Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation
The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-10-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/4/188 |
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| author | Jun-Sheng Duan Di-Chen Hu Ming Li |
| author_facet | Jun-Sheng Duan Di-Chen Hu Ming Li |
| author_sort | Jun-Sheng Duan |
| collection | DOAJ |
| description | The impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> satisfying <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>. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of <i>t</i>, which converges rapidly for a small <i>t</i>. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large <i>t</i>. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small <i>t</i> and large <i>t</i> were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of <i>t</i>. The results include that of the integer-order cases, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo></mrow></semantics></math></inline-formula> 0, 1 and 2. |
| first_indexed | 2024-03-10T04:05:03Z |
| format | Article |
| id | doaj.art-23b19e814dfa40afb403f20977a59a46 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T04:05:03Z |
| publishDate | 2021-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-23b19e814dfa40afb403f20977a59a462023-11-23T08:23:30ZengMDPI AGFractal and Fractional2504-31102021-10-015418810.3390/fractalfract5040188Comparison of Two Different Analytical Forms of Response for Fractional Oscillation EquationJun-Sheng Duan0Di-Chen Hu1Ming Li2School of Sciences, Shanghai Institute of Technology, Shanghai 201418, ChinaSchool of Sciences, Shanghai Institute of Technology, Shanghai 201418, ChinaOcean College, Zhejiang University, Hangzhou 310012, ChinaThe impulse response of the fractional oscillation equation was investigated, where the damping term was characterized by means of the Riemann–Liouville fractional derivative with the order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> satisfying <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>. Two different analytical forms of the response were obtained by using the two different methods of inverse Laplace transform. The first analytical form is a series composed of positive powers of <i>t</i>, which converges rapidly for a small <i>t</i>. The second form is a sum of a damped harmonic oscillation with negative exponential amplitude and a decayed function in the form of an infinite integral, where the infinite integral converges rapidly for a large <i>t</i>. Furthermore, the Gauss–Laguerre quadrature formula was used for numerical calculation of the infinite integral to generate an analytical approximation to the response. The asymptotic behaviours for a small <i>t</i> and large <i>t</i> were obtained from the two forms of response. The second form provides more details for the response and is applicable for a larger range of <i>t</i>. The results include that of the integer-order cases, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo></mrow></semantics></math></inline-formula> 0, 1 and 2.https://www.mdpi.com/2504-3110/5/4/188fractional calculusfractional oscillatorfractional differential equationimpulse responseLaplace transform |
| spellingShingle | Jun-Sheng Duan Di-Chen Hu Ming Li Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation Fractal and Fractional fractional calculus fractional oscillator fractional differential equation impulse response Laplace transform |
| title | Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation |
| title_full | Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation |
| title_fullStr | Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation |
| title_full_unstemmed | Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation |
| title_short | Comparison of Two Different Analytical Forms of Response for Fractional Oscillation Equation |
| title_sort | comparison of two different analytical forms of response for fractional oscillation equation |
| topic | fractional calculus fractional oscillator fractional differential equation impulse response Laplace transform |
| url | https://www.mdpi.com/2504-3110/5/4/188 |
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