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**Numerical methods for ordinary differential equations.
2nd revised ed.**
*(English)*
Zbl 1167.65041

Hoboken, NJ: John Wiley & Sons (ISBN 978-0-470-72335-7/hbk; 978-0-470-75376-7/ebook). xix, 463 p. (2008).

This book is the second edition of the book, first published in 2003 by J.Wiley and sons, 435 pp [Zbl 1040.65057]. It consists of 5 chapters, each of which consists of several sections. It has a bibliography pp.453-458, and an Index, pp.459-463.

Chapter 1. Differential and difference equations, consists of 5 sections. It contains some introductory background material.

Chapter 2. Numerical differential equation methods, consists of 8 sections. It contains a general overview of the numerical methods for ordinary differential equations (ODE). These methods include the Euler method, the Runge-Kutta methods, multistep and hybrid methods, and an introduction to the implementation of these methods.

Chapter 3. Runge-Kutta methods, consists of 10 sections. It contains a detailed analysis of these methods, explicit and implicit Runge-Kutta methods, stability and implementation of these methods, and their algebraic properties. A separate section deals with the implementation issues.

Chapter 4. Linear multistep methods, consists of 7 sections. The order of the linear multistep methods, their error estimates, stability analysis, and implementation issues are discussed in detail.

Chapter 5. General linear methods, consists of 6 sections. It treats some of the known methods, in particular, the Runge-Kutta methods, the linear multistep methods, some known unconvential methods, and some recently discovered methods as general linear methods. The stability analysis of general linear methods, the order of these methods and the error analysis of these methods are discussed, and design criteria for general linear methods are proposed. The author is one of the well-known contributors to the numerical methods for solving ODE. Part of the results, presented in this book, belong to the author.

The book is a useful contribution to the literature. It is written both to the students and specialists in the field of numerical solution of ODE.

Chapter 1. Differential and difference equations, consists of 5 sections. It contains some introductory background material.

Chapter 2. Numerical differential equation methods, consists of 8 sections. It contains a general overview of the numerical methods for ordinary differential equations (ODE). These methods include the Euler method, the Runge-Kutta methods, multistep and hybrid methods, and an introduction to the implementation of these methods.

Chapter 3. Runge-Kutta methods, consists of 10 sections. It contains a detailed analysis of these methods, explicit and implicit Runge-Kutta methods, stability and implementation of these methods, and their algebraic properties. A separate section deals with the implementation issues.

Chapter 4. Linear multistep methods, consists of 7 sections. The order of the linear multistep methods, their error estimates, stability analysis, and implementation issues are discussed in detail.

Chapter 5. General linear methods, consists of 6 sections. It treats some of the known methods, in particular, the Runge-Kutta methods, the linear multistep methods, some known unconvential methods, and some recently discovered methods as general linear methods. The stability analysis of general linear methods, the order of these methods and the error analysis of these methods are discussed, and design criteria for general linear methods are proposed. The author is one of the well-known contributors to the numerical methods for solving ODE. Part of the results, presented in this book, belong to the author.

The book is a useful contribution to the literature. It is written both to the students and specialists in the field of numerical solution of ODE.

Reviewer: Alexander G. Ramm (Manhattan)

### MSC:

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |