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Solving Ordinary Differential Equations

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Numerical Methods and Optimization

Abstract

Differential equations play a crucial role in the simulation of physical systems, and most of them can only be solved numerically. Ordinary differential equations (or ODEs) have only one independent variable and are the simplest case by far. Various procedures are described and illustrated for getting ODEs in state-space form (as required by most ODE solvers). Initial-value problems (IVPs), where the value of the state is known at the start of the simulation are then considered. The pros and cons of the main explicit and implicit single-step and multistep methods for solving IVPs are described, and procedures for assessing local and global errors and for tuning step-size based on the local behavior of the solution are explained. Boundary-value problems (BVPs) are finally considered, and in particular two-endpoint BVPs, where partial information is available on the initial and final states. Since most methods for solving BVPs for ODEs extend to solving partial differential equations, this latter part also serves as an introduction to the next chapter.

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Authors and Affiliations

  1. Laboratoire des Signaux et Systèmes, CNRS-SUPÉLEC-Université Paris-Sud, 91192, Gif-sur-Yvette, France

    Éric Walter

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  1. Éric Walter
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Walter, É. (2014). Solving Ordinary Differential Equations. In: Numerical Methods and Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-07671-3_12

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  • DOI: https://doi.org/10.1007/978-3-319-07671-3_12

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  • Print ISBN: 978-3-319-07670-6

  • Online ISBN: 978-3-319-07671-3

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