## Table of contents for Numerical solution of ordinary differential equations : for classical, relativistic and nano systems / Donald Greenspan.

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```1       Euler's Method  1
11      Introduction  1
1.2     Euler's Method  I
1.3     Convergence of Euler's Method* 5
1.4     Remarks 8
"1.5    Exercises 9
2       Runge-Kutta Methods 11
2.1     Introduction  11
2.2     A Runge-Kutta Formula 11
2.3     Higher-Order Runge-Kutta Formulas 15
2.4      Kutta's Fourth-Order Formula 22
25      Kutta's Formulas for Systems of First-Order Equations 23
2.6     Kutta's Formulas for Second-Order Differential Equations 26
2.7      Application - The Nonlinear Pendulum  28
2.8      Application - Impulsive Forces 31
2.9     Exercises 34
3       The Method of Taylor Expansions 37
3.1     Introduction  37
3.2     First-Order Problems 37
33      Systems of First-Order Equations 40
3.4     Second-Order Initial Value Problems 41
3.5     Application- The van der Pol Oscillator 43
3.6     Exercises 45
4       Large Second-Order Systems with Application to Nano Systems 49
4.1     Introduction  49
4.2     The N-Body Problem  49
4 3     Classical Molecular Potentials 50
4.4     Molecular Mechanics 52
4.5     The Leap Frog Formulas 52
4.6     Equations of Motion for Argon Vapor 53
4.7     A Cavity Problem  54
4.8     Computational Considerations 56
4.9     Examples of Primary Vortex Generation  56
4.10    Examples of Turbulent Flow  59
4.11    Remark   61
4.12    Molecular Formulas for Air 62
4.13    A Cavity Problem  63
4.14    Initial Data 64
4.15    Examples of Primary Vortex Generation  65
4.16    Turbulent Flow  66
4.17    Colliding Microdrops of Water Vapor 70
4.18    Remarks 72
4.19    Exercises 74
5       Completely Conservative, Covariant Numerical Methodology 77
5.1    Introduction  77
5.2     Mathematical Considerations 77
5.3     Numerical Methodology  78
5 4     Conservation Laws 79
5.5     Covariance 82
5.6     Application -A Spinning Top on a Smooth Horizontal Plane 85
5.7     Application - Calogero and Toda Hamiltonian Systems 103
5 8     Remarks 108
5.9     Exercises 109
6       nstabity  111
6.1     Introduction  I11
6.2    Instability Analysis 111
6.3     Numerical Solution of Mildly Nonlinear Autonomous Systems  122
6.4     Exercises 130
7       Numerica Solution of Tindiagonal Linear Algebraic Systems and Related
Nonlinear Systems 133
7.1    Introduction  133
7.2     Tridiagonal Systems 133
"7.3    The Direct Method  136
7.4     The Newton-Lieberstein Method   137
"7.5    Exercises 140
8       Approximate Solution of Boundary Value Problems  143
81      Introduction  143
8.2     Approximate Differentiation  143
8.3     Numerical Solution of Boundary Value Problems Using Difference
Equations 144
8 4     Upwind Differencing  148
8.5     Mildly Nonlinear Boundary Value Problems 150
8.6     Theoretical Support* 152
8.7     Application - Approximation of Airy Functions 155
8.8     Exercises 156
9       Special Relativistic Motion  159
9.1     Introduction  159
92      InertialFrames  160
9.3     The Lorentz Transformation  161
9.4     Rod Contraction and Time Dilation  161
9.5     Relativistic Particle Motion. 163
9.6     Covariance  163
9.7     Particle Motion  165
9.8     Numerical Methodology   166
9.9     Relativistic Harmonic Oscillation  169
9.10    Computational Covariance  170
9.11    Remarks   174
9.12    Exercises 175
10      Specia Topics  177
101     Introduction  177
10 2    Solving Boundary Value Problems by Initial Value Techniques 177
10.3    Solving Initial Value Problems by Boundary Value Techniques 178
10.4    Predictor-Corrector Methods 179
10.5    Multistep Methods 180
10.6    Other Methods 180
10.7    Consistency* 181
10.8    Differential Eigenvalue Problems 182
109     Chaos* 184
10.10   Contact Mechanics 184

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Library of Congress subject headings for this publication: Differential equations Numerical solutions