Research Progresses of Wavelet Methods and Their Applications in Mechanics

LIU Xiaojing; ZHOU Youhe; WANG Jizeng

doi:10.21656/1000-0887.420388

Volume 43 Issue 1

Jan. 2022

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LIU Xiaojing, ZHOU Youhe, WANG Jizeng. Research Progresses of Wavelet Methods and Their Applications in Mechanics[J]. Applied Mathematics and Mechanics, 2022, 43(1): 1-13. doi: 10.21656/1000-0887.420388

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Research Progresses of Wavelet Methods and Their Applications in Mechanics

doi: 10.21656/1000-0887.420388

Key Laboratory of Mechanics on Disaster and Environment in Western China, the Ministry of Education of China, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, P.R.China

Received Date: 2021-12-09
Accepted Date: 2021-12-30
Rev Recd Date: 2021-12-29

Available Online: 2021-12-31

Publish Date: 2022-01-01

Abstract

Abstract

The wavelet theory shows very unique time-frequency localization and multi-resolution analysis ability in signal processing and function approximation. The wavelet basis function has excellent mathematical properties such as orthogonality, compactness, low-pass filtering and interpolation, which endows the wavelet analysis theory with great application potential in the fields of computational mathematics and computational mechanics, and creates new opportunities for breakthrough development in these fields. Since the 1990s, a large number of studies have proved that the numerical method based on the wavelet theory has very obvious advantages in solving differential equations, but at the same time, have exposed some limitations of numerical calculation application caused by the wavelet basis function itself and its unique approximation method. In order to promote the innovative application of the wavelet theory in the fields of computational mathematics and mechanics and provide researchers with a new research perspective, the development background of the wavelet analysis and the research history of methods based on the wavelet theory were reviewed, and the numerical method problems were emphasized and the research progresses made in recent years discussed. The conclusions and comments may provide a meaningful reference for the further development and improvement of quantitative mathematical solution methods based on the wavelet theory and applications in mechanics as well as solutions of a wide range of engineering problems.
- wavelet analysis,
- multiresolution analysis,
- mechanics problem,
- irregular domain,
- strong nonlinearity,
- high-order differential equation

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References

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