Fractional Definite Integral

Ortigueira, Manuel Duarte; Machado, J.A.T.

doi:10.3390/fractalfract1010002

Cited by 15 publications

(13 citation statements)

References 15 publications

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“…Anti-derivative The operator that is simultaneously the left and right inverse of the derivative will be called anti-derivative. It will be used to compute the definite integral through the Barrow formula [20]. This should be not confused with the negative order derivative, that needs not to be inverse of a derivative.…”

Section: Glossary and Assumptionsmentioning

confidence: 99%

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Fractional Derivatives: The Perspective of System Theory

Ortigueira

Machado

2019

Mathematics

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Section: Glossary and Assumptionsmentioning

confidence: 99%

“…We change also the nomenclature, using left for forward and right for backward. Therefore, (20) and (21)…”

Section: Third Unificationmentioning

confidence: 99%

Fractional Derivatives: The Perspective of System Theory

Ortigueira

Machado

2019

Mathematics

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“…One of them was, until recently, the non-existence of the definition of the "fractional definite integral". This gap was filled by Ortigueira and Machado [5]. Here, we tried to fill in another gap, by introducing a definition of the fractional line integral.…”

Section: Introductionmentioning

confidence: 99%

“…With the integral introduced here, the Green theorem, for example, can be generalised. Here, we took advantage of the results stated in [5] to propose a fractional line integral. We used directional derivatives.…”

Section: Introductionmentioning

confidence: 99%

Fractional Line Integral

Bengochea

Ortigueira

2021

Mathematics

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This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

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“…A new generalization for this definition can be found in [14]. In the rest of this paper, we use the Caputo idea of the fractional derivative.…”

Section: Introductionmentioning

confidence: 99%

Exponential form for Lyapunov function and stability analysis of the fractional differential equations

Sene¹

2018

J. Math. Computer Sci.

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This paper deals with an exponential form for Lyapunov function, in perspective to analyze the Lyapunov characterization of the Mittag-Leffler stability and the asymptotic stability for the fractional differential equations. In addition, a new Lyapunov characterization of Mittag-Leffler stability for fractional differential equations will be introduced. The exponential form will be used to prove the Lyapunov characterization of several stability notions, used in fractional differential equations. In this paper, the Caputo fractional derivative operator will be used to do the studies.

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Fractional Definite Integral

Cited by 15 publications

References 15 publications

Fractional Derivatives: The Perspective of System Theory

Fractional Derivatives: The Perspective of System Theory

Fractional Line Integral

Exponential form for Lyapunov function and stability analysis of the fractional differential equations

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