Mathematical Modeling of Linear Fractional Oscillators
Abstract
:1. Introduction
2. Preliminary Material on Fractional Calculus
3. Generalization of Newton’s Second Law
4. Formulation of the Problem
5. Solution Method
5.1. Explicit Non-Local Finite-Difference Scheme
- ,
- ,
- .
5.2. Fractional Adams–Bashforth–Moulton Method
5.3. Computational Accuracy Analysis
6. Forced Oscillations of Linear Fractional Oscillators
7. Research of Some Types of LFO
7.1. An Analogue of a Harmonic Oscillator
7.2. Fractional Oscillator Mathieu
7.3. Airy Fractional Oscillator
8. Discussion
9. Conclusions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
LHO | Linear Hereditary Oscillator |
ENFDS | Explicit Non-local Finite-Difference Scheme |
ABM | Adams–Bashforth–Moulton method |
ES | Exact Solution |
LFO | Linear Fractional Oscillators |
AFC | Amplitude–Frequency Characteristics |
PFC | Phase-Frequency Characteristics |
Q-factor | Quality Factor |
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N | h | ||||
---|---|---|---|---|---|
10 | 1/10 | 0.0561528 | 0.0039560 | - | - |
20 | 1/20 | 0.0340440 | 0.0009317 | 0.721956135 | 2.0861050 |
40 | 1/40 | 0.0195531 | 0.0002362 | 0.800003094 | 1.9798565 |
80 | 1/80 | 0.0108341 | 0.0000628 | 0.851815427 | 1.9100427 |
160 | 1/160 | 0.0058603 | 0.0000171 | 0.886538801 | 1.8748556 |
320 | 1/320 | 0.0031176 | 0.0000047 | 0.910518254 | 1.8577979 |
640 | 1/640 | 0.0016380 | 0.0000014 | 0.92851899 | 1.7975624 |
N | h | ||||
---|---|---|---|---|---|
10 | 1/10 | 0.025634 | 0.0011076 | - | - |
20 | 1/20 | 0.015479 | 0.0002546 | 0.727709917 | 2.120946173 |
40 | 1/40 | 0.008987 | 0.0000635 | 0.784362481 | 2.004652815 |
80 | 1/80 | 0.005045 | 0.0000167 | 0.832949245 | 1.923159876 |
160 | 1/160 | 0.002761 | 0.0000045 | 0.86953863 | 1.881091919 |
320 | 1/320 | 0.001484 | 0.0000013 | 0.895926947 | 1.860912083 |
640 | 1/640 | 0.000786 | 0.0000004 | 0.916645745 | 1.78891439 |
N | h | ||||
---|---|---|---|---|---|
10 | 1/10 | 0.00758 | 0.001049 | - | - |
20 | 1/20 | 0.00253 | 0.000232 | 1.585196 | 2.179457 |
40 | 1/40 | 0.00093 | 0.000054 | 1.445025 | 2.093444 |
80 | 1/80 | 0.00038 | 0.000013 | 1.296494 | 2.046768 |
160 | 1/160 | 0.00017 | 0.000003 | 1.177333 | 2.023954 |
320 | 1/320 | 0.00008 | 0.0000008007 | 1.100742 | 2.012096 |
640 | 1/640 | 0.00003739 | 0.0000001994 | 1.058277 | 2.005656 |
N | h | ||||
---|---|---|---|---|---|
10 | 1/10 | 0.0962543446 | 0.0974594278 | - | - |
20 | 1/20 | 0.0483529972 | 0.0450644425 | 0.993246342 | 1.112812209 |
40 | 1/40 | 0.0243287977 | 0.0216596566 | 0.990940293 | 1.056979173 |
80 | 1/80 | 0.0122257234 | 0.0106471450 | 0.992745193 | 1.024543742 |
160 | 1/160 | 0.0061384979 | 0.0052867956 | 0.993962258 | 1.010001173 |
320 | 1/320 | 0.0030783418 | 0.0026378633 | 0.995732241 | 1.003023747 |
640 | 1/640 | 0.0015461838 | 0.0013190516 | 0.993441601 | 0.9998688 |
N | h | ||||
---|---|---|---|---|---|
10 | 1/10 | 0.1220802888 | 0.0833919341 | - | - |
20 | 1/20 | 0.0632933442 | 0.0376498054 | 0.947704578 | 1.147265441 |
40 | 1/40 | 0.0326304379 | 0.0179210448 | 0.955835448 | 1.070987659 |
80 | 1/80 | 0.0167613673 | 0.0087578726 | 0.961078508 | 1.033002381 |
160 | 1/160 | 0.0085801061 | 0.0043373867 | 0.966072448 | 1.013754391 |
320 | 1/320 | 0.0043785606 | 0.0021615271 | 0.970538809 | 1.004775148 |
640 | 1/640 | 0.0022275871 | 0.0010803221 | 0.974974836 | 1.000589405 |
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Parovik, R. Mathematical Modeling of Linear Fractional Oscillators. Mathematics 2020, 8, 1879. https://doi.org/10.3390/math8111879
Parovik R. Mathematical Modeling of Linear Fractional Oscillators. Mathematics. 2020; 8(11):1879. https://doi.org/10.3390/math8111879
Chicago/Turabian StyleParovik, Roman. 2020. "Mathematical Modeling of Linear Fractional Oscillators" Mathematics 8, no. 11: 1879. https://doi.org/10.3390/math8111879
APA StyleParovik, R. (2020). Mathematical Modeling of Linear Fractional Oscillators. Mathematics, 8(11), 1879. https://doi.org/10.3390/math8111879