Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions
<p>The <math display="inline"><semantics> <mi>Γ</mi> </semantics></math> path considered for impulse response computation of <math display="inline"><semantics> <mrow> <mi>h</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 2
<p>Function <math display="inline"><semantics> <mrow> <mi>μ</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 3
<p>Spatial representation of the diffusive part of a fractional model.</p> "> Figure 4
<p>Frequency response of <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>.</p> "> Figure 5
<p>Various representations of fractional models exhibiting the same problem: infinite memory.</p> ">
Abstract
:1. Introduction
- -
- infinite as they are distributed
- -
- and infinite as they are defined (the parameter distribution) in an infinite spatial domain.
2. A physical and Systemic Analysis of Fractional Models
2.1. Model Definition
2.2. Poles and Time Constants Distribution and Infinite Memory
2.3. Spatial Definition and Infinite Memory
3. Other Modelling Solutions
3.1. Kernels with Limited Memory
3.2. Volterra Integro-Differential Equations
3.3. Time Delay Models
3.4. Nonlinear Models
3.5. Time-Varying Models
3.6. Diffusion Equation with Spatially Variable Coefficients
4. Concluding Remarks
Funding
Conflicts of Interest
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Sabatier, J. Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions. Symmetry 2021, 13, 1099. https://doi.org/10.3390/sym13061099
Sabatier J. Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions. Symmetry. 2021; 13(6):1099. https://doi.org/10.3390/sym13061099
Chicago/Turabian StyleSabatier, Jocelyn. 2021. "Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions" Symmetry 13, no. 6: 1099. https://doi.org/10.3390/sym13061099