Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations

Author: Uri M. Ascher

Publisher: SIAM

Published: 1998-01-01

Total Pages: 305

ISBN-13: 9781611971392

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Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field's leading authorities, it provides a unified presentation of initial value and boundary value problems in ODEs as well as differential-algebraic equations. The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. This book is a practical and mathematically well-informed introduction that emphasizes basic methods and theory, issues in the use and development of mathematical software, and examples from scientific engineering applications. Topics requiring an extensive amount of mathematical development, such as symplectic methods for Hamiltonian systems, are introduced, motivated, and included in the exercises, but a complete and rigorous mathematical presentation is referenced rather than included.

Solving Ordinary Differential Equations II
Solving Ordinary Differential Equations II

Author: Ernst Hairer

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 615

ISBN-13: 3662099470

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"Whatever regrets may be, we have done our best." (Sir Ernest Shackleton, turning back on 9 January 1909 at 88°23' South.) Brahms struggled for 20 years to write his first symphony. Compared to this, the 10 years we have been working on these two volumes may even appear short. This second volume treats stiff differential equations and differential alge braic equations. It contains three chapters: Chapter IV on one-step (Runge Kutta) methods for stiff problems, Chapter Von multistep methods for stiff problems, and Chapter VI on singular perturbation and differential-algebraic equations. Each chapter is divided into sections. Usually the first sections of a chapter are of an introductory nature, explain numerical phenomena and exhibit numerical results. Investigations of a more theoretieal nature are presented in the later sections of each chapter. As in Volume I, the formulas, theorems, tables and figures are numbered consecutively in each section and indicate, in addition, the section num ber. In cross references to other chapters the (latin) chapter number is put first. References to the bibliography are again by "author" plus "year" in parentheses. The bibliography again contains only those papers which are discussed in the text and is in no way meant to be complete.

Ordinary Differential Equations and Linear Algebra
Ordinary Differential Equations and Linear Algebra

Author: Todd Kapitula

Publisher: SIAM

Published: 2015-11-17

Total Pages: 308

ISBN-13: 1611974097

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Ordinary differential equations (ODEs) and linear algebra are foundational postcalculus mathematics courses in the sciences. The goal of this text is to help students master both subject areas in a one-semester course. Linear algebra is developed first, with an eye toward solving linear systems of ODEs. A computer algebra system is used for intermediate calculations (Gaussian elimination, complicated integrals, etc.); however, the text is not tailored toward a particular system. Ordinary Differential Equations and Linear Algebra: A Systems Approach systematically develops the linear algebra needed to solve systems of ODEs and includes over 15 distinct applications of the theory, many of which are not typically seen in a textbook at this level (e.g., lead poisoning, SIR models, digital filters). It emphasizes mathematical modeling and contains group projects at the end of each chapter that allow students to more fully explore the interaction between the modeling of a system, the solution of the model, and the resulting physical description.

Numerical Solution of Initial-value Problems in Differential-algebraic Equations
Numerical Solution of Initial-value Problems in Differential-algebraic Equations

Author: K. E. Brenan

Publisher: SIAM

Published: 1996-01-01

Total Pages: 268

ISBN-13: 9781611971224

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Many physical problems are most naturally described by systems of differential and algebraic equations. This book describes some of the places where differential-algebraic equations (DAE's) occur. The basic mathematical theory for these equations is developed and numerical methods are presented and analyzed. Examples drawn from a variety of applications are used to motivate and illustrate the concepts and techniques. This classic edition, originally published in 1989, is the only general DAE book available. It not only develops guidelines for choosing different numerical methods, it is the first book to discuss DAE codes, including the popular DASSL code. An extensive discussion of backward differentiation formulas details why they have emerged as the most popular and best understood class of linear multistep methods for general DAE's. New to this edition is a chapter that brings the discussion of DAE software up to date. The objective of this monograph is to advance and consolidate the existing research results for the numerical solution of DAE's. The authors present results on the analysis of numerical methods, and also show how these results are relevant for the solution of problems from applications. They develop guidelines for problem formulation and effective use of the available mathematical software and provide extensive references for further study.

Numerical Solution of Boundary Value Problems for Ordinary Differential Equations
Numerical Solution of Boundary Value Problems for Ordinary Differential Equations

Author: Uri M. Ascher

Publisher: SIAM

Published: 1994-12-01

Total Pages: 620

ISBN-13: 9781611971231

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This book is the most comprehensive, up-to-date account of the popular numerical methods for solving boundary value problems in ordinary differential equations. It aims at a thorough understanding of the field by giving an in-depth analysis of the numerical methods by using decoupling principles. Numerous exercises and real-world examples are used throughout to demonstrate the methods and the theory. Although first published in 1988, this republication remains the most comprehensive theoretical coverage of the subject matter, not available elsewhere in one volume. Many problems, arising in a wide variety of application areas, give rise to mathematical models which form boundary value problems for ordinary differential equations. These problems rarely have a closed form solution, and computer simulation is typically used to obtain their approximate solution. This book discusses methods to carry out such computer simulations in a robust, efficient, and reliable manner.

Finite Difference Methods for Ordinary and Partial Differential Equations
Finite Difference Methods for Ordinary and Partial Differential Equations

Author: Randall J. LeVeque

Publisher: SIAM

Published: 2007-01-01

Total Pages: 356

ISBN-13: 9780898717839

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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

Numerical Methods for Ordinary Differential Equations
Numerical Methods for Ordinary Differential Equations

Author: David F. Griffiths

Publisher: Springer Science & Business Media

Published: 2010-11-11

Total Pages: 274

ISBN-13: 0857291483

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Numerical Methods for Ordinary Differential Equations is a self-contained introduction to a fundamental field of numerical analysis and scientific computation. Written for undergraduate students with a mathematical background, this book focuses on the analysis of numerical methods without losing sight of the practical nature of the subject. It covers the topics traditionally treated in a first course, but also highlights new and emerging themes. Chapters are broken down into `lecture' sized pieces, motivated and illustrated by numerous theoretical and computational examples. Over 200 exercises are provided and these are starred according to their degree of difficulty. Solutions to all exercises are available to authorized instructors. The book covers key foundation topics: o Taylor series methods o Runge--Kutta methods o Linear multistep methods o Convergence o Stability and a range of modern themes: o Adaptive stepsize selection o Long term dynamics o Modified equations o Geometric integration o Stochastic differential equations The prerequisite of a basic university-level calculus class is assumed, although appropriate background results are also summarized in appendices. A dedicated website for the book containing extra information can be found via www.springer.com