Stability Of Numerical Solution Of Systems Of Ordinary Differential Equations

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Numerical Solution of Ordinary Differential Equations

Author : Kendall Atkinson
Publisher : John Wiley & Sons
ISBN 13 : 1118164520
Total Pages : 272 pages
Book Rating : 4.1/5 (181 download)

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Book Synopsis Numerical Solution of Ordinary Differential Equations by : Kendall Atkinson

Download or read book Numerical Solution of Ordinary Differential Equations written by Kendall Atkinson and published by John Wiley & Sons. This book was released on 2011-10-24 with total page 272 pages. Available in PDF, EPUB and Kindle. Book excerpt: A concise introduction to numerical methodsand the mathematicalframework neededto understand their performance Numerical Solution of Ordinary Differential Equationspresents a complete and easy-to-follow introduction to classicaltopics in the numerical solution of ordinary differentialequations. The book's approach not only explains the presentedmathematics, but also helps readers understand how these numericalmethods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringingtogether and categorizing different types of problems in order tohelp readers comprehend the applications of ordinary differentialequations. In addition, the authors' collective academic experienceensures a coherent and accessible discussion of key topics,including: Euler's method Taylor and Runge-Kutta methods General error analysis for multi-step methods Stiff differential equations Differential algebraic equations Two-point boundary value problems Volterra integral equations Each chapter features problem sets that enable readers to testand build their knowledge of the presented methods, and a relatedWeb site features MATLAB® programs that facilitate theexploration of numerical methods in greater depth. Detailedreferences outline additional literature on both analytical andnumerical aspects of ordinary differential equations for furtherexploration of individual topics. Numerical Solution of Ordinary Differential Equations isan excellent textbook for courses on the numerical solution ofdifferential equations at the upper-undergraduate and beginninggraduate levels. It also serves as a valuable reference forresearchers in the fields of mathematics and engineering.

Stability of Numerical Solution of Systems of Ordinary Differential Equations

Author : John Jacob Kohfeld
Publisher :
ISBN 13 :
Total Pages : 104 pages
Book Rating : 4.:/5 (183 download)

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Book Synopsis Stability of Numerical Solution of Systems of Ordinary Differential Equations by : John Jacob Kohfeld

Download or read book Stability of Numerical Solution of Systems of Ordinary Differential Equations written by John Jacob Kohfeld and published by . This book was released on 1963 with total page 104 pages. Available in PDF, EPUB and Kindle. Book excerpt: The background for this paper is the use of quadrature formulas for the solution of ordinary differential equations. If we know the values of the dependent variable for which we are solving, and its derivative, at several equally spaced points, i.e., at values of the independent variable separated by equal intervals, we may use a quadrature formula to integrate the values of the derivative, so that we may obtain an approximate value of the dependent variable at the next point. The differential equation is then used to evaluate the derivative at the new point. This procedure is then repeated to evaluate the dependent variable and its derivative at point after point. The accuracy of this method is limited by the accuracy of the quadrature formula used. In order to improve the accuracy of the solution one may use an open-type quadrature formula to "predict" the value of the dependent variable at the next point, then calculate the derivative, and now use a more accurate closed-type formula to "correct" the value of the dependent variable. This procedure is the basis of "Milne's method". It has been shown that an error introduced at a step propagates itself approximately according to a linear combination of the solutions of a linear difference equation associated with the corrector . The solutions of this difference equation consist of an approximation to the solution of the differential equation and in some cases one or more extraneous solutions. If one or more of the latter increases as the process is repeated from step to step, the method is called instable. Remedies for instability include periodic use of special quadrature formulas called "stabilizers". This has been treated in the case of fifth-order formulas by Milne and Reynolds. In this paper the idea is extended to formulas of seventh order.

Numerical Methods for Ordinary Differential Equations

Author : David F. Griffiths
Publisher : Springer Science & Business Media
ISBN 13 : 0857291483
Total Pages : 274 pages
Book Rating : 4.8/5 (572 download)

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Book Synopsis Numerical Methods for Ordinary Differential Equations by : David F. Griffiths

Download or read book Numerical Methods for Ordinary Differential Equations written by David F. Griffiths and published by Springer Science & Business Media. This book was released on 2010-11-11 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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

Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations

Author : S. S. Artemiev
Publisher : Walter de Gruyter
ISBN 13 : 3110944669
Total Pages : 185 pages
Book Rating : 4.1/5 (19 download)

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Book Synopsis Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations by : S. S. Artemiev

Download or read book Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations written by S. S. Artemiev and published by Walter de Gruyter. This book was released on 2011-02-11 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text deals with numerical analysis of systems of both ordinary and stochastic differential equations. It covers numerical solution problems of the Cauchy problem for stiff ordinary differential equations (ODE) systems by Rosenbrock-type methods (RTMs).

Numerical Solution of Ordinary Differential Equations

Author :
Publisher : Academic Press
ISBN 13 : 0080955835
Total Pages : 317 pages
Book Rating : 4.0/5 (89 download)

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Book Synopsis Numerical Solution of Ordinary Differential Equations by :

Download or read book Numerical Solution of Ordinary Differential Equations written by and published by Academic Press. This book was released on 1971-03-31 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank matrix approximations; hybrid methods based on a combination of iterative procedures and best operator approximation; andmethods for information compression and filtering under condition that a filter model should satisfy restrictions associated with causality and different types of memory.As a result, the book represents a blend of new methods in general computational analysis,and specific, but also generic, techniques for study of systems theory ant its particularbranches, such as optimal filtering and information compression.- Best operator approximation,- Non-Lagrange interpolation,- Generic Karhunen-Loeve transform- Generalised low-rank matrix approximation- Optimal data compression- Optimal nonlinear filtering

On the Stability of Numerical Solutions of Ordinary Differential Equations

Author : Robert N. Lea
Publisher :
ISBN 13 :
Total Pages : 20 pages
Book Rating : 4.:/5 (31 download)

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Book Synopsis On the Stability of Numerical Solutions of Ordinary Differential Equations by : Robert N. Lea

Download or read book On the Stability of Numerical Solutions of Ordinary Differential Equations written by Robert N. Lea and published by . This book was released on 1967 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Stability of Numerical Methods for Delay Differential Equations

Author : Jiaoxun Kuang
Publisher : Elsevier
ISBN 13 : 9787030163172
Total Pages : 312 pages
Book Rating : 4.1/5 (631 download)

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Book Synopsis Stability of Numerical Methods for Delay Differential Equations by : Jiaoxun Kuang

Download or read book Stability of Numerical Methods for Delay Differential Equations written by Jiaoxun Kuang and published by Elsevier. This book was released on 2005 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Distributed by Elsevier Science on behalf of Science Press. Available internationally for the first time, this book introduces the basic concepts and theory of the stability of numerical methods for solving differential equations, with emphasis on delay differential equations and basic techniques for proving stability of numerical methods. It is a desirable reference for engineers and academic researchers and can also be used by graduate students in mathematics, physics, and engineering. Emphasis on the stability of numerical methods for solving delay differential equations, which is vital for engineers and researchers applying these mathematical models Introduces basic concepts and theory as well as basic techniques for readers to apply in practice Can be used as for graduate courses or as a reference book for researchers and engineers in related areas Written by leading mathematicians from Shanghai Normal University in China

Numerical Solution of Ordinary Differential Equations

Author : L.F. Shampine
Publisher : Routledge
ISBN 13 : 1351427555
Total Pages : 632 pages
Book Rating : 4.3/5 (514 download)

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Book Synopsis Numerical Solution of Ordinary Differential Equations by : L.F. Shampine

Download or read book Numerical Solution of Ordinary Differential Equations written by L.F. Shampine and published by Routledge. This book was released on 2018-10-24 with total page 632 pages. Available in PDF, EPUB and Kindle. Book excerpt: This new work is an introduction to the numerical solution of the initial value problem for a system of ordinary differential equations. The first three chapters are general in nature, and chapters 4 through 8 derive the basic numerical methods, prove their convergence, study their stability and consider how to implement them effectively. The book focuses on the most important methods in practice and develops them fully, uses examples throughout, and emphasizes practical problem-solving methods.

Modern Numerical Methods for Ordinary Differential Equations

Author : G. Hall
Publisher : Oxford University Press, USA
ISBN 13 :
Total Pages : 358 pages
Book Rating : 4.:/5 (44 download)

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Book Synopsis Modern Numerical Methods for Ordinary Differential Equations by : G. Hall

Download or read book Modern Numerical Methods for Ordinary Differential Equations written by G. Hall and published by Oxford University Press, USA. This book was released on 1976 with total page 358 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Numerical Methods for Initial Value Problems in Ordinary Differential Equations

Author : Simeon Ola Fatunla
Publisher : Academic Press
ISBN 13 : 1483269264
Total Pages : 308 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Numerical Methods for Initial Value Problems in Ordinary Differential Equations by : Simeon Ola Fatunla

Download or read book Numerical Methods for Initial Value Problems in Ordinary Differential Equations written by Simeon Ola Fatunla and published by Academic Press. This book was released on 2014-05-10 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: Numerical Method for Initial Value Problems in Ordinary Differential Equations deals with numerical treatment of special differential equations: stiff, stiff oscillatory, singular, and discontinuous initial value problems, characterized by large Lipschitz constants. The book reviews the difference operators, the theory of interpolation, first integral mean value theorem, and numerical integration algorithms. The text explains the theory of one-step methods, the Euler scheme, the inverse Euler scheme, and also Richardson's extrapolation. The book discusses the general theory of Runge-Kutta processes, including the error estimation, and stepsize selection of the R-K process. The text evaluates the different linear multistep methods such as the explicit linear multistep methods (Adams-Bashforth, 1883), the implicit linear multistep methods (Adams-Moulton scheme, 1926), and the general theory of linear multistep methods. The book also reviews the existing stiff codes based on the implicit/semi-implicit, singly/diagonally implicit Runge-Kutta schemes, the backward differentiation formulas, the second derivative formulas, as well as the related extrapolation processes. The text is intended for undergraduates in mathematics, computer science, or engineering courses, andfor postgraduate students or researchers in related disciplines.

Numerical Methods for Ordinary Differential Equations

Author : J. C. Butcher
Publisher : John Wiley & Sons
ISBN 13 : 1119121515
Total Pages : 544 pages
Book Rating : 4.1/5 (191 download)

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Book Synopsis Numerical Methods for Ordinary Differential Equations by : J. C. Butcher

Download or read book Numerical Methods for Ordinary Differential Equations written by J. C. Butcher and published by John Wiley & Sons. This book was released on 2016-07-11 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world’s leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numerical schemes to a source of new and practical algorithms in their own right. As the founder of general linear method research, John Butcher has been a leading contributor to its development; his special role is reflected in the text. The book is written in the lucid style characteristic of the author, and combines enlightening explanations with rigorous and precise analysis. In addition to these anticipated features, the book breaks new ground by including the latest results on the highly efficient G-symplectic methods which compete strongly with the well-known symplectic Runge-Kutta methods for long-term integration of conservative mechanical systems. This third edition of Numerical Methods for Ordinary Differential Equations will serve as a key text for senior undergraduate and graduate courses in numerical analysis, and is an essential resource for research workers in applied mathematics, physics and engineering.

Solving Ordinary Differential Equations II

Author : Ernst Hairer
Publisher : Springer Science & Business Media
ISBN 13 : 3662099470
Total Pages : 615 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Solving Ordinary Differential Equations II by : Ernst Hairer

Download or read book Solving Ordinary Differential Equations II written by Ernst Hairer and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 615 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.

Numerical Methods for Differential Equations

Author : J.R. Dormand
Publisher : CRC Press
ISBN 13 : 1351092006
Total Pages : 349 pages
Book Rating : 4.3/5 (51 download)

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Book Synopsis Numerical Methods for Differential Equations by : J.R. Dormand

Download or read book Numerical Methods for Differential Equations written by J.R. Dormand and published by CRC Press. This book was released on 2018-05-04 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: With emphasis on modern techniques, Numerical Methods for Differential Equations: A Computational Approach covers the development and application of methods for the numerical solution of ordinary differential equations. Some of the methods are extended to cover partial differential equations. All techniques covered in the text are on a program disk included with the book, and are written in Fortran 90. These programs are ideal for students, researchers, and practitioners because they allow for straightforward application of the numerical methods described in the text. The code is easily modified to solve new systems of equations. Numerical Methods for Differential Equations: A Computational Approach also contains a reliable and inexpensive global error code for those interested in global error estimation. This is a valuable text for students, who will find the derivations of the numerical methods extremely helpful and the programs themselves easy to use. It is also an excellent reference and source of software for researchers and practitioners who need computer solutions to differential equations.

The Concept of Stability in Numerical Mathematics

Author : Wolfgang Hackbusch
Publisher : Springer Science & Business Media
ISBN 13 : 3642393861
Total Pages : 202 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis The Concept of Stability in Numerical Mathematics by : Wolfgang Hackbusch

Download or read book The Concept of Stability in Numerical Mathematics written by Wolfgang Hackbusch and published by Springer Science & Business Media. This book was released on 2014-02-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book, the author compares the meaning of stability in different subfields of numerical mathematics. Concept of Stability in numerical mathematics opens by examining the stability of finite algorithms. A more precise definition of stability holds for quadrature and interpolation methods, which the following chapters focus on. The discussion then progresses to the numerical treatment of ordinary differential equations (ODEs). While one-step methods for ODEs are always stable, this is not the case for hyperbolic or parabolic differential equations, which are investigated next. The final chapters discuss stability for discretisations of elliptic differential equations and integral equations. In comparison among the subfields we discuss the practical importance of stability and the possible conflict between higher consistency order and stability.

Stability of Linear Delay Differential Equations

Author : Dimitri Breda
Publisher : Springer
ISBN 13 : 149392107X
Total Pages : 162 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Stability of Linear Delay Differential Equations by : Dimitri Breda

Download or read book Stability of Linear Delay Differential Equations written by Dimitri Breda and published by Springer. This book was released on 2014-10-21 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents the authors' recent work on the numerical methods for the stability analysis of linear autonomous and periodic delay differential equations, which consist in applying pseudospectral techniques to discretize either the solution operator or the infinitesimal generator and in using the eigenvalues of the resulting matrices to approximate the exact spectra. The purpose of the book is to provide a complete and self-contained treatment, which includes the basic underlying mathematics and numerics, examples from population dynamics and engineering applications, and Matlab programs implementing the proposed numerical methods. A number of proofs is given to furnish a solid foundation, but the emphasis is on the (unifying) idea of the pseudospectral technique for the stability analysis of DDEs. It is aimed at advanced students and researchers in applied mathematics, in dynamical systems and in various fields of science and engineering, concerned with delay systems. A relevant feature of the book is that it also provides the Matlab codes to encourage the readers to experience the practical aspects. They could use the codes to test the theory and to analyze the performances of the methods on the given examples. Moreover, they could easily modify them to tackle the numerical stability analysis of their own delay models.

Numerical Analysis Of Ordinary Differential Equations And Its Applications

Author : Taketomo Mitsui
Publisher : World Scientific
ISBN 13 : 9814500569
Total Pages : 240 pages
Book Rating : 4.8/5 (145 download)

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Book Synopsis Numerical Analysis Of Ordinary Differential Equations And Its Applications by : Taketomo Mitsui

Download or read book Numerical Analysis Of Ordinary Differential Equations And Its Applications written by Taketomo Mitsui and published by World Scientific. This book was released on 1995-10-12 with total page 240 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book collects original articles on numerical analysis of ordinary differential equations and its applications. Some of the topics covered in this volume are: discrete variable methods, Runge-Kutta methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, differential-algebraic and delay-differential equations, and stochastic initial value problems.

Numerical Analysis of Ordinary Differential Equations and Its Applications

Author : Taketomo Mitsui
Publisher : World Scientific
ISBN 13 : 9789810222291
Total Pages : 244 pages
Book Rating : 4.2/5 (222 download)

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Book Synopsis Numerical Analysis of Ordinary Differential Equations and Its Applications by : Taketomo Mitsui

Download or read book Numerical Analysis of Ordinary Differential Equations and Its Applications written by Taketomo Mitsui and published by World Scientific. This book was released on 1995 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book collects original articles on numerical analysis of ordinary differential equations and its applications. Some of the topics covered in this volume are: discrete variable methods, Runge-Kutta methods, linear multistep methods, stability analysis, parallel implementation, self-validating numerical methods, analysis of nonlinear oscillation by numerical means, differential-algebraic and delay-differential equations, and stochastic initial value problems.