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