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**Computer methods for ordinary differential equations and differential-algebraic equations.**
*(English)*
Zbl 0908.65055

Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xvii, 314 p. (1998).

Ordinary differential equations (ODEs) have become widely used in applied science and this explains the need for a book which aims at giving a practical understanding of numerical methods for different branches of ODEs without presenting all the mathematical proofs. The book is organized in parts covering respectively the numerical solution of ordinary differential equations (initial value problems), boundary value problems as well as differential algebraic equations (DAEs).

The part on initial value problems addresses general concepts such as convergence and stability issues followed by Runge-Kutta methods and multistep methods. A fair amount of attention is given to implementational issues such as error estimation, step-size control and the modified Newton iteration. For boundary value problems shooting methods and finite difference methods are described and again many implementational issues are discussed. The last part on DAEs has a much more extensive introduction than the other parts and treats concepts such as index and invariants.

The chapter on numerical methods extends the methods from the section on initial value problems and describes some of the problems and their possible solutions. All in all the book, which also contains many examples and pointers to software, is excellent as an introduction to the field and definitely suitable for introductory courses at senior undergraduate or beginning graduate level.

The part on initial value problems addresses general concepts such as convergence and stability issues followed by Runge-Kutta methods and multistep methods. A fair amount of attention is given to implementational issues such as error estimation, step-size control and the modified Newton iteration. For boundary value problems shooting methods and finite difference methods are described and again many implementational issues are discussed. The last part on DAEs has a much more extensive introduction than the other parts and treats concepts such as index and invariants.

The chapter on numerical methods extends the methods from the section on initial value problems and describes some of the problems and their possible solutions. All in all the book, which also contains many examples and pointers to software, is excellent as an introduction to the field and definitely suitable for introductory courses at senior undergraduate or beginning graduate level.

Reviewer: C.Bendtsen (Lyngby)

### MSC:

65Lxx | Numerical methods for ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

65-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis |

### Keywords:

ordinary differential equations; initial value problems; boundary value problems; differential algebraic equations; convergence; stability; Runge-Kutta methods; multistep methods; error estimation; step-size control; Newton iteration; shooting methods; finite difference methods; index
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\textit{U. M. Ascher} and \textit{L. R. Petzold}, Computer methods for ordinary differential equations and differential-algebraic equations. Philadelphia, PA: SIAM (1998; Zbl 0908.65055)