On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

Sierociuk, Dominik; Malesza, Wiktor; Macias, Michał

doi:10.1007/s00034-014-9895-1

Cited by 52 publications

(16 citation statements)

References 24 publications

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“…[7,23,138]. Especially, more and more mathematical physical equations have been solved by using computationally efficient numerical methods [73,95,124].…”

Section: Numerical Methods For Vo-fdesmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

259

138

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Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

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“…[7,23,138]. Especially, more and more mathematical physical equations have been solved by using computationally efficient numerical methods [73,95,124].…”

Section: Numerical Methods For Vo-fdesmentioning

confidence: 99%

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Sun

Chang

Zhang

et al. 2019

FCAA

259

138

View full text Add to dashboard Cite

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“…Several authors [36,47,48,[50][51][52][53][54][55] have defined the VO Grünwald-Letnikov (VO-GL) operator corresponding to the general-order variation α(t, τ ) as…”

Section: (A) Definitions Of Variable-order Differ-integral Operatorsmentioning

confidence: 99%

Applications of variable-order fractional operators: a review

2020

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Variable-order fractional operators were conceived and mathematically formalized only in recent years. The possibility of formulating evolutionary governing equations has led to the successful application of these operators to the modelling of complex real-world problems ranging from mechanics, to transport processes, to control theory, to biology. Variable-order fractional calculus (VO-FC) is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Recognizing this untapped potential, the scientific community has been intensively exploring applications of VO-FC to the modelling of engineering and physical systems. This review is intended to serve as a starting point for the reader interested in approaching this fascinating field. We provide a concise and comprehensive summary of the progress made in the development of VO-FC analytical and computational methods with application to the simulation of complex physical systems. More specifically, following a short introduction of the fundamental mathematical concepts, we present the topic of VO-FC from the point of view of practical applications in the context of scientific modelling.

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“…In such a case, the fractional variable order (FVO) operators have to be used. To give a deeper insight into FVO calculus, four switching schemes, equivalent to four definitions of FVO derivatives, are presented in [30][31][32]. The switching strategies introduced, given unambiguously, classify and identify ways of changing the order of derivatives (integrals).…”

Section: Introductionmentioning

confidence: 99%

MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm

Macias

Sierociuk

Malesza

2022

Sensors

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This paper is devoted to identifying parameters of fractional order noises with application to noises obtained from MEMS accelerometer. The analysis and parameters estimation will be based on the Triple Estimation algorithm, which can simultaneously estimate state, fractional order, and parameter estimates. The capability of the Triple Estimation algorithm to fractional noises estimation will be confirmed by the sets of numerical analyses for fractional constant and variable order systems with Gaussian noise input signal. For experimental data analysis, the MEMS sensor SparkFun MPU9250 Inertial Measurement Unit (IMU) was used with data obtained from the accelerometer in x, y and z-axes. The experimental results clearly show the existence of fractional noise in this MEMS’ noise, which can be essential information in the design of filtering algorithms, for example, in inertial navigation.

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On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

Cited by 52 publications

References 24 publications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications

Applications of variable-order fractional operators: a review

MEMS Accelerometer Noises Analysis Based on Triple Estimation Fractional Order Algorithm

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