Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- 17 Initial Value Problems for Ordinary Differential Equations
- 18 Single-Step Methods
- 19 Runge–Kutta Methods
- 20 Linear Multi-step Methods
- 21 Stiff Systems of Ordinary Differential Equations and Linear Stability
- 22 Galerkin Methods for Initial Value Problems
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
17 - Initial Value Problems for Ordinary Differential Equations
from Part IV - Initial Value Problems for Ordinary Differential Equations
Published online by Cambridge University Press: 29 September 2022
Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgements
- Symbols
- Part I Numerical Linear Algebra
- Part II Constructive Approximation Theory
- Part III Nonlinear Equations and Optimization
- Part IV Initial Value Problems for Ordinary Differential Equations
- 17 Initial Value Problems for Ordinary Differential Equations
- 18 Single-Step Methods
- 19 Runge–Kutta Methods
- 20 Linear Multi-step Methods
- 21 Stiff Systems of Ordinary Differential Equations and Linear Stability
- 22 Galerkin Methods for Initial Value Problems
- Part V Boundary and Initial Boundary Value Problems
- Appendix A Linear Algebra Review
- Appendix B Basic Analysis Review
- Appendix C Banach Fixed Point Theorem
- Appendix D A (Petting) Zoo of Function Spaces
- References
- Index
Summary
This chapter presents all the needed theoretical background regarding the initial value problem for a first order ordinary differential equation in finite dimensions. Local and global existence, uniqueness, and continuous dependence on data are presented. The discussion then turns to stability of solutions. We discuss the flow map and the Alekseev-Grobner Lemma. Dissipative equations. and a discussion of Lyapunov stability of fixed points conclude the chapter.
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- Classical Numerical AnalysisA Comprehensive Course, pp. 509 - 524Publisher: Cambridge University PressPrint publication year: 2022