A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses
The fractional oscillator equation with the sinusoidal excitation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msup><mi>x</mi><mo>″</mo></msup>...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-11-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/6/12/692 |
| _version_ | 1797458338590490624 |
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| author | Jun-Sheng Duan Yu-Jie Lan Ming Li |
| author_facet | Jun-Sheng Duan Yu-Jie Lan Ming Li |
| author_sort | Jun-Sheng Duan |
| collection | DOAJ |
| description | The fractional oscillator equation with the sinusoidal excitation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msup><mi>x</mi><mo>″</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mspace width="0.166667em"></mspace><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>k</mi><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>F</mi><mo form="prefix">sin</mo><mrow><mo>(</mo><mi>ω</mi><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>ω</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>W</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the response to excitation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is a superposition of three parts: the steady periodic oscillation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>W</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense. |
| first_indexed | 2024-03-09T16:35:40Z |
| format | Article |
| id | doaj.art-da1edd133cfa48f186a2d5f4cdf4df7b |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-09T16:35:40Z |
| publishDate | 2022-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-da1edd133cfa48f186a2d5f4cdf4df7b2023-11-24T14:57:01ZengMDPI AGFractal and Fractional2504-31102022-11-0161269210.3390/fractalfract6120692A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo SensesJun-Sheng Duan0Yu-Jie Lan1Ming Li2School of Sciences, Shanghai Institute of Technology, Shanghai 201418, ChinaSchool of Sciences, Shanghai Institute of Technology, Shanghai 201418, ChinaOcean College, Zhejiang University, Zhoushan 316021, ChinaThe fractional oscillator equation with the sinusoidal excitation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><msup><mi>x</mi><mo>″</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>b</mi><mspace width="0.166667em"></mspace><msubsup><mi>D</mi><mi>t</mi><mi>α</mi></msubsup><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>k</mi><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>F</mi><mo form="prefix">sin</mo><mrow><mo>(</mo><mi>ω</mi><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>ω</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>2</mn></mrow></semantics></math></inline-formula> is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>W</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>∪</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, the response to excitation, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is a superposition of three parts: the steady periodic oscillation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mi>W</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense.https://www.mdpi.com/2504-3110/6/12/692fractional calculusfractional oscillatorWeyl fractional derivativeCaputo fractional derivativeLaplace transform |
| spellingShingle | Jun-Sheng Duan Yu-Jie Lan Ming Li A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses Fractal and Fractional fractional calculus fractional oscillator Weyl fractional derivative Caputo fractional derivative Laplace transform |
| title | A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses |
| title_full | A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses |
| title_fullStr | A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses |
| title_full_unstemmed | A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses |
| title_short | A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses |
| title_sort | comparative study of responses of fractional oscillator to sinusoidal excitation in the weyl and caputo senses |
| topic | fractional calculus fractional oscillator Weyl fractional derivative Caputo fractional derivative Laplace transform |
| url | https://www.mdpi.com/2504-3110/6/12/692 |
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