Bilateral Tempered Fractional Derivatives

Ortigueira, Manuel Duarte; Bengochea, Gabriel

doi:10.3390/sym13050823

Cited by 9 publications

(6 citation statements)

References 37 publications

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“…Therefore, it would be important to make an evaluation of the quality of the interconversions. As a continuation, it would be interesting to study the interconversions of causal tempered systems [61], as well as the ones based on two-sided derivatives [38,44,62,63].…”

Section: Discussionmentioning

confidence: 99%

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

Ortigueira¹,

Magin

2022

Fractal Fract

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The equivalence of continuous-/discrete-time autoregressive-moving average (ARMA) systems is considered in this paper. For the integer-order cases, the interrelations between systems defined by continuous-time (CT) differential and discrete-time (DT) difference equations are found, leading to formulae relating partial fractions of the continuous and discrete transfer functions. Simple transformations are presented to allow interconversions between both systems, recovering formulae obtained with the impulse invariant method. These transformations are also used to formulate a covariance equivalence. The spectral correspondence implied by the bilinear (Tustin) transformation is used to study the equivalence between the two types of systems. The general fractional CT/DT ARMA systems are also studied by considering two DT differential fractional autoregressive-moving average (FARMA) systems based on the nabla/delta and bilinear derivatives. The interrelations CT/DT are also considered, paying special attention to the systems defined by the bilinear derivatives.

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Section: Discussionmentioning

confidence: 99%

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

Ortigueira¹,

Magin

2022

Fractal Fract

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“…The bilateral tempered fractional differences are somehow more involved. They can be obtained from the results introduced in [112]. We will not do it here.…”

Section: Definitionmentioning

confidence: 99%

Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms

Ortigueira

2023

Fractal Fract

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Differences are introduced as outputs of linear systems called differencers, being considered two classes: shift and scale-invariant. Several types are presented, namely: nabla and delta, bilateral, tempered, bilinear, stretching, and shrinking. Both continuous and discrete-time differences are described. ARMA-type systems based on differencers are introduced and exemplified. In passing, the incorrectness of the usual delta difference is shown.

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“…[16][17][18] Tempered fractional calculus is so important that it has been rediscovered at least twice under different names, as generalised proportional fractional calculus 19 and as substantial fractional calculus. [20][21][22] Other variants of fractional calculus using the same tempered concept have also been studied, such as tempered versions of the Riesz derivative and fractional Laplacian, [23][24][25] which are of great interest to mathematicians.…”

Section: Introductionmentioning

confidence: 99%

On tempered fractional calculus with respect to functions and the associated fractional differential equations

Mali

Kucche

Fernandez

et al. 2022

Math Methods in App Sciences

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The prime aim of the present paper is to continue developing the theory of tempered fractional integrals and derivatives of a function with respect to another function. This theory combines the tempered fractional calculus with the Ψ-fractional calculus, both of which have found applications in topics including continuous time random walks. After studying the basic theory of the Ψ-tempered operators, we prove mean value theorems and Taylor's theorems for both Riemann-Liouville-type and Caputo-type cases of these operators. Furthermore, we study some non-linear fractional differential equations involving Ψ-tempered derivatives, proving existence-uniqueness theorems by using the Banach contraction principle and proving stability results by using Grönwall type inequalities.

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Bilateral Tempered Fractional Derivatives

Abstract: The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative.

Cited by 9 publications

References 37 publications

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

On the Equivalence between Integer- and Fractional Order-Models of Continuous-Time and Discrete-Time ARMA Systems

Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms

On tempered fractional calculus with respect to functions and the associated fractional differential equations

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