On The Instability Of Methods For The Integration Of Ordinary Differential Equations
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On The Instability Of Methods For The Integration Of Ordinary Differential Equations
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Author : Heinz Rutishauser
language : en
Publisher:
Release Date : 1956
On The Instability Of Methods For The Integration Of Ordinary Differential Equations written by Heinz Rutishauser and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1956 with Differential equations categories.
Examples and a criterion for stability of integration methods is provided. The criterion is applied to well-known integration formulas.
An Operational Unification Of Finite Difference Methods For The Numerical Integration Of Ordinary Differential Equations
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Author : Harvard Lomax
language : en
Publisher:
Release Date : 1967
An Operational Unification Of Finite Difference Methods For The Numerical Integration Of Ordinary Differential Equations written by Harvard Lomax and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1967 with Differential equations categories.
One purpose of this report is to present a mathematical procedure which can be used to study and compare various numerical methods for integrating ordinary differential equations. This procedure is relatively simple, mathematically rigorous, and of such a nature that matters of interest in digital computations, such as machine memory and running time, can be weighed against the accuracy and stability provided by the method under consideration. Briefly, the procedure is as follows: (1) Find a single differential equation that is sufficiently representative (this is fully defined in the report) of an arbitrary number of nonhomogeneous, linear, ordinary differential equations with constant coefficients. (2) Solve this differential equation exactly. (3) Choose any given numerical method, use it -- in its entirety -- to reduce the differential equation to difference equations, and, by means of operational techniques, solve the latter exactly. (4) Study and compare the results of (2) and (3). Conceptually there is nothing new in this procedure, but the particular development presented in this report does not appear to have been carried out before. Another purpose is to use the procedure just described to analyze a variety of numerical methods, ranging from classical, predictor-corrector systems to Runge-Kutta techniques and including various combinations of the two.
General Linear Methods For Ordinary Differential Equations
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Author : Zdzislaw Jackiewicz
language : en
Publisher: John Wiley & Sons
Release Date : 2009-08-14
General Linear Methods For Ordinary Differential Equations written by Zdzislaw Jackiewicz and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-08-14 with Mathematics categories.
Learn to develop numerical methods for ordinary differential equations General Linear Methods for Ordinary Differential Equations fills a gap in the existing literature by presenting a comprehensive and up-to-date collection of recent advances and developments in the field. This book provides modern coverage of the theory, construction, and implementation of both classical and modern general linear methods for solving ordinary differential equations as they apply to a variety of related areas, including mathematics, applied science, and engineering. The author provides the theoretical foundation for understanding basic concepts and presents a short introduction to ordinary differential equations that encompasses the related concepts of existence and uniqueness theory, stability theory, and stiff differential equations and systems. In addition, a thorough presentation of general linear methods explores relevant subtopics such as pre-consistency, consistency, stage-consistency, zero stability, convergence, order- and stage-order conditions, local discretization error, and linear stability theory. Subsequent chapters feature coverage of: Differential equations and systems Introduction to general linear methods (GLMs) Diagonally implicit multistage integration methods (DIMSIMs) Implementation of DIMSIMs Two-step Runge-Kutta (TSRK) methods Implementation of TSRK methods GLMs with inherent Runge-Kutta stability (IRKS) Implementation of GLMs with IRKS General Linear Methods for Ordinary Differential Equations is an excellent book for courses on numerical ordinary differential equations at the upper-undergraduate and graduate levels. It is also a useful reference for academic and research professionals in the fields of computational and applied mathematics, computational physics, civil and chemical engineering, chemistry, and the life sciences.
An Engineering Expos Of Numerical Integration Of Ordinary Differential Equations
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Author : John L. Engvall
language : en
Publisher:
Release Date : 1966
An Engineering Expos Of Numerical Integration Of Ordinary Differential Equations written by John L. Engvall and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1966 with Differential equations categories.
Numerical Methods For Ordinary Differential Equations
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Author : David F. Griffiths
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-11-11
Numerical Methods For Ordinary Differential Equations written by David F. Griffiths and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-11 with Mathematics categories.
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
Construction Of Integration Formulas For Initial Value Problems
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Author : P.J. Van Der Houwen
language : en
Publisher: Elsevier
Release Date : 2012-12-02
Construction Of Integration Formulas For Initial Value Problems written by P.J. Van Der Houwen and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-02 with Mathematics categories.
Construction of Integration Formulas for Initial Value Problems provides practice-oriented insights into the numerical integration of initial value problems for ordinary differential equations. It describes a number of integration techniques, including single-step methods such as Taylor methods, Runge-Kutta methods, and generalized Runge-Kutta methods. It also looks at multistep methods and stability polynomials. Comprised of four chapters, this volume begins with an overview of definitions of important concepts and theorems that are relevant to the construction of numerical integration methods for initial value problems. It then turns to a discussion of how to convert two-point and initial boundary value problems for partial differential equations into initial value problems for ordinary differential equations. The reader is also introduced to stiff differential equations, partial differential equations, matrix theory and functional analysis, and non-linear equations. The order of approximation of the single-step methods to the differential equation is considered, along with the convergence of a consistent single-step method. There is an explanation on how to construct integration formulas with adaptive stability functions and how to derive the most important stability polynomials. Finally, the book examines the consistency, convergence, and stability conditions for multistep methods. This book is a valuable resource for anyone who is acquainted with introductory calculus, linear algebra, and functional analysis.
Numerical Methods For Ordinary Differential Equations
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Author : J. C. Butcher
language : en
Publisher: John Wiley & Sons
Release Date : 2016-08-29
Numerical Methods For Ordinary Differential Equations written by J. C. Butcher and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-08-29 with Mathematics categories.
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.
A Proof Of The Instability Of Backward Difference Multistep Methods For The Numerical Integration Of Ordinary Differential Equations
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Author : Colin W. Cryer
language : en
Publisher:
Release Date : 1971
A Proof Of The Instability Of Backward Difference Multistep Methods For The Numerical Integration Of Ordinary Differential Equations written by Colin W. Cryer and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971 with categories.
It is shown that the backward difference multistep method summation, m = 1 to q of (1/m(del sup m)(y sup p))=h(f sub p) for the numerical integration of y'(x) = f(x, y) is stable in the sense of Dahlquist iff 1 = or
The Numerical Integration Of Ordinary Differential Equations
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Author : T. E. Hull
language : en
Publisher:
Release Date : 1966
The Numerical Integration Of Ordinary Differential Equations written by T. E. Hull and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1966 with Differential equations categories.
Numerical Methods For Initial Value Problems In Ordinary Differential Equations
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Author : Simeon Ola Fatunla
language : en
Publisher: Academic Press
Release Date : 2014-05-10
Numerical Methods For Initial Value Problems In Ordinary Differential Equations written by Simeon Ola Fatunla and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-05-10 with Mathematics categories.
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.