具有多项式非线性项的四阶常微分方程的数值解法

摘要

本文主要研究常系数和变系数非线性四阶常微分方程多解计算的数值方法。主要包括关于微分方程的特征函数展开离散化方法和关于离散化方程组的同伦延拓法。
　　在第一部分中，对四阶常微分方程采用特征函数展开法进行离散，并分别得到关于离散误差的H1-范数估计和L2-范数估计。用数值算例验证特征函数展开法的收敛速度。
　　在第二部分中，从数值上寻找具立方非线性的四阶常微分方程的多解。对于用特征函数展开法得到的离散化方程组，构造了快速求其全部解的多项式同伦。此种同伦方法的思想是逐次求解自由度逐渐增加的离散化多项式方程组，前一步的解代入后一步的初始方程组，然后只需求解最后一个多项式即可得到后一步初始方程组的全部解，然后再用路径跟踪求得后一步目标方程组的全部解，由此形成一个递归过程来求最细离散水平上方程组的全部解。提出有限差分过滤子和牛顿过滤子以剔除离散化方程组解集中可能出现的伪解。这些过滤子基于误差估计。用数值例子验证所提同伦方法的效率。
　　最后，提出求解变系数的三次和五次非线性四阶常微分方程的对称同伦方法。对特征函数展开离散化方程组，分析离散化多项式方程组解集的对称性。基于这种对称性，选取简单的四阶常微分方程作为初始问题，并将其在特征子空间中离散。此种离散化子方程组很容易求解。然后，将这些子方程组按块组装作成初始方程组，构造对称同伦，求解目标离散化方程组。由于只需跟踪代表解路径，对称同伦可以节省计算量。应用牛顿过滤子以剔除可能的伪解。数值实验表明，所构造的对称同伦是高效的。

目录

声明

ABSTRACT

摘要

CONTENTS

List of Figures

List of Tables

List of Abbreviations and Notations

1 Introduction

1．1 Background and Significance of Study

1．1．1 Classical Models for Fourth-Order ODEs

1．1．2 History of Multiple Solutions

1．2 Methods for Solving Fourth-Order ODEs

1．3 Statement of the Problem and Motivation

1．4 Objectives of the Study

1．5 Organization of the Thesis

2 An Overview of the Numerical Methods

2．1 Traditional Discretization Methods for Differential Equations

2．1．1 Finite Differences Method

2．1．2 Finite Elements Method

2．1．3 Boundary Element Method

2．1．4 Eigenfunction Expansion Method

2．2 Solving System of Polynomial Equations

2．2．1 Newton’S Method

2．2．2 Homotopy Continuation Method

2．3 Concepts from Functional Analysis

3 Eiqention Expansion Uescretization and Error Estimation

3．1．1 Fourth-Order ODEs with Simply Supported Boundary Conditions

3．1．2 Fourth-Order ODEs with Nonhomogeneous Simply Supported Boundary Conditions

3．1．3 Fourth-Order ODEs with Cantilever Boundary Conditions

3．1．4 Fourth-Order ODEs with Nonhomogeneous Cantilever Boundary Conditions

3．1．5 Fourth-Order ODEs with Three-point Boundary Conditions

3．2 Error Analysis of the Eigenfunction Expansion Method

3．3 Numerical Experiments

4 Extension Homotopy Method for Solving the EEM Descretized Problem

4．1 Introduction

4．2 Construction of the Extension Homotopy

4．3 Filters for Removing Spurious Solutions

4．3．1 The Finite Difference Filter

4．3．2 The Newton Method Filter

4．4 Numerical Experiments

4．4．1 Efficiency of the Extension Homotopy

4．5 Conclusions of this Chapter

5 Symmetric Homotopy Method for Solving the EEM Descretized Problem

5．1 Introduction

5．2 Symmetry Group for the Solution Set of the Discretized Problem

5．3 Construction of the Symmetric Homotopy

5．3．1 The Symmetric Homotopy for Cubic Polynomial Nonlinearity

5．3．2 The Additional Symmetry for Odd Cubic Nonlinearity

5．3．3 The Symmetric Homotopy for Quintic Polynomial Nonlinearity

5．3．4 The Additional Symmetry for Odd Quintic Nonlinearity

5．4 Numerical Experiments

5．5 Conclusions of this Chapter

6 Summary and Further Research

6．1 Summary

6．2 Future Research

References

Published Academic Articles during PhD period

Acknowledgements

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