| ABSTRACT | 第7页 |
| 摘要 | 第8-15页 |
| List of Abbreviations and Notations | 第15-16页 |
| 1 Introduction | 第16-25页 |
| 1.1 Background and Significance of Study | 第16-21页 |
| 1.1.1 Classical Models for Fourth-Order ODEs | 第16-19页 |
| 1.1.2 History of Multiple Solutions | 第19-21页 |
| 1.2 Methods for Solving Fourth-Order ODEs | 第21-22页 |
| 1.3 Statement of the Problem and Motivation | 第22页 |
| 1.4 Objectives of the Study | 第22-23页 |
| 1.5 Organization of the Thesis | 第23-25页 |
| 2 An Overview of the Numerical Methods | 第25-36页 |
| 2.1 Traditional Discretization Methods for Differential Equations | 第25-29页 |
| 2.1.1 Finite Differences Method | 第25-26页 |
| 2.1.2 Finite Elements Method | 第26-27页 |
| 2.1.3 Boundary Element Method | 第27-28页 |
| 2.1.4 Eigenfunction Expansion Method | 第28-29页 |
| 2.2 Solving System of Polynomial Equations | 第29-33页 |
| 2.2.1 Newton's Method | 第31-32页 |
| 2.2.2 Homotopy Continuation Method | 第32-33页 |
| 2.3 Concepts from Functional Analysis | 第33-36页 |
| 3 Eigenfunction Expansion Descretization and Error Estimation | 第36-53页 |
| 3.1 Discretization of Fourth-Order ODEs by Eigenfunction Expansion Method | 第36-46页 |
| 3.1.1 Fourth-Order ODEs with Simply Supported Boundary Conditions | 第36-40页 |
| 3.1.2 Fourth-Order ODEs with Nonhomogeneous Simply Supported Boundary Con-ditions | 第40-42页 |
| 3.1.3 Fourth-Order ODEs with Cantilever Boundary Conditions | 第42-43页 |
| 3.1.4 Fourth-Order ODEs with Nonhomogeneous Cantilever Boundary Conditions | 第43-45页 |
| 3.1.5 Fourth-Order ODEs with Three-point Boundary Conditions | 第45-46页 |
| 3.2 Error Analysis of the Eigenfunction Expansion Method | 第46-50页 |
| 3.3 Numerical Experiments | 第50-53页 |
| 4 Extension Homotopy Method for Solving the EEM Descretized Problem | 第53-67页 |
| 4.1 Introduction | 第53-55页 |
| 4.2 Construction of the Extension Homotopy | 第55-57页 |
| 4.3 Filters for Removing Spurious Solutions | 第57-58页 |
| 4.3.1 The Finite Difference Filter | 第57-58页 |
| 4.3.2 The Newton Method Filter | 第58页 |
| 4.4 Numerical Experiments | 第58-65页 |
| 4.4.1 Efficiency of the Extension Homotopy | 第58-65页 |
| 4.5 Conclusions of this Chapter | 第65-67页 |
| 5 Symmetric Homotopy Method for Solving the EEM Descretized Problem | 第67-86页 |
| 5.1 Introduction | 第67-70页 |
| 5.2 Symmetry Group for the Solution Set of the Discretized Problem | 第70-74页 |
| 5.3 Construction of the Symmetric Homotopy | 第74-78页 |
| 5.3.1 The Symmetric Homotopy for Cubic Polynomial Nonlinearity | 第75-76页 |
| 5.3.2 The Additional Symmetry for Odd Cubic Nonlinearity | 第76页 |
| 5.3.3 The Symmetric Homotopy for Quintic Polynomial Nonlinearity | 第76-77页 |
| 5.3.4 The Additional Symmetry for Odd Quintic Nonlinearity | 第77-78页 |
| 5.4 Numerical Experiments | 第78-85页 |
| 5.5 Conclusions of this Chapter | 第85-86页 |
| 6 Summary and Further Research | 第86-89页 |
| 6.1 Summary | 第86-87页 |
| 6.2 Future Research | 第87-89页 |
| References | 第89-101页 |
| Published Academic Articles during PhD period | 第101-102页 |
| Acknowledgements | 第102-103页 |
| Author Information | 第103页 |
