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.
Donald Greenspan is Professor of Mathematics at the University of Texas, where he received the Distinguished Research Award in 1983. An experienced lecturer, he has authored 200 papers and 14 books, many of them textbooks on computational mathematics. His assignments included positions at Harvard, Stanford, Berkeley and Princeton.
Preface. 1 Euler's Method. 1.1 Introduction. 1.2 Euler's Method. 1.3 Convergence of Euler's Method. 1.4 Remarks. 1.5 Exercises. 2 Runge-Kutta Methods. 2.1 Introduction. 2.2 A Runge-Kutta Formula. 2.3 Higher-Order Runge-Kutta Formulas. 2.4 Kutta's Fourth-Order Formula. 2.5 Kutta's Formulas for Systems of First-Order Equations. 2.6 Kutta's Formulas for Second-Order Differential Equations. 2.7 Application - The Nonlinear Pendulum. 2.8 Application - Impulsive Forces. 2.9 Exercises. 3 The Method of Taylor Expansions. 3.1 Introduction. 3.2 First-Order Problems. 3.3 Systems of First-Order Equations. 3.4 Second-Order Initial Value Problems. 3.5 Application - The van der Pol Oscillator. 3.6 Exercises. 4 Large Second-Order Systems with Application to Nano Systems. 4.1 Introduction. 4.2 The N-Body Problem. 4.3 Classical Molecular Potentials. 4.4 Molecular Mechanics. 4.5 The Leap Frog Formulas. 4.6 Equations of Motion for Argon Vapor. 4.7 A Cavity Problem. 4.8 Computational Considerations. 4.9 Examples of Primary Vortex Generation. 4.10 Examples of Turbulent Flow. 4.11 Remark. 4.12 Molecular Formulas for Air. 4.13 A Cavity Problem. 4.14 Initial Data. 4.15 Examples of Primary Vortex Generation. 4.16 Turbulent Flow. 4.17 Colliding Microdrops of Water Vapor. 4.18 Remarks. 4.19 Exercises. 5 Completely Conservative, Covariant Numerical Methodology. 5.1 Introduction. 5.2 Mathematical Considerations. 5.3 Numerical Methodology. 5.4 Conservation Laws. 5.5 Covariance. 5.6 Application - A Spinning Top on a Smooth Horizontal Plane. 5.7 Application - Calogero and Toda Hamiltonian Systems. 5.8 Remarks. 5.9 Exercises. 6 Instability. 6.1 Introduction. 6.2 Instability Analysis. 6.3 Numerical Solution of Mildly Nonlinear Autonomous Systems. 6.4 Exercises. 7 Numerical Solution of Tridiagonal Linear Algebraic Systems and Related Nonlinear Systems. 7.1 Introduction. 7.2 Tridiagonal Systems. 7.3 The Direct Method. 7.4 The Newton-Lieberstein Method. 7.5 Exercises. 8 Approximate Solution of Boundary Value Problems. 8.1 Introduction. 8.2 Approximate Differentiation. 8.3 Numerical Solution of Boundary Value Problems Using Difference Equations. 8.4 Upwind Differencing. 8.5 Mildly Nonlinear Boundary Value Problems. 8.6 Theoretical Support. 8.7 Application - Approximation of Airy Functions. 8.8 Exercises. 9 Special Relativistic Motion. 9.1 Introduction. 9.2 Inertial Frames. 9.3 The Lorentz Transformation. 9.4 Rod Contraction and Time Dilation. 9.5 Relativistic Particle Motion. 9.6 Covariance. 9.7 Particle Motion. 9.8 Numerical Methodology. 9.9 Relativistic Harmonic Oscillation. 9.10 Computational Covariance. 9.11 Remarks. 9.12 Exercises. 10 Special Topics. 10.1 Introduction. 10.2 Solving Boundary Value Problems by Initial Value Techniques. 10.3 Solving Initial Value Problems by Boundary Value Techniques. 10.4 Predictor-CorrectorMethods. 10.5 Multistep Methods. 10.6 Other Methods. 10.7 Consistency. 10.8 Differential Eigenvalue Problems. 10.9 Chaos. 10.10 Contact Mechanics. Appendix. A Basic Matrix Operations. Solutions to Selected Exercises. References. Index.