Models of dielectric relaxation based on completely monotone functions
TL;DR: In this paper, the relaxation properties of dielectric materials are described, in the frequency domain, according to one of the several models proposed over the years: Kohlrausch-Williams-Watts, Cole-Cole, ColeDavidson, Havriliak-Negami (with its modified version) and Excess wing model.
References
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Journal ArticleDOI
An Introduction to Probability Theory and Its Applications
David A. Freedman,William Feller +1 more
Book
Theory and Applications of Fractional Differential Equations
TL;DR: In this article, the authors present a method for solving Fractional Differential Equations (DFE) using Integral Transform Methods for Explicit Solutions to FractionAL Differentially Equations.
An Introduction To Probability Theory And Its Applications
TL;DR: A First Course in Probability (8th ed.) by S. Ross is a lively text that covers the basic ideas of probability theory including those needed in statistics.
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