Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem
In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a...
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-10-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2504-3110/7/10/739 |
| _version_ | 1797573778514903040 |
|---|---|
| author | Xinxin Sun Ailing Zhu Zhe Yin Pengfei Ji |
| author_facet | Xinxin Sun Ailing Zhu Zhe Yin Pengfei Ji |
| author_sort | Xinxin Sun |
| collection | DOAJ |
| description | In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a convergence order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>τ</mi><mo>+</mo><msup><mi>τ</mi><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>4</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The validity of the scheme is verified by numerical examples, the effects of fractional derivative order and damping coefficient on beam vibration are analyzed and the superiority of the fractional order model has been demonstrated by comparing with the traditional damping model. |
| first_indexed | 2024-03-10T21:13:50Z |
| format | Article |
| id | doaj.art-8451b59a12824379a39e518dab3fea3c |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T21:13:50Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-8451b59a12824379a39e518dab3fea3c2023-11-19T16:34:36ZengMDPI AGFractal and Fractional2504-31102023-10-0171073910.3390/fractalfract7100739Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration ProblemXinxin Sun0Ailing Zhu1Zhe Yin2Pengfei Ji3School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaSchool of Mathematics and Statistics, Shandong Normal University, Jinan 250358, ChinaIn this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a convergence order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>τ</mi><mo>+</mo><msup><mi>τ</mi><mrow><mn>3</mn><mo>−</mo><mi>α</mi></mrow></msup><mo>+</mo><msup><mi>h</mi><mn>4</mn></msup><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>. The validity of the scheme is verified by numerical examples, the effects of fractional derivative order and damping coefficient on beam vibration are analyzed and the superiority of the fractional order model has been demonstrated by comparing with the traditional damping model.https://www.mdpi.com/2504-3110/7/10/739Hermite finite element methoderror estimatetime fractional ordernumerical simulationdamping beam |
| spellingShingle | Xinxin Sun Ailing Zhu Zhe Yin Pengfei Ji Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem Fractal and Fractional Hermite finite element method error estimate time fractional order numerical simulation damping beam |
| title | Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem |
| title_full | Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem |
| title_fullStr | Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem |
| title_full_unstemmed | Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem |
| title_short | Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem |
| title_sort | hermite finite element method for time fractional order damping beam vibration problem |
| topic | Hermite finite element method error estimate time fractional order numerical simulation damping beam |
| url | https://www.mdpi.com/2504-3110/7/10/739 |
| work_keys_str_mv | AT xinxinsun hermitefiniteelementmethodfortimefractionalorderdampingbeamvibrationproblem AT ailingzhu hermitefiniteelementmethodfortimefractionalorderdampingbeamvibrationproblem AT zheyin hermitefiniteelementmethodfortimefractionalorderdampingbeamvibrationproblem AT pengfeiji hermitefiniteelementmethodfortimefractionalorderdampingbeamvibrationproblem |