Journal ArticleDOI
Remarks on fractional derivatives
Changpin Li,Weihua Deng +1 more
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TLDR
This paper further discusses the properties of three kinds of fractional derivatives: the Grunwald-Letnikov derivative, the Riemann-Liouville derivative and the Caputo derivative, and compares them with the classical derivative.About:
This article is published in Applied Mathematics and Computation.The article was published on 2007-04-01. It has received 553 citations till now. The article focuses on the topics: Grünwald–Letnikov derivative & Derivative (chemistry).read more
Citations
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Technical communique: Mittag-Leffler stability of fractional order nonlinear dynamic systems
TL;DR: The definition of Mittag-Leffler stability is proposed and the fractional Lyapunov direct method is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional orders systems.
Journal ArticleDOI
Ergodic properties of fractional Brownian-Langevin motion
TL;DR: Fractional Brownian motion as a model for recent experiments of subdiffusion of mRNA in the cell is briefly discussed, and a comparison with the continuous-time random-walk model is made.
Journal ArticleDOI
A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems
TL;DR: In this article, a comparative analysis of integer-order derivative, constant-order fractional derivative and two types of variable order fractional derivatives in characterizing the memory property of complex systems is presented.
Journal ArticleDOI
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion
TL;DR: The time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on are studied.
Journal ArticleDOI
Stability analysis of Caputo fractional-order nonlinear systems revisited
TL;DR: An extension of Lyapunov direct method for fractional-order nonlinear systems using Bihari's and Bellman-Gronwall's inequality and a proof of comparison theorem are proposed in this article.
References
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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.
Journal ArticleDOI
Analysis of Fractional Differential Equations
Kai Diethelm,Neville J. Ford +1 more
TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.
Book
A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations
TL;DR: In this paper, an Adams-type predictor-corrector method for the numerical solution of fractional differential equations is discussed, which may be used both for linear and nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator) too.
Proceedings Article
Stability results for fractional differential equations with applications to control processing
TL;DR: In this article, stability results for finite-dimensional linear fractional differential systems in state-space form are given for both internal and external stability, and the main qualitative result is that stabilities are guaranteed iff the roots of some polynomial lie outside the closed angular sector.