Wavelet Numerical Method and Its Applications in Nonlinear Problems (Engineering Applications of Computational Methods, 6)

✍ Scribed by You-He Zhou

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 494
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book summarizes the basic theory of wavelets and some related algorithms in an easy-to-understand language from the perspective of an engineer rather than a mathematician. In this book, the wavelet solution schemes are systematically established and introduced for solving general linear and nonlinear initial boundary value problems in engineering, including the technique of boundary extension in approximating interval-bounded functions, the calculation method for various connection coefficients, the single-point Gaussian integration method in calculating the coefficients of wavelet expansions and unique treatments on nonlinear terms in differential equations. At the same time, this book is supplemented by a large number of numerical examples to specifically explain procedures and characteristics of the method, as well as detailed treatments for specific problems. Different from most of the current monographs focusing on the basic theory of wavelets, it focuses on the use of wavelet-based numerical methods developed by the author over the years. Even for the necessary basic theory of wavelet in engineering applications, this book is based on the author’s own understanding in plain language, instead of a relatively difficult professional mathematical description. This book is very suitable for students, researchers and technical personnel who only want to need the minimal knowledge of wavelet method to solve specific problems in engineering.

✦ Table of Contents

Foreword
Preface
Acknowledgments
Contents
About the Author
1 Introduction
1.1 Brief Review of Solution Methods for Linear Systems
1.2 Origination of Nonlinear Science and Some Challenges
1.3 Main Solution Methods for Nonlinear Problems
1.3.1 Analytical Methods
1.3.2 Numerical Methods
1.3.3 Examples of Main Program of Solution Methods for Nonlinear Problems
1.4 Brief Review of Wavelet Methods
References
2 Mathematical Framework of Compactly Supported Orthogonal Wavelets
2.1 Essentials of Compactly Supported Orthogonal Wavelets
2.2 Conditions for Constructing an Orthogonal Wavelet
2.2.1 General Conditions on Filter Coefficients from Orthogonality
2.2.2 Properties on Moments of Scaling and Wavelet Functions
2.2.3 Generalized Gaussian Integral for Calculating Expansion Coefficients
2.3 Numerical Generation of Orthogonal Wavelets
2.3.1 Determination of Filter Coefficients
2.3.2 Generation of Scaling and Wavelet Functions
2.3.3 Examples of Compactly Supported Orthogonal Wavelets
2.3.4 Analysis for Decomposition and Reconstruction Calculations
2.4 Spectrum Characteristics of the Orthogonal Wavelets
2.4.1 Essentials of Spectrum Analysis
2.4.2 Spectrum Characteristics of Compactly Supported Orthogonal Wavelets
2.4.3 Spectrum Characteristics of Ideal Wavelets
2.4.4 Spectrum Characteristic of the Generalized Coiflets
2.5 Calculations for Derivatives, Integrations, and Connection Coefficients of the Orthogonal Base Scaling Functions
2.5.1 Calculation of Derivatives of Scaling Function
2.5.2 Calculation of Integral of Scaling Function
2.5.3 Calculation of Connection Coefficients
Appendix 2.1 Moment Relationships of Orthogonal Scaling Functions
Appendix 2.2 Moment Relationships of Scaling Functions of Coiflets
Appendix 2.3 Condition on Filer Coefficients from Vanishing Moments
References
3 Essentials to Solving Nonlinear Boundary-Value Problems
3.1 Governing Equations of 1D Nonlinear Boundary-Value Problems
3.2 Solution Methods from 1D Linear Ordinary Differential Equations
3.2.1 General Solutions of the Ordinary Differential Equations with Constant Coefficients
3.2.2 Solution Method of Homogeneous Linear Ordinary Differential Equations with Constant Coefficients
3.3 Essentials to Approximate Solutions in Mathematics
3.4 Closedness Concepts of Approximate Solutions for Nonlinear Boundary-Value Problems
3.4.1 Examples of Non-closed Solutions to Nonlinear Problems in Approximation
3.4.2 Nonlinear Problems with Non-integer Power Nonlinearity
3.4.3 Concepts of Strong Nonlinearity and Weak Nonlinearity
3.5 Closed Wavelet-Based Solution for Solving 1D Nonlinear Boundary-Value Problems
3.5.1 Expansion of Nonlinear Operator Terms
3.5.2 Wavelet-Based Solution Program of 1D Nonlinear Boundary-Value Problems
3.5.3 Some Discussions for the Wavelet-Based Approximate Solution Program
3.6 Wavelet Closed Solution Method for 2D and 3D Nonlinear Boundary-Value Problems
3.6.1 Two-Dimensional Generalized Coiflets
3.6.2 Three-Dimensional Generalized Coiflets
3.6.3 Closed Spatial Discretization for Initial-Boundary-Value Problems in 3D Space
3.6.4 Application Example—Closed Decomposition or Solution to the N–S Equations in Fluid Mechanics
References
4 Error Analysis and Boundary Extension
4.1 Error Estimation of Approximation of a Function
4.1.1 Truncation Error Analysis of the Wavelet-Based Approximation
4.1.2 Error Analysis to the Gaussian Integral for Decomposition Coefficients
4.2 Error Estimations in Other Applications of the Generalized Coiflets
4.2.1 Error Analysis to the Decomposition Coefficient of Derivatives
4.2.2 Error Analysis to Decomposition Coefficients of Nonlinear Functions
4.2.3 Error Analysis to the Approximation of Integral Functions
4.3 Boundary Extension Technology and Its Error Estimation
4.3.1 General Criterion for Boundary Extension Based on Error Analysis
4.3.2 Boundary Extension Arithmetic Using Lagrange Polynomial Functions
4.3.3 Numerical Test Examples of Approximations to a Known Function in a Finite Region
References
5 Wavelet-Based Solutions for Linear Boundary-Value Problems
5.1 One-Dimensional Boundary-Value Problems
5.1.1 The Wavelet Approximation Incorporating Boundary Extension
5.1.2 Galerkin-Wavelet Solution Program
5.1.3 Numerical Solution Results of 1D Poisson Equation
5.2 2D and 3D Boundary-Value Problems
5.2.1 Galerkin-Wavelet Solution Program
5.2.2 Numerical Solution Results of 2D Laplace and Poisson Problem
5.2.3 Numerical Solution Results of 3D Poisson Equation
5.3 Deflection of Thin Rectangular Plate with Variable Thickness
5.3.1 Differential Equation with Variable Coefficients
5.3.2 Wavelet-Based Solution Program
5.3.3 Numerical Solution Results
Appendix 5.1 Calculation of Connection Coefficients of Modified Basis Function
References
6 Wavelet-Based Laplace Transformation for Initial- and Boundary-Value Problems
6.1 Essentials of Laplace Transformation
6.1.1 Laplace Transform of a Function and its Inverse Transform
6.1.2 Laplace Transforms of Function Derivative and Integral
6.2 Wavelet-Based Laplace Transforms
6.2.1 Quantitative Spectrum Feature of Scaling Function Employed
6.2.2 Fourier Transform and Inverse Transform Based on Wavelet
6.2.3 Laplace Transform and Inverse Transform Based on Wavelet
6.3 Application Examples: Numerical Solution of A Fractionally Damped Dynamic System
6.3.1 Dynamic Equations with Fractional Damping
6.3.2 Numerical Solution for Nonlinear Fractional Dynamic System with Single–Degree-of-Freedom
6.3.3 Numerical Solution for the Multi-term Time-Fractional Diffusion-Wave Equation
6.3.4 Numerical Solution for Nonlinear Fractional Diffusion-Wave Equation
References
7 Wavelet-Based Solutions for Boundary-Value Problems
7.1 Expansion of Nonlinear Operator Equation in One Dimension and Error Estimations
7.2 Galerkin-Wavelet Solution Program
7.3 Numerical Solution Examples of Application to 1D Nonlinear Problems
7.3.1 Solution Results of 1D Bratu Equation with Exponential Nonlinearity
7.3.2 Solution Results of 1D Boundary-Value Problem with Sine Nonlinearity
7.3.3 Numerical Solution of 2D Bratu Equation
References
8 Space–Time Fully Decoupled Wavelet-Based Solution to Nonlinear Problems
8.1 Galerkin-Wavelet Solution Program
8.1.1 Spatial Discretization by Wavelet
8.1.2 Time Integration to the Induced Ordinary Differential Equations
8.1.3 Some Remarks
8.2 Numerical Solution to 1D Nonlinear Equations with Initial-Boundary-Value Conditions
8.2.1 1D Nonlinear Klein–Gordon Equation with Initial-Boundary-Value Conditions
8.2.2 Numerical Solution to 1D sine–Gordon Equations
8.2.3 Interaction Between Solitary Wave and Inclusion
8.3 Dynamic Control of Piezoelectric Thin Beam-Type Plates with Large Deflection
8.3.1 Governing Equations of Laminated Beam-Type Plates
8.3.2 Sensors and Actuators Designed by the Wavelet-Based Method
8.3.3 Simulation Results of Feedback Control with Piezoelectric Film Sensors and Actuators
8.4 Multidimensional Nonlinear Schrödinger Equations
8.4.1 Schrödinger’s Governing Equation
8.4.2 Solution of the Generalized Nonlinear Schrödinger Equation
8.4.3 Numerical Examples
References
9 Applications to Nonlinear Solid Mechanics
9.1 Large Deflection and Post-buckling Tracking of Beams
9.1.1 Strong Nonlinear Solution of Deflection in Post-buckling Path
9.1.2 Solution of Large Deflection of Flexible Beam with Immovably Simple Supports
9.1.3 Solution Results to Elastic Line Equation of Flexible Rod with Material Nonlinearity
9.2 Solution of Axisymmetric Deflection of Von Kármán Circular Plate with Strong Nonlinearity
9.2.1 Essential Equations with Two Coupled Unknown Functions
9.2.2 Wavelet-Based Solution Arithmetic
9.2.3 Numerical Results and Discussions
9.3 Wavelet-Based Solution to Von Kármán Equations of Rectangular Thin Plates
9.3.1 Essential Equations
9.3.2 Arithmetic of Wavelet-Based Solution Method
9.3.3 Examples of Numerical Solution
9.4 Solution to Other Nonlinear Problems of Beam and Plate Structures
9.4.1 Deflection of Rectangular Thin Plate with Third-Order Power Nonlinearity
9.4.2 Nonlinear Free Vibration of Beam with Immovably Simple Supports
9.4.3 Nonlinear Forced Vibration of Beam with Immovably Simple Supports
Appendix 9.1 Wavelet Numerical Integration Method with Modified Scaling Function
Appendix 9.2 An Analytical Solution of Elastic Line Equation of Flexible Rod
Appendix 9.3 Approximate Theoretical Analyses on the Nonlinear Free Vibration of Beams
References
10 Applications to Laminar Flows in Nonlinear Fluid Mechanics
10.1 Burgers’ Equation
10.1.1 Governing Equations with Initial-Boundary-Value Conditions
10.1.2 Ordinary Differential Equations Gained by Wavelet-Based Space Expansion
10.1.3 Solution Results to 1D and 2D Burgers’ Equations
10.2 2D Poiseuille and Couette Laminar Flow
10.2.1 Governing Equations with Initial-Boundary-Value Conditions
10.2.2 High-Order Splitting Methods for Incompressible Flows
10.2.3 Numerical Solution Results
10.3 2D Cavity Laminar Flow
10.3.1 Essential Equations in the Calculations
10.3.2 Numerical Solution Results
10.4 Strong Nonlinear Problem of 2D Vortex Emerging Interaction
10.4.1 Governing Equations for the Problem
10.4.2 Arithmetic of Wavelet-Based Solution Method
10.4.3 Results of Numerical Solution
References
11 Extended Wavelet Methods to 2D Irregular Domain and Local Refinement
11.1 Wavelet Multiresolution Approximation with Targeted Interpolation
11.1.1 Brief Overview of the Interpolating Wavelet Approximation
11.1.2 Wavelet Approximation of Functions Defined on Finite Domain
11.1.3 Modified Multiresolution Approximation Defined on a Finite Domain
11.1.4 Construction of the Wavelet Targeted Interpolation
11.1.5 Some Essential Attributes of the Wavelet Targeted Interpolation
11.2 Wavelet Multiresolution Solution to Elasticity Problems
11.2.1 Node Generation and Pre-Processing
11.2.2 Variational Formulation of Plane Elasticity Problems
11.2.3 Calculation of Stiffness Matrix
11.2.4 Error Analysis
11.3 Numerical Examples
11.3.1 Patch Test
11.3.2 Test of Convergence and Stability Against Irregular Nodes
11.3.3 Infinite Plate with a Circular Hole
11.3.4 Semi-infinite Plate Subjected to Concentrated Edge Load
11.3.5 Semi-infinite Plate Subjected to a Uniform Local Loading
11.3.6 Bridge Pier
11.3.7 Corner Brace
11.3.8 Automotive Wheel
11.3.9 Stress Intensity Factors (SIFs) of Shear Edge Crack
11.3.10 Crack Propagation in a Rectangular Plate
11.4 Summarized Remarks
Appendix 11.1 Proof of Essential Properties of Interpolating Wavelet
Appendix 11.2 Multiresolution Decomposition of Interpolating Wavelet
Appendix 11.3 Error Estimation of the Interpolating Wavelet Approximation Defined on the Whole Space
Appendix 11.4 Error Estimation of the Interpolating Wavelet Approximation Defined on a Finite Domain
Appendix 11.5 Proof of the Interpolating Property for the Modified Multiresolution Approximation
Appendix 11.6 Error Estimation of the Modified Interpolating Multiresolution Approximation
Appendix 11.7 Construction of the Targeted Interpolation Based on Interpolating Wavelet
References
12 Brief Introduction in Applications of Other Groups
12.1 Deep Improvement for Homotopy Analysis Method (HAM) by the Generalized Coiflets
12.2 Applications of the Generalized Coiflets and Relevant Method in Random Dynamic Problems
12.3 Applications of the Generalized Coiflets in Dynamic Control Systems and Others
References
Index