An Atlas of Functions

Oldham, Keit; Myland, Jan C.; Spanier, Jerome

doi:10.1007/978-0-387-48807-3

Cited by 330 publications

(256 citation statements)

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“…The function g(α) = (t + a) −α j is Lipschitz continuous for t > −a which leads to the Lipschitz continuity of g 5 (α) and g 6 (α) using assumption 2. For the functions g 3 (α) and g 4 (α), properties on the gamma and digamma functions along with the reflection formulas [20] are used to conclude about their Lipschitz continuity.…”

Section: Convergence Of the Algorithmmentioning

confidence: 99%

Parameters and fractional differentiation orders estimation for linear continuous-time non-commensurate fractional order systems

Belkhatir

Laleg‐Kirati

2018

Systems & Control Letters

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This paper proposes a two-stage estimation algorithm to solve the problem of joint estimation of the parameters and the fractional differentiation orders of a linear continuous-time fractional system with non-commensurate orders. The proposed algorithm combines the modulating functions and the first-order Newton methods. Sufficient conditions ensuring the convergence of the method are provided. An error analysis in the discrete case is performed. Moreover, the method is extended to the joint estimation of smooth unknown input and fractional differentiation orders. The performance of the proposed approach is illustrated with different numerical examples. Furthermore, a potential application of the algorithm is proposed which consists in the estimation of the differentiation orders of a fractional neurovascular model along with the neural activity considered as input for this model.

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Section: Convergence Of the Algorithmmentioning

confidence: 99%

Parameters and fractional differentiation orders estimation for linear continuous-time non-commensurate fractional order systems

Belkhatir

Laleg‐Kirati

2018

Systems & Control Letters

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“…During the two centuries since the method's discovery, linear regression has been thoroughly assimilated into the repertoire of the physical, chemical and social sciences and the method is discussed in a great many publications [see, for example, [3][4][5][6][7][8][9]. Linear least squares regression is the workhorse in the chemical laboratory which is used not only for determination of the slope and intercept of the regressed line, but, also, used to demonstrate a technique of removing bad data points from the fit when the residuals are analysed.…”

Section: Introductionmentioning

confidence: 99%

Linear regressions to which the standard formulas do not apply

Mashkina

Oldham

2015

ChemTexts

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“…The interested reader is referred to the references of Podlubny [15], Kilbas [6], Samko and al [18] for further information and applications. Other recent works on this theory can be founded in [14,13,18,17]. Moreover, the investigation for the existence solutions of fractional differential equations have been developed very quickly, see for instance [1,3,4,5].…”

Section: Introductionmentioning

confidence: 99%