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The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type
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TLDR
In this paper, the existence and uniqueness results for Riemann-Liouville Fractional Differential Equations are presented. But they do not cover the special cases of fractional calculus.Abstract:
Fundamentals of Fractional Calculus.- Riemann-Liouville Differential and Integral Operators.- Caputo's Approach.- Mittag-Leffler Functions.- Theory of Fractional Differential Equations.- Existence and Uniqueness Results for Riemann-Liouville Fractional Differential Equations.- Single-Term Caputo Fractional Differential Equations: Basic Theory and Fundamental Results.- Single-Term Caputo Fractional Differential Equations: Advanced Results for Special Cases.- Multi-Term Caputo Fractional Differential Equations.read more
Citations
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A new Definition of Fractional Derivative without Singular Kernel
Michele Caputo,Mauro Fabrizio +1 more
TL;DR: In this article, the authors present a new definition of fractional derivative with a smooth kernel, which takes on two different representations for the temporal and spatial variable, for which it is more convenient to work with the Fourier transform.
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Recent history of fractional calculus
TL;DR: A survey of the major documents and events in the area of fractional calculus that took place since 1974 up to the present date can be found in this article, where the authors report some of the most important documents and major events.
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Lyapunov functions for fractional order systems
TL;DR: A new lemma for the Caputo fractional derivatives, when 0 α 1 , is proposed, which has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.
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A Caputo fractional derivative of a function with respect to another function
TL;DR: A numerical method, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented and the efficiency and applicability of the method are shown.
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A new approach to generalized fractional derivatives
TL;DR: In this article, the authors presented a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form.
References
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