A sixth-order wavelet integral collocation method for solving nonlinear boundary value problems in three dimensions

Hou, Zhichun; Weng, Jiong; Liu, Xiaojing; Zhou, Youhe; Wang, Jizeng

doi:10.1007/s10409-021-09039-x

Cited by 9 publications

(16 citation statements)

References 31 publications

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“…[140,141] Chen and colleagues established WFEMs using interval B-spline wavelets and effectively applied them to the dynamic and static analysis of beam, plate, and shell structures. [3,117,[118][119][120][121][122][123][124][125][126][127][128][129][130][131] Their research demonstrated that the WFEM has high precision and can yield usable results with fewer nodes.…”

Section: Wavelet Finite Element Methodsmentioning

confidence: 99%

“…In recent years, researchers have proposed a Wavelet Integral Collocation Method (WICM) for solving high-order differential equations. [125,127,129] The basic idea is to take the highest-order derivative appearing in the differential equation as the fundamental solution quantity. Then, using the wavelet integral method, the remaining lower-order derivatives and the unknown function in the equation are approximated.…”

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

“…For example [127,129] , consider the Poisson equation defined on a complex domain, Ω, whose boundary is shown in Figure 4 {…”

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

“…Currently, several approaches have been developed to address the handling of irregular domains in wavelet methods, including Wavelet FEM (WFEM), [115][116][117][118][119] VDM, [114,120,121] Interpolation WGM (IWGM), [122][123][124] and Wavelet Integral Collocation Method (WICM). [125][126][127][128]…”

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

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Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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The quantitative study of many physical problems ultimately boils down to solving various partial differential equations (PDEs). Wavelet analysis, known as the “mathematical microscope”, has been hailed for its excellent Multiresolution Analysis (MRA) capabilities and its basis functions that possess various desirable mathematical qualities such as orthogonality, compact support, low‐pass filtering, and interpolation. These properties make wavelets a powerful tool for efficiently solving these PDEs. Over the past 30 years, numerical methods such as wavelet Galerkin methods, wavelet collocation methods, wavelet finite element methods, and wavelet integral collocation methods have been proposed and successfully applied in the quantitative analysis of various physical problems. This article will start from the fundamental theory of wavelet MRA and provide a brief summary of the advantages and limitations of various numerical methods based on wavelet bases. The objective of this article is to assist researchers in choosing the appropriate numerical methodologies for their particular physical issues. Furthermore, it will explore prospective advancements in wavelet‐based techniques, offering valuable insights for researchers committed to enhancing wavelet numerical methods in the field of computational physics.

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Section: Wavelet Finite Element Methodsmentioning

confidence: 99%

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

“…For example [127,129] , consider the Poisson equation defined on a complex domain, Ω, whose boundary is shown in Figure 4 {…”

Section: Wavelet Integral Collocation Methodsmentioning

confidence: 99%

Section: Handling Of Irregular Regionsmentioning

confidence: 99%

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Application of Wavelet Methods in Computational Physics

Wang,

Liu,

Zhou

2024

Annalen der Physik

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“…Xu [21] used new reproducing kernel-based collocation method to get numerical solutions of BVPs. In [22], nonlinear boundary value problems in three dimensions was researched by using sixth-order wavelet integral collocation method. Feng [23] considered existence and multiplicity of positive solutions for a singular third-order three-point boundary value problem.…”

Section: Introductionmentioning

confidence: 99%

A new algorithm of boundary value problems based on improved wavelet basis and the reproducing kernel theory

Zhang,

Lin

2023

Math Methods in App Sciences

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Relying upon Legendre polynomials, an improved multiwavelet orthonormal basis is constructed in the reproducing kernel space. We study boundary value problems (BVPs) based on the improved wavelet orthonormal basis and construct the ‐best approximate solution. Convergence and the stability are discussed; this method has higher order convergence. The main advantage of the wavelet based our method is its efficiency and simple applicability for a variety of boundary conditions. Some numerical tests even including singular BVPs reveal the stability and efficiency of the new algorithm.

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