Nonstandard finite difference models of differential equations

목차 (Table of Contents)

• CONTENTS
• 1. Introduction = 1
• 1.1 Numerical Integration = 1
• 1.2 Standard Finite-Difference Modeling Rules = 2
• 1.3 Examples = 4
• 1.4 Critique = 13
• References = 14
• 2. Numerical Instabilities = 17
• 2.1 Introduction = 17
• 2.2 Decay Equation = I8
• 2.3 Harmonic Oscillator = 29
• 2.4 Logistic Differential Equation = 35
• 2.5 Unidirectional Wave Equation = 51
• 2.6 Burgers' Equation = 58
• 2.7 Summary = 60
• References = 65
• 3. Nonstandard Finite- Difference Schemes = 68
• 3.1 Introduction = 68
• 3.2 Exact Finite-Difference Schemes = 70
• 3.3 Examples of Exact Schemes = 72
• 3.4 Nonstandard Modeling Rules = 81
• 3.5 Best Finite-Difference Schemes = 85
• References = 90
• 4. First-Order ODE's = 93
• 4.1 Introduction = 93
• 4.2 A New Finite-Difference Scheme = 94
• 4.3 Examples = 98
• 4.4 Nonstandard Schemes = 106
• 4.5 Discussion = 115
• References = 119
• 5. Second-Order, Nonlinear Oscillator Equations = 120
• 5.1 Introduction = 120
• 5.2 Mathematical Preliminaries = 122
• 5.3 Conservative Oscillators = 124
• 5.4 Limit-Cycle Oscillators = 132
• 5.5 General Oscillator Equations = 137
• 5.6 Response of a Linear System = 138
• References = 141
• 6. Two First-Order, Coupled Ordinary Differential Equations = 144
• 6.1 Introduction = 144
• 6.2 Background = 146
• 6.3 Exact Scheme for Linear Ordinary Differential Equations = 147
• 6.4 Nonlinear Equations = 150
• 6.5 Examples = 151
• 6.6 Summary =. 162
• References = 163
• 7. Partial Differential Equations = 165
• 7.1 Introduction = 165
• 7.2 Wave Equations = 166
• 7.3 Diffusion Equations = 173
• 7.4 Burgers' Type Equations = 182
• 7.5 Discussion = 188
• References = 189
• 8. Schr$$\ddot o$$dinger Differential Equations = 193
• 8.1 Introduction = 193
• 8.2 Schr$$\ddot o$$dinger Ordinary Differential Equations = 194
• 8.3 Schr$$\ddot o$$dinger Partial Differential Equations = 198
• References = 213
• 9. Summary and Discussion = 217
• 9.1 R$$\acute e$$sum$$\acute e$$ = 217
• 9.2 Nonstandard Modeling Rules Revisited = 219
• 9.3 Two Examples = 223
• 9.4 Future Directions = 229
• References = 230
• Appendix A: Difference Equations = 232
• Appendix B: Linear Stability Analysis = 236
• Appendix C: Discrete WKB Method = 239
• Bibliography = 241
• Index = 247