Two‐sided and regularised Riesz‐Feller derivatives

Ortigueira, Manuel Duarte

doi:10.1002/mma.5720

Cited by 34 publications

(40 citation statements)

References 31 publications

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“…Fractional derivatives here should be non-local and symmetric. The so-called two-sided fractional derivatives by [19] satisfy all these requirements and its special version, the Riesz derivative, was used in all bifurcation investigations.…”

Section: Discussionmentioning

confidence: 99%

Fractional Derivatives and Dynamical Systems in Material Instability

Béda

2020

Fractal Fract

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Loss of stability is studied extensively in nonlinear investigations, and classified as generic bifurcations. It requires regularity, being connected with non-locality. Such behavior comes from gradient terms in constitutive equations. Most fractional derivatives are non-local, thus by using them in defining strain, a non-local strain appears. In such a way, various versions of non-localities are obtained by using various types of fractional derivatives. The study aims for constitutive modeling via instability phenomena, that is, by observing the way of loss of stability of material, we can be informed about the proper form of its mathematical model.

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Section: Discussionmentioning

confidence: 99%

Fractional Derivatives and Dynamical Systems in Material Instability

Béda

2020

Fractal Fract

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“…Definition 1. In [33], we introduced formally a general two-sided fractional derivative (TSFD), 0 D β θ , through its Fourier transform…”

Section: The Two-sided Fractional Derivativesmentioning

confidence: 99%

“…The inverse Fourier transform computation of (2) is not important here (see, [33]). In Table 2 we present the most interesting definitions of the two-sided derivatives together with the corresponding Fourier transform.…”

Section: The Two-sided Fractional Derivativesmentioning

confidence: 99%

“…In this paper, we follow the work described in our previous paper [1] where a deep study on the tempered one-sided derivative was performed. Therefore, we intend here to enlarge the results we obtained previously by combining them with the two-sided derivatives studied in [33]. This approach intends to show that the TRD is not really a fractional derivative according to the criterion introduced in [34].…”

Section: Introductionmentioning

confidence: 97%

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Bilateral Tempered Fractional Derivatives

Ortigueira¹,

Bengochea²

2021

Preprint

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“…where α, β < 0. For α, β > 0, the integrals are singular and need to be regularized [62,63]. If 0 < α, β ≤ 1, then we can write…”

Section: On the 2d Fractional Derivativesmentioning

confidence: 99%

Revisiting the 1D and 2D Laplace Transforms

Ortigueira

Machado

2020

Mathematics

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The paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. The case of fractional-order systems is also included. General two-dimensional linear systems are introduced and the corresponding transfer function is defined.

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Two‐sided and regularised Riesz‐Feller derivatives

Abstract: The two-sided derivatives, the Riesz-Feller potentials, and their interrelations are studied in this paper. A general integral formulation for two-sided derivatives and anti-derivatives is introduced. The integer order cases are studied.Regularised Riesz-Feller derivatives are proposed.

Cited by 34 publications

References 31 publications

Fractional Derivatives and Dynamical Systems in Material Instability

Fractional Derivatives and Dynamical Systems in Material Instability

Bilateral Tempered Fractional Derivatives

Revisiting the 1D and 2D Laplace Transforms

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