Boundary value problems for the diffusion equation of the variable order in differential and difference settings

Alikhanov, Anatoly A.

doi:10.1016/j.amc.2012.10.029

Cited by 44 publications

(32 citation statements)

References 17 publications

(29 reference statements)

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“…Those Lyapunov functions are quadratic (positive definite) and are accompanied by a new Lemma that allows proving that their fractional derivatives are negative semidefinite or definite so it can be used along with the fractional extension of the Lyapunov direct method to determine stability or asymptotical stability. A similar result can be found in Corollary 1 of [8] which is a consequence of an expression, introduced in that paper, for the Caputo derivative of the product of two functions.…”

Section: Hernández-martínez)supporting

confidence: 69%

“…The following Lemma, which generalizes Lemma 1 in [7] and Corollary 1 in [8], will subsequently allow us to extend Lyapunov type results for nonlinear and time-varying fractional order systems.…”

Section: A New Caputo Fractional Derivative Resultsmentioning

confidence: 96%

“…In the interesting articles [7,8], some relevant results for the study of fractional order systems are introduced. The following Lemma, which generalizes Lemma 1 in [7] and Corollary 1 in [8], will subsequently allow us to extend Lyapunov type results for nonlinear and time-varying fractional order systems.…”

Section: A New Caputo Fractional Derivative Resultsmentioning

confidence: 99%

“…This work gives a generalization of Lemma 1 of [7] and Corollary 1 of [8], that allows building a broader family of Lyapunov functions and determine stability in Caputo fractional systems. As a consequence of this extension, we give in Proposition 1 sufficient conditions for a Caputo fractional system to be stable or asymptotically stable.…”

Section: Discussionmentioning

confidence: 98%

“…Motivated by the previous articles, we extend the results of [7,8] by presenting a new Lemma for the Caputo fractional derivative, which enables us to build polynomial Lyapunov functions of greater order. We also give a similar but more general result than the Proposition 5 of [13].…”

Section: Hernández-martínez)mentioning

confidence: 95%

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