| 摘要 | 第1-4页 |
| ABSTRACT | 第4-8页 |
| Chapter 1 Introduction | 第8-13页 |
| ·The development of spline functions to solve differential equations | 第8-9页 |
| ·The main work presented in this thesis | 第9-13页 |
| Chapter 2 Univariate Polynomail Spline Functions | 第13-21页 |
| ·Polynomial spline functions | 第13页 |
| ·Cubic spline functions and interpolation error analysis | 第13-15页 |
| ·Cubic spline functions and relations | 第13-14页 |
| ·Interpolation error of cubic splines | 第14-15页 |
| ·Quartic spline functions and interpolation error analysis | 第15-16页 |
| ·Quartic spline functions and relations | 第15-16页 |
| ·Interpolation error of quartic splines | 第16页 |
| ·Quintic polynomial spline functions and interpolation error analysis | 第16-18页 |
| ·Quintic polynomial spline functions and relations | 第16-17页 |
| ·Interpolation error of quintic splines | 第17-18页 |
| ·Sextic spline functions and interpolation error analysis | 第18-21页 |
| ·Sextic spline functions and relations | 第18-19页 |
| ·Interpolation error of sextic splines | 第19-21页 |
| Chapter 3 Splines Methods for Solving Singularly-Perturbed Boundary-Value Problems with Constant Coefficients | 第21-36页 |
| ·The quartic spline method | 第21-25页 |
| ·The method | 第21-22页 |
| ·Convergence analysis | 第22-25页 |
| ·The sextic splines method | 第25-29页 |
| ·The method | 第25-26页 |
| ·Convergence analysis | 第26-29页 |
| ·Numerical experiments | 第29-33页 |
| ·Discussion | 第33-36页 |
| Chapter 4 Polynomial Spline Approaches for Solving Second-Order Boundary-Value Problems with Neumann Conditions | 第36-45页 |
| ·The difference scheme at interior nodal points | 第36-38页 |
| ·The difference schemes on the boundary and near the boundary | 第38-40页 |
| ·Convergence analysis | 第40-43页 |
| ·Numerical examples and discussion | 第43-45页 |
| Chapter 5 An Unconditionally Stable Spline Difference Scheme of O(k~2 + h~4) for Solving the Second Order One-Dimensional Linear Hyperbolic Equations | 第45-56页 |
| ·The spline difference scheme | 第45-47页 |
| ·Stability analysis | 第47-48页 |
| ·Numerical examples and discussion | 第48-56页 |
| Chapter 6 A Semi-Discretization Method Based on Quartic Splines for Solving One-Space-Dimensional Hyperbolic Equations | 第56-63页 |
| ·Two schemes based on the C~3 quartic splines | 第56-60页 |
| ·Stability analysis | 第60-61页 |
| ·Numerical examples and discussion | 第61-63页 |
| Chapter 7 An Unconditionally Stable Spline Method for Solving Parabolic Equations | 第63-70页 |
| ·A quartic spline method | 第63-67页 |
| ·Stability analysis | 第67页 |
| ·Numerical examples | 第67-70页 |
| Chapter 8 A Spline Difference Method for the One-Dimensional Heat Equation with Nonlocal Boundary Conditions | 第70-75页 |
| ·Quartic splines based difference scheme | 第70-72页 |
| ·Difference schemes at the endpoints | 第72-73页 |
| ·Numerical examples | 第73-75页 |
| References | 第75-83页 |
| Publications of the Author | 第83-86页 |
| Acknowledgements | 第86页 |
