2019
DOI: 10.1007/s10915-019-00935-0
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Fractional Sensitivity Equation Method: Application to Fractional Model Construction

Abstract: Fractional differential equations provide a tractable mathematical framework to describe anomalous behavior in complex physical systems, yet they introduce new sensitive model parameters, i.e. derivative orders, in addition to model coefficients. We formulate a sensitivity analysis of fractional models by developing a fractional sensitivity equation method. We obtain the adjoint fractional sensitivity equations, in which we present a fractional operator associated with logarithmic-power law kernel. We further … Show more

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Cited by 21 publications
(17 citation statements)
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“…Phase‐field models derived from free‐energy potentials with nonlocal effects were first discussed by Giacomin and Lebowitz 54,55 with recent contributions from Abels et al, 56 and Ainsworth and Mao 53 . Fractional‐order models for structural analysis have been developed, 57 to which corresponding fractional uncertainty/sensitivity analyses can be formulated via operator‐based uncertainty quantification 58 and fractional sensitivity equation method 59 …”
Section: Numerical Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…Phase‐field models derived from free‐energy potentials with nonlocal effects were first discussed by Giacomin and Lebowitz 54,55 with recent contributions from Abels et al, 56 and Ainsworth and Mao 53 . Fractional‐order models for structural analysis have been developed, 57 to which corresponding fractional uncertainty/sensitivity analyses can be formulated via operator‐based uncertainty quantification 58 and fractional sensitivity equation method 59 …”
Section: Numerical Resultsmentioning
confidence: 99%
“…53 Fractional-order models for structural analysis have been developed, 57 to which corresponding fractional uncertainty/sensitivity analyses can be formulated via operator-based uncertainty quantification 58 and fractional sensitivity equation method. 59…”
Section: F I G U R E 17mentioning
confidence: 99%
“…Fractional calculus introduces well-established mathematical tools for an accurate description of anomalous phenomena, ubiquitous in a wide range of applications from bio-tissues (Ionescu et al 2017;Naghibolhosseini & Long 2018) and material science (Meral, Royston & Magin 2010;Suzuki & Zayernouri 2021;Suzuki et al 2021a) to vibration (Suzuki et al 2021b), porous media (Xie & Fang 2019;Zaky, Hendy & Macías-Díaz 2020;Samiee et al 2020b). As an alternative approach to standard methods, they leverage their inherent potentials in representing long-range interactions, self-similar structures, sharp peaks and memory effects in a variety of applications (see Kharazmi & Zayernouri 2019;Burkovska, Glusa & D'Elia 2020). This potential is substantially indicated by power-law or logarithmic kernels of convolution type in the corresponding fractional operators.…”
Section: Preliminaries On Tempered Fractional Calculusmentioning
confidence: 99%
“…They typically convert the problem into an optimization problem, and then, formulate a suitable estimator by minimizing an objective function, see e.g. 715 . Here, we overcome this difficulty by developing physics-informed neural networks (PINNs) both for integer-order and fractional-order models 16 .…”
mentioning
confidence: 99%