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Abstract Wavelet analysis is a mathematical branch developed in the past several decades, which is known as the so-called "numerical microscope". Wavelets have the unique mathematical property of multiresolution analysis. When using wavelet and scaling functions as basis, they have excellent mathematical characteristics of orthogonality, compactness, symmetry, low-pass filter, approximate linear phase and interpolation etc. These properties brought new opportunities to developing advanced numerical techniques on accurately and efficiently solving differential equations in nonlinear mechanics problems. Since the 1990s, numerical methods such as wavelet Galerkin method, wavelet collocation method, wavelet finite element method and wavelet boundary element method etc. have been constructed and successfully applied to the quantitative research of mechanical problems. Most importantly, wavelet analysis provides a totally new way to develop robust and adaptive methods for efficiently solving mechanical problems with large local gradients, and to propose closure algorithms to uniformly solving problems with strong nonlinearity. Problems with these two types of features are usually very difficult to deal with by using most traditional methods. Starting from the review of historical background and theory of multiresolution analysis, this review systematically discusses how specific mathematic property of the wavelets can merit high efficiency and accuracy of the wavelet-based method, and why the Coiflet-based method is a good choice in developing advanced numerical algorithms for solving nonlinear differential equations. Also, this paper analyzes the existing numerical methods related to wavelets and summarizes the advantages, disadvantages and possible development directions of wavelet based methods. Especially, this paper discusses the closed-form numerical algorithm based on the Coiflets for solving nonlinear mechanical problems in detail. An example on the shallow water equation demonstrates that such a method has the ability to capture major pattern characteristics of the solution even under very coarse space-time meshes. Our ultimate goal is to eventually provide a valuable reference to the further development of wavelet methods and their applications in various complex mechanics problems.
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Received: 30 March 2017
Published: 28 August 2017
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2017, Vol. 38 
