Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type
The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-01-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/11/3/558 |
| _version_ | 1797623899513421824 |
|---|---|
| author | Valentine Aleksandrovich Kim Roman Ivanovich Parovik Zafar Ravshanovich Rakhmonov |
| author_facet | Valentine Aleksandrovich Kim Roman Ivanovich Parovik Zafar Ravshanovich Rakhmonov |
| author_sort | Valentine Aleksandrovich Kim |
| collection | DOAJ |
| description | The article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann–Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one. |
| first_indexed | 2024-03-11T09:35:25Z |
| format | Article |
| id | doaj.art-50651d6e14fb454f9c1f094de79c2803 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:35:25Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-50651d6e14fb454f9c1f094de79c28032023-11-16T17:21:19ZengMDPI AGMathematics2227-73902023-01-0111355810.3390/math11030558Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville TypeValentine Aleksandrovich Kim0Roman Ivanovich Parovik1Zafar Ravshanovich Rakhmonov2International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4, Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, RussiaInternational Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4, Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, RussiaFaculty of Applied Mathematics and Intelligent Technologies, National University of Uzbekistan Named after Mirzo Ulugbek, 4 Universitetskaya St., Tashkent 100174, UzbekistanThe article considers an implicit finite-difference scheme for the Duffing equation with a derivative of a fractional variable order of the Riemann–Liouville type. The issues of stability and convergence of an implicit finite-difference scheme are considered. Test examples are given to substantiate the theoretical results. Using the Runge rule, the results of the implicit scheme are compared with the results of the explicit scheme. Phase trajectories and oscillograms for a Duffing oscillator with a fractional derivative of variable order of the Riemann–Liouville type are constructed, chaotic modes are detected using the spectrum of maximum Lyapunov exponents and Poincare sections. Q-factor surfaces, amplitude-frequency and phase-frequency characteristics are constructed for the study of forced oscillations. The results of the study showed that the implicit finite-difference scheme shows more accurate results than the explicit one.https://www.mdpi.com/2227-7390/11/3/558Duffing oscillatorRunge ruleRiemann-Liouville operatorGrunwald-Letnikov operatoramplitude-frequency responsephase-frequency response |
| spellingShingle | Valentine Aleksandrovich Kim Roman Ivanovich Parovik Zafar Ravshanovich Rakhmonov Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type Mathematics Duffing oscillator Runge rule Riemann-Liouville operator Grunwald-Letnikov operator amplitude-frequency response phase-frequency response |
| title | Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type |
| title_full | Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type |
| title_fullStr | Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type |
| title_full_unstemmed | Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type |
| title_short | Implicit Finite-Difference Scheme for a Duffing Oscillator with a Derivative of Variable Fractional Order of the Riemann-Liouville Type |
| title_sort | implicit finite difference scheme for a duffing oscillator with a derivative of variable fractional order of the riemann liouville type |
| topic | Duffing oscillator Runge rule Riemann-Liouville operator Grunwald-Letnikov operator amplitude-frequency response phase-frequency response |
| url | https://www.mdpi.com/2227-7390/11/3/558 |
| work_keys_str_mv | AT valentinealeksandrovichkim implicitfinitedifferenceschemeforaduffingoscillatorwithaderivativeofvariablefractionalorderoftheriemannliouvilletype AT romanivanovichparovik implicitfinitedifferenceschemeforaduffingoscillatorwithaderivativeofvariablefractionalorderoftheriemannliouvilletype AT zafarravshanovichrakhmonov implicitfinitedifferenceschemeforaduffingoscillatorwithaderivativeofvariablefractionalorderoftheriemannliouvilletype |