Download or read book Analysis of Discretization Methods for Ordinary Differential Equations written by Hans J. Stetter and published by Springer Science & Business Media. This book was released on 2013-03-12 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Finite Difference Methods for Ordinary and Partial Differential Equations written by Randall J. LeVeque and published by SIAM. This book was released on 2007-01-01 with total page 356 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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
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.
Download or read book Numerical Analysis written by James M. Ortega and published by SIAM. This book was released on 1990-01-01 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
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.
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:
Download or read book Discretization in Differential Equations and Enclosures written by Ernst Adams and published by . This book was released on 1987 with total page 266 pages. Available in PDF, EPUB and Kindle. Book excerpt:
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).
Download or read book Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations written by Uri M. Ascher and published by SIAM. This book was released on 1998-01-01 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Fractional Differential Equations written by Angelamaria Cardone and published by Springer Nature. This book was released on 2023-06-16 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book A First Course in Ordinary Differential Equations written by Martin Hermann and published by Springer Science & Business. This book was released on 2014-04-22 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents a modern introduction to analytical and numerical techniques for solving ordinary differential equations (ODEs). Contrary to the traditional format—the theorem-and-proof format—the book is focusing on analytical and numerical methods. The book supplies a variety of problems and examples, ranging from the elementary to the advanced level, to introduce and study the mathematics of ODEs. The analytical part of the book deals with solution techniques for scalar first-order and second-order linear ODEs, and systems of linear ODEs—with a special focus on the Laplace transform, operator techniques and power series solutions. In the numerical part, theoretical and practical aspects of Runge-Kutta methods for solving initial-value problems and shooting methods for linear two-point boundary-value problems are considered. The book is intended as a primary text for courses on the theory of ODEs and numerical treatment of ODEs for advanced undergraduate and early graduate students. It is assumed that the reader has a basic grasp of elementary calculus, in particular methods of integration, and of numerical analysis. Physicists, chemists, biologists, computer scientists and engineers whose work involves solving ODEs will also find the book useful as a reference work and tool for independent study. The book has been prepared within the framework of a German–Iranian research project on mathematical methods for ODEs, which was started in early 2012.
Download or read book Numerical Approximation of Ordinary Differential Problems written by Raffaele D'Ambrosio and published by Springer Nature. This book was released on 2023-09-26 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Advanced Numerical and Semi Analytical Methods for Differential Equations written by Snehashish Chakraverty and published by John Wiley & Sons. This book was released on 2019-04-10 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Numerical Solution of Boundary Value Problems for Ordinary Differential Equations written by Uri M. Ascher and published by SIAM. This book was released on 1994-12-01 with total page 620 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations written by Uri M. Ascher and published by SIAM. This book was released on 1998-08-01 with total page 304 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains all the material necessary for a course on the numerical solution of differential equations.
Download or read book Scientific Computing with Ordinary Differential Equations written by Peter Deuflhard and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 498 pages. Available in PDF, EPUB and Kindle. Book excerpt: Well-known authors; Includes topics and results that have previously not been covered in a book; Uses many interesting examples from science and engineering; Contains numerous homework exercises; Scientific computing is a hot and topical area
