Analysis Of Discretization Methods For Ordinary Differential Equations

Author : Hans J. Stetter
Publisher : Springer Science & Business Media
Page : 407 pages
File Size : 42,6 Mb
Release : 2013-03-12
Category : Mathematics
ISBN : 9783642654718

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Analysis of Discretization Methods for Ordinary Differential Equations by Hans J. Stetter Pdf

Due to the fundamental role of differential equations in science and engineering it has long been a basic task of numerical analysts to generate numerical values of solutions to differential equations. Nearly all approaches to this task involve a "finitization" of the original differential equation problem, usually by a projection into a finite-dimensional space. By far the most popular of these finitization processes consists of a reduction to a difference equation problem for functions which take values only on a grid of argument points. Although some of these finite difference methods have been known for a long time, their wide applica bility and great efficiency came to light only with the spread of electronic computers. This in tum strongly stimulated research on the properties and practical use of finite-difference methods. While the theory or partial differential equations and their discrete analogues is a very hard subject, and progress is consequently slow, the initial value problem for a system of first order ordinary differential equations lends itself so naturally to discretization that hundreds of numerical analysts have felt inspired to invent an ever-increasing number of finite-difference methods for its solution. For about 15 years, there has hardly been an issue of a numerical journal without new results of this kind; but clearly the vast majority of these methods have just been variations of a few basic themes. In this situation, the classical text book by P.

Author : Randall J. LeVeque
Publisher : SIAM
Page : 356 pages
File Size : 53,7 Mb
Release : 2007-01-01
Category : Mathematics
ISBN : 0898717833

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Finite Difference Methods for Ordinary and Partial Differential Equations by Randall J. LeVeque Pdf

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.

Author : Taketomo Mitsui,Yoshitane Shinohara
Publisher : World Scientific
Page : 244 pages
File Size : 53,6 Mb
Release : 1995
Category : Mathematics
ISBN : 9810222297

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Numerical Analysis of Ordinary Differential Equations and Its Applications by Taketomo Mitsui,Yoshitane Shinohara Pdf

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.

Author : David F. Griffiths,Desmond J. Higham
Publisher : Springer Science & Business Media
Page : 271 pages
File Size : 42,9 Mb
Release : 2010-11-11
Category : Mathematics
ISBN : 9780857291486

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Numerical Methods for Ordinary Differential Equations by David F. Griffiths,Desmond J. Higham Pdf

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

Author : Ernst Adams
Publisher : Unknown
Page : 266 pages
File Size : 48,9 Mb
Release : 1987
Category : Boundary value problems
ISBN : UOM:39015015701454

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Discretization in Differential Equations and Enclosures by Ernst Adams Pdf

Author : G. Hall,James Murray Watt
Publisher : Oxford University Press, USA
Page : 358 pages
File Size : 42,9 Mb
Release : 1976
Category : Boundary value problems
ISBN : UCAL:B4406402

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Modern Numerical Methods for Ordinary Differential Equations by G. Hall,James Murray Watt Pdf

Author : Raffaele D'Ambrosio
Publisher : Springer Nature
Page : 391 pages
File Size : 49,9 Mb
Release : 2023-09-26
Category : Mathematics
ISBN : 9783031313431

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Numerical Approximation of Ordinary Differential Problems by Raffaele D'Ambrosio Pdf

This book is focused on the numerical discretization of ordinary differential equations (ODEs), under several perspectives. The attention is first conveyed to providing accurate numerical solutions of deterministic problems. Then, the presentation moves to a more modern vision of numerical approximation, oriented to reproducing qualitative properties of the continuous problem along the discretized dynamics over long times. The book finally performs some steps in the direction of stochastic differential equations (SDEs), with the intention of offering useful tools to generalize the techniques introduced for the numerical approximation of ODEs to the stochastic case, as well as of presenting numerical issues natively introduced for SDEs. The book is the result of an intense teaching experience as well as of the research carried out in the last decade by the author. It is both intended for students and instructors: for the students, this book is comprehensive and rather self-contained; for the instructors, there is material for one or more monographic courses on ODEs and related topics. In this respect, the book can be followed in its designed path and includes motivational aspects, historical background, examples and a software programs, implemented in Matlab, that can be useful for the laboratory part of a course on numerical ODEs/SDEs. The book also contains the portraits of several pioneers in the numerical discretization of differential problems, useful to provide a framework to understand their contributes in the presented fields. Last, but not least, rigor joins readability in the book.

Author : James M. Ortega
Publisher : SIAM
Page : 214 pages
File Size : 45,8 Mb
Release : 1990-01-01
Category : Mathematics
ISBN : 1611971322

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Numerical Analysis by James M. Ortega Pdf

This book addresses some of the basic questions in numerical analysis: convergence theorems for iterative methods for both linear and nonlinear equations; discretization error, especially for ordinary differential equations; rounding error analysis; sensitivity of eigenvalues; and solutions of linear equations with respect to changes in the data.

Author : Uri M. Ascher,Linda R. Petzold
Publisher : SIAM
Page : 304 pages
File Size : 46,7 Mb
Release : 1998-01-01
Category : Mathematics
ISBN : 9781611971392

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations by Uri M. Ascher,Linda R. Petzold Pdf

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.

Author : Uri M. Ascher,Linda R. Petzold
Publisher : SIAM
Page : 304 pages
File Size : 44,8 Mb
Release : 1998-08-01
Category : Mathematics
ISBN : 9780898714128

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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations by Uri M. Ascher,Linda R. Petzold Pdf

This book contains all the material necessary for a course on the numerical solution of differential equations.

Author : Angelamaria Cardone,Marco Donatelli,Fabio Durastante,Roberto Garrappa,Mariarosa Mazza,Marina Popolizio
Publisher : Springer Nature
Page : 152 pages
File Size : 50,5 Mb
Release : 2023-06-16
Category : Mathematics
ISBN : 9789811977169

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Fractional Differential Equations by Angelamaria Cardone,Marco Donatelli,Fabio Durastante,Roberto Garrappa,Mariarosa Mazza,Marina Popolizio Pdf

The content of the book collects some contributions related to the talks presented during the INdAM Workshop "Fractional Differential Equations: Modelling, Discretization, and Numerical Solvers", held in Rome, Italy, on July 12–14, 2021. All contributions are original and not published elsewhere. The main topic of the book is fractional calculus, a topic that addresses the study and application of integrals and derivatives of noninteger order. These operators, unlike the classic operators of integer order, are nonlocal operators and are better suited to describe phenomena with memory (with respect to time and/or space). Although the basic ideas of fractional calculus go back over three centuries, only in recent decades there has been a rapid increase in interest in this field of research due not only to the increasing use of fractional calculus in applications in biology, physics, engineering, probability, etc., but also thanks to the availability of new and more powerful numerical tools that allow for an efficient solution of problems that until a few years ago appeared unsolvable. The analytical solution of fractional differential equations (FDEs) appears even more difficult than in the integer case. Hence, numerical analysis plays a decisive role since practically every type of application of fractional calculus requires adequate numerical tools. The aim of this book is therefore to collect and spread ideas mainly coming from the two communities of numerical analysts operating in this field - the one working on methods for the solution of differential problems and the one working on the numerical linear algebra side - to share knowledge and create synergies. At the same time, the book intends to realize a direct bridge between researchers working on applications and numerical analysts. Indeed, the book collects papers on applications, numerical methods for differential problems of fractional order, and related aspects in numerical linear algebra. The target audience of the book is scholars interested in recent advancements in fractional calculus.

Author : S. S. Artemiev,T. A. Averina
Publisher : Walter de Gruyter
Page : 185 pages
File Size : 55,5 Mb
Release : 2011-02-11
Category : Mathematics
ISBN : 9783110944662

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Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations by S. S. Artemiev,T. A. Averina Pdf

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).

Author : Snehashish Chakraverty,Nisha Mahato,Perumandla Karunakar,Tharasi Dilleswar Rao
Publisher : John Wiley & Sons
Page : 265 pages
File Size : 53,7 Mb
Release : 2019-04-10
Category : Mathematics
ISBN : 9781119423430

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Advanced Numerical and Semi-Analytical Methods for Differential Equations by Snehashish Chakraverty,Nisha Mahato,Perumandla Karunakar,Tharasi Dilleswar Rao Pdf

Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. Featuring both traditional and recent methods, Advanced Numerical and Semi Analytical Methods for Differential Equations begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: Discusses various methods for solving linear and nonlinear ODEs and PDEs Covers basic numerical techniques for solving differential equations along with various discretization methods Investigates nonlinear differential equations using semi-analytical methods Examines differential equations in an uncertain environment Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered Advanced Numerical and Semi Analytical Methods for Differential Equations is an excellent text for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analytically.

Author : Uri M. Ascher,Robert M. M. Mattheij,Robert D. Russell
Publisher : SIAM
Page : 620 pages
File Size : 52,8 Mb
Release : 1994-12-01
Category : Mathematics
ISBN : 1611971233

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Numerical Solution of Boundary Value Problems for Ordinary Differential Equations by Uri M. Ascher,Robert M. M. Mattheij,Robert D. Russell Pdf

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.

Author : Zdzislaw Jackiewicz
Publisher : John Wiley & Sons
Page : 500 pages
File Size : 45,6 Mb
Release : 2009-08-14
Category : Mathematics
ISBN : 9780470522158

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General Linear Methods for Ordinary Differential Equations by Zdzislaw Jackiewicz Pdf

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.