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Fractional Behaviours Modelling

Part of the Intelligent Systems, Control and Automation: Science and Engineering Book Series (volume 101)

DOI: 10.1007/978-3-030-96749-9

ISBN: 978-3-030-96748-2

Publisher: Springer International Publishing - Springer Journals

Mar 19, 2022

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Introduction

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 1 –4

[After a period of very intense progression and unity in the field of fractional calculus that lasted more than 40 years, different trends seem now to set in. Here one designates by fractional calculus all the approaches involving fractional differentiation and integration operators in...
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Power-Law Type Dynamic Behaviours

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 5 –12

[The designation “fractional behaviour” used in the title of this book is actually a shorthand for “long memoryMemory power-law type behaviours” which expresses well that the studied behaviours admit, in a limited frequency or time domain a power-law type response.]
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Fractional Order Models

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 13 –39

[To model power-law type behaviours, many practitioners naturally think to fractional models, models described by differential equations or partial derivatives involving non-integer order derivatives in place of classical derivatives.]
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Introduction of New Kernels

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 41 –53

[Drawback on the singularity of the kernel appearing in the definition of fractional derivatives and integrals (see the previous chapter) has led some authors to propose definitions based on non-singular kernelsNon-singular kernels.]
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Volterra Equation

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 55 –68

[In the field of automatic control, the literature abounds with papers based on a generalisation of the classical state space description denoted “fractional pseudo state space description”.]
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Non-linear Models

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 69 –96

[This chapter proposes another alternative to fractional models for fractional behaviours modelling.]
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Partial Differential Equations with Spatially Variable Coefficients

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 97 –117

[One more time, for all these reasons, alternative models must be found to model anomalous diffusion processes that exhibit power-law type long memoryMemory behaviours, as was done recently for fractional differential equations.]
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Conclusion

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

Pages: 119 –120

[The first lesson of this book is that we must not confuse fractional behaviour and fractional models.]

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