This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic systems. The examples are carefully explained and compiled into an algorithm, each of which is presented independent of a specific programming language. Each chapter is rounded off with exercises.
Author(s): Donald Greenspan
Series: Physics Textbook
Edition: 1
Publisher: Wiley-VCH
Year: 2006
Language: English
Pages: 217
Numerical Solution of Ordinary Differential Equations for Classical, Relativistic and Nano Systems......Page 2
Contents......Page 8
Preface......Page 12
1.2 Euler’s Method......Page 14
1.3 Convergence of Euler’s Method*......Page 18
1.4 Remarks......Page 21
1.5 Exercises......Page 22
2.2 A Runge–Kutta Formula......Page 24
2.3 Higher-Order Runge–Kutta Formulas......Page 28
2.4 Kutta’s Fourth-Order Formula......Page 35
2.5 Kutta’s Formulas for Systems of First-Order Equations......Page 36
2.6 Kutta’s Formulas for Second-Order Differential Equations......Page 39
2.7 Application – The Nonlinear Pendulum......Page 41
2.8 Application – Impulsive Forces......Page 44
2.9 Exercises......Page 47
3.2 First-Order Problems......Page 50
3.3 Systems of First-Order Equations......Page 53
3.4 Second-Order Initial Value Problems......Page 54
3.5 Application – The van der Pol Oscillator......Page 56
3.6 Exercises......Page 58
4.2 The N-Body Problem......Page 62
4.3 Classical Molecular Potentials......Page 63
4.5 The Leap Frog Formulas......Page 65
4.6 Equations of Motion for Argon Vapor......Page 66
4.7 A Cavity Problem......Page 67
4.9 Examples of Primary Vortex Generation......Page 69
4.10 Examples of Turbulent Flow......Page 72
4.11 Remark......Page 74
4.12 Molecular Formulas for Air......Page 75
4.13 A Cavity Problem......Page 76
4.14 Initial Data......Page 77
4.15 Examples of Primary Vortex Generation......Page 78
4.16 Turbulent Flow......Page 79
4.17 Colliding Microdrops of Water Vapor......Page 83
4.18 Remarks......Page 85
4.19 Exercises......Page 87
5.2 Mathematical Considerations......Page 90
5.3 Numerical Methodology......Page 91
5.4 Conservation Laws......Page 92
5.5 Covariance......Page 95
5.6 Application – A Spinning Top on a Smooth Horizontal Plane......Page 98
5.7 Application – Calogero and Toda Hamiltonian Systems......Page 116
5.8 Remarks......Page 121
5.9 Exercises......Page 122
6.2 Instability Analysis......Page 124
6.3 Numerical Solution of Mildly Nonlinear Autonomous Systems......Page 135
6.4 Exercises......Page 143
7.2 Tridiagonal Systems......Page 146
7.3 The Direct Method......Page 149
7.4 The Newton–Lieberstein Method......Page 150
7.5 Exercises......Page 153
8.2 Approximate Differentiation......Page 156
8.3 Numerical Solution of Boundary Value Problems Using Difference Equations......Page 157
8.4 Upwind Differencing......Page 161
8.5 Mildly Nonlinear Boundary Value Problems......Page 163
8.6 Theoretical Support*......Page 165
8.7 Application – Approximation of Airy Functions......Page 168
8.8 Exercises......Page 169
9.1 Introduction......Page 172
9.2 Inertial Frames......Page 173
9.4 Rod Contraction and Time Dilation......Page 174
9.6 Covariance......Page 176
9.7 Particle Motion......Page 178
9.8 Numerical Methodology......Page 179
9.9 Relativistic Harmonic Oscillation......Page 182
9.10 Computational Covariance......Page 183
9.11 Remarks......Page 187
9.12 Exercises......Page 188
10.2 Solving Boundary Value Problems by Initial Value Techniques......Page 190
10.3 Solving Initial Value Problems by Boundary Value Techniques......Page 191
10.4 Predictor-CorrectorMethods......Page 192
10.6 Other Methods......Page 193
10.7 Consistency*......Page 194
10.8 Differential Eigenvalue Problems......Page 195
10.10 Contact Mechanics......Page 197
Appendix A Basic Matrix Operations......Page 200
Solutions to Selected Exercises......Page 204
References......Page 210
Index......Page 216
