Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours
Abstract
:1. Introduction
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- to illustrate these drawbacks and to show that alternative solutions exist for power law-type long memory behaviours modelling;
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- to clarify the limits and benefits of fractional models.
2. Power Law-Type Long Memory Behaviours
2.1. Spectral Density and Autocorrelation Functions of the Input Output Signals of an LTI System
2.2. Power Law Concept Extended to LTI Systems
- Its impulse response slowly decays with respect to time according to:
- For a white noise input of variance , its output autocorrelation function is:
- For a white noise input of variance , its output power spectral density is:
3. Sand Heap Growth: An Example of Power Law-Type Long Memory Behaviour
3.1. System Description
3.2. Fractional Modelling of the Sand Pile Growth
4. Drawbacks of Fractional Modelling
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- the temperature measure is done at the point where the heat flux is applied; or
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- an infinite dimension medium is considered.
5. Another Possible Model
6. Beyond Fractional Models
6.1. Some Classes of Non-Linear Models
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- With the RSA process (as for the sand pile process), if the flow is stopped, then the surface filling stops. If the flow restarts, the surface filling restarts from the same state. Such behaviour cannot be reproduced with a classical linear fractional model whose output relaxes for a null input.
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- With a fractional modelling approach, an infinite dimensional model is obtained, requiring the entire model past knowledge for a proper initialisation. However, in practice, such knowledge is not required. Initialisation of the RSA process only requires the knowledge of the density and a uniform distribution of the disks on the surface. Exact knowledge of the position of all the disks on the surface is not necessary, and thus not all the process history is required.
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- It permits an accurate fitting of the RSA process kinetic in spite of its power law behaviour;
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- It takes into account some non-linear behaviours in relation to the flow of incoming disks (or particles for the case of a real adsorption process);
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- Its state is only of one dimension, and its initialisation only requires knowledge of the covered density;
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- Its implementation does not require any approximation step.
6.2. Distributed Time Delay Models
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- In Relation (43), the variable can be viewed as a real state and a physical meaning can be associated to it;
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- There is no longer any ambiguity in the operator used for the definition of Relation (43) (in Equation (32), Caputo’s, Riemann–Liouville, or another can be chosen);
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- Kernel in Relation (43) is not singular, unlike the definition of fractional derivative in Equation (32);
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- The memory of Model (43) is of finite length;
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- Initialisation of Model (43) requires knowledge of its state on a finite length and is well defined.
6.3. First Kind Volterra Equations
- Adapting the kernel in Relation (48) (see also [29], it is possible to produce, with the same kind of equation, power law behaviours of various types (denoted explicit, implicit), but also many other long memory behaviours;
- In Relation (48), if , is a matrix of kernels such that , thus permitting great flexibility in the tuning of Relation (48). The case comes closer to the non-commensurate fractional pseudo-state space representation case, but it should be remembered that physical interpretations invalidate this kind of model [18].
7. Conclusions
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- Even if fractional models permit an accurate fitting of power law-type input–output behaviours, they can give birth to disconnected issues of the system considered (initialisation, dimension, interpretations, …)
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- simpler more physical models can be obtained if we try to understand the physical origin of the behaviour.
Author Contributions
Funding
Conflicts of Interest
References
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Sabatier, J.; Farges, C.; Tartaglione, V. Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours. Mathematics 2020, 8, 196. https://doi.org/10.3390/math8020196
Sabatier J, Farges C, Tartaglione V. Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours. Mathematics. 2020; 8(2):196. https://doi.org/10.3390/math8020196
Chicago/Turabian StyleSabatier, Jocelyn, Christophe Farges, and Vincent Tartaglione. 2020. "Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours" Mathematics 8, no. 2: 196. https://doi.org/10.3390/math8020196
APA StyleSabatier, J., Farges, C., & Tartaglione, V. (2020). Some Alternative Solutions to Fractional Models for Modelling Power Law Type Long Memory Behaviours. Mathematics, 8(2), 196. https://doi.org/10.3390/math8020196