A New Look at the Initial Condition Problem
Abstract
:1. Introduction
- Let . The analytic function defined by
- The bilateral Laplace transform is defined by
2. On the Fractional Derivatives
3. Some Myths and Contradictions of Fractional Calculus
3.1. Methaphysique Derivatives
- Riemann-Liouville derivative
- Caputo derivativeThis case must be studied with care. If the integration starts at , as usually done,The derivative of the unit step being zero is a negative result that was used in [50], to show that the Caputo derivative is useless for modelling circuits with fractional capacitors, since the results are contradicted by laboratory experiments.
3.2. RL and C Initial Conditions
3.2.1. Incoherences
3.2.2. Outfit Results
- The CMLF solves Equation (22) for the RL derivative,
- The natural IC is which originates the appearence of the term in contradiction with Equation (16),
- Relation Equation (23) can be written as
- We need two IC, independently of or not.
- Instead of the IC used in the previous sub-section, we need and , for both RL and C derivatives.
4. Redefining the Problem
4.1. Systems and Differential Equations
4.2. or ?
4.3. Another Look at the IC of the Integer Order Systems
4.4. Fractional Order Systems
4.5. From the Observable Canonical Form
4.6. A Bucket of Cold Water?
- Acquisitions of past input and output in a given interval, we can
- Design a preditor by the Wiener–Hopf method of a Kalman filter [61],
- Use other functional extrapolation methods, such as polynomial or by splines.
- Sampling the input and output signals
- Use a discrete-time Kalman filter [61],
5. Conclusions
Funding
Conflicts of Interest
Abbreviations
ARMA | autoregressive-moving average |
BLT | bilateral Laplace transform |
C | Caputo |
CMLF | causal Mittag-Leffler function |
FARMA | fractional autoregressive-moving average |
FC | Fractional calculus |
GL | Grünwald–Letnikov |
IC | initial-conditions |
L | Liouville |
LC | Liouville–Caputo |
MLF | Mittag-Leffler function |
RL | Riemann-Liouville |
ULT | unilateral Laplace transform |
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Ortigueira, M.D. A New Look at the Initial Condition Problem. Mathematics 2022, 10, 1771. https://doi.org/10.3390/math10101771
Ortigueira MD. A New Look at the Initial Condition Problem. Mathematics. 2022; 10(10):1771. https://doi.org/10.3390/math10101771
Chicago/Turabian StyleOrtigueira, Manuel D. 2022. "A New Look at the Initial Condition Problem" Mathematics 10, no. 10: 1771. https://doi.org/10.3390/math10101771