On Fractional Operators and Their Classifications
Dumitru Baleanu,Arran Fernandez +1 more
- Vol. 7, Iss: 9, pp 830
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TLDR
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.Abstract:
Fractional calculus dates its inception to a correspondence between Leibniz and L’Hopital in 1695, when Leibniz described “paradoxes” and predicted that “one day useful consequences will be drawn” from them. In today’s world, the study of non-integer orders of differentiation has become a thriving field of research, not only in mathematics but also in other parts of science such as physics, biology, and engineering: many of the “useful consequences” predicted by Leibniz have been discovered. However, the field has grown so far that researchers cannot yet agree on what a “fractional derivative” can be. In this manuscript, we suggest and justify the idea of classification of fractional calculus into distinct classes of operators.read more
Citations
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On a Fractional Operator Combining Proportional and Classical Differintegrals
TL;DR: The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviors by fractional differential equations as discussed by the authors. But it is not a suitable operator for modeling the Mittag-Leffler function.
Journal ArticleDOI
Collocation methods for terminal value problems of tempered fractional differential equations
TL;DR: In this article, a class of tempered fractional differential equations with terminal value problems are investigated and Discretized collocation methods on piecewise polynomials spaces are proposed for solving these equations.
Journal ArticleDOI
A fractional derivative with two singular kernels and application to a heat conduction problem
TL;DR: In this article, a new notion of fractional derivative involving two singular kernels is proposed and some properties related to this new operator are established and some examples are provided, and a numerical algorithm based on a Picard iteration for approximating the solutions is given.
Journal ArticleDOI
A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators
TL;DR: An analogue of the classical Mittag-Leffler function which is applied to two variables, and its basic properties are defined, and an associated fractional integral operator which has many interesting properties is defined.
Journal ArticleDOI
Fractional calculus in the sky
TL;DR: Fractional calculus has been a very useful tool for tackling the dynamics of complex systems from various branches of science and engineering as discussed by the authors, and it has been used for many applications in the last 300 years.
References
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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.
Book
An Introduction to the Fractional Calculus and Fractional Differential Equations
Kenneth S. Miller,Bertram Ross +1 more
TL;DR: The Riemann-Liouville Fractional Integral Integral Calculus as discussed by the authors is a fractional integral integral calculus with integral integral components, and the Weyl fractional calculus has integral components.
Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Book
Applications Of Fractional Calculus In Physics
TL;DR: An introduction to fractional calculus can be found in this paper, where Butzer et al. present a discussion of fractional fractional derivatives, derivatives and fractal time series.