Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations
<p>Comparison between box-type and exponential enclosures of the Mittag-Leffler function <math display="inline"><semantics> <mrow> <msub> <mi>E</mi> <mrow> <mi>ν</mi> <mo>,</mo> <mn>1</mn> </mrow> </msub> <mfenced separators="" open="(" close=")"> <mrow> <mi>λ</mi> <msup> <mi>t</mi> <mi>ν</mi> </msup> </mrow> </mfenced> </mrow> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mn>0</mn> <mo>;</mo> <mspace width="0.277778em"/> <mn>1</mn> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <munder> <mi>t</mi> <mo>¯</mo> </munder> <mo>=</mo> <mi>k</mi> <mo>·</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mover> <mi>t</mi> <mo>¯</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>·</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>∈</mo> <mo>{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>9</mn> <mo>}</mo> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>;</mo> <mspace width="0.277778em"/> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> </semantics></math>: (<b>a</b>) Box-type enclosure of the Mittag-Leffler function; (<b>b</b>) Exponential enclosure of the Mittag-Leffler function; (<b>c</b>) Overestimation of the lower enclosure bound; (<b>d</b>) Overestimation of the upper enclosure bound.</p> "> Figure 2
<p>Evolution of the parameter <math display="inline"><semantics> <msup> <mi>η</mi> <mo>*</mo> </msup> </semantics></math> as a function of <math display="inline"><semantics> <msup> <mover accent="true"> <mi>t</mi> <mo>˜</mo> </mover> <mo>*</mo> </msup> </semantics></math> for different values of the fractional differentiation order <math display="inline"><semantics> <mi>ν</mi> </semantics></math>.</p> "> Figure 3
<p>Illustration of the two considered guaranteed enclosure methods for the quotient of two Mittag-Leffler functions with uncertain parameters <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>λ</mi> <mn>1</mn> </msub> </mfenced> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>∈</mo> <mfenced separators="" open="[" close="]"> <msub> <mi>λ</mi> <mn>2</mn> </msub> </mfenced> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>: (<b>a</b>) Box-type enclosures vs. exact range for the quotient (<a href="#FD36-fractalfract-06-00567" class="html-disp-formula">36</a>); (<b>b</b>) Exponential enclosures vs. exact range for the quotient (<a href="#FD36-fractalfract-06-00567" class="html-disp-formula">36</a>).</p> "> Figure 4
<p>Basic fractional-order equivalent circuit model of batteries according to [<a href="#B25-fractalfract-06-00567" class="html-bibr">25</a>].</p> "> Figure 5
<p>Use of box-type enclosures for the evaluation of the iteration Formula (<a href="#FD31-fractalfract-06-00567" class="html-disp-formula">31</a>) for the computation of guaranteed pseudo-state enclosures: (<b>a</b>) State of charge <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) State of charge <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) Voltage <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>d</b>) Voltage <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>e</b>) Interval diameter for the enclosure of <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>; (<b>f</b>) Interval diameter for the enclosure of <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 6
<p>Use of exponential enclosures, intersected with the box-type ones, for the evaluation of the iteration Formula (<a href="#FD34-fractalfract-06-00567" class="html-disp-formula">34</a>) for the computation of guaranteed pseudo-state enclosures: (<b>a</b>) State of charge <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>b</b>) State of charge <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>c</b>) Voltage <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>d</b>) Voltage <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>; (<b>e</b>) Interval diameter for the enclosure of <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>; (<b>f</b>) Interval diameter for the enclosure of <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p> "> Figure 7
<p>Comparison of the required number of iterations for both box-type and exponential enclosures: (<b>a</b>) Required iterations for the initialization <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (box-type enclosure); (<b>b</b>) Required iterations for the initialization <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (box-type enclosure); (<b>c</b>) Required iterations for the initialization <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (exponential enclosure); (<b>d</b>) Required iterations for the initialization <math display="inline"><semantics> <mrow> <mi mathvariant="bold">x</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mfenced open="[" close="]"> <mi mathvariant="bold">x</mi> </mfenced> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> (exponential enclosure).</p> ">
Abstract
:1. Introduction
2. Fundamentals of Verified Mittag-Leffler-Type Pseudo-State Enclosures for Fractional Differential Equations
2.1. System Models under Consideration
2.2. Linear Scalar System Models
2.3. Mittag-Leffler Functions as Pseudo-State Enclosures for Fractional-Order Differential Equations
3. Exponential Enclosures for Fractional-Order System Models
- The replacement of the solution representation given so far by Mittag-Leffler functions by exponential functions; or
- The introduction of exponential enclosures for the interval evaluation of the Mittag-Leffler function instead of the currently employed box-type representations.
3.1. Exponential Pseudo-State Enclosures
3.2. Exponential Enclosures of the Mittag-Leffler Function
- ;
- ;
- for ; and
- for .
3.3. Iterative Pseudo-State Enclosures for Box-Type and Exponential Representations of Mittag-Leffler Functions
4. Simulation Results
4.1. Simplified Fractional-Order Battery Model
4.2. Simulation with the Help of Box-Type Enclosures
4.3. Simulation with the Help of Exponential Enclosures
5. Conclusions and Outlook on Future Work
Funding
Data Availability Statement
Conflicts of Interest
References
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Rauh, A. Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations. Fractal Fract. 2022, 6, 567. https://doi.org/10.3390/fractalfract6100567
Rauh A. Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations. Fractal and Fractional. 2022; 6(10):567. https://doi.org/10.3390/fractalfract6100567
Chicago/Turabian StyleRauh, Andreas. 2022. "Exponential Enclosures for the Verified Simulation of Fractional-Order Differential Equations" Fractal and Fractional 6, no. 10: 567. https://doi.org/10.3390/fractalfract6100567