Numerical Approximation Of Ordinary Differential Problems
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Numerical Approximation Of Partial Differential Equations
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Author : Alfio Quarteroni
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-02-11
Numerical Approximation Of Partial Differential Equations written by Alfio Quarteroni 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 2009-02-11 with Mathematics categories.
Everything is more simple than one thinks but at the same time more complex than one can understand Johann Wolfgang von Goethe To reach the point that is unknown to you, you must take the road that is unknown to you St. John of the Cross This is a book on the numerical approximation ofpartial differential equations (PDEs). Its scope is to provide a thorough illustration of numerical methods (especially those stemming from the variational formulation of PDEs), carry out their stability and convergence analysis, derive error bounds, and discuss the algorithmic aspects relative to their implementation. A sound balancing of theoretical analysis, description of algorithms and discussion of applications is our primary concern. Many kinds of problems are addressed: linear and nonlinear, steady and time-dependent, having either smooth or non-smooth solutions. Besides model equations, we consider a number of (initial-) boundary value problems of interest in several fields of applications. Part I is devoted to the description and analysis of general numerical methods for the discretization of partial differential equations. A comprehensive theory of Galerkin methods and its variants (Petrov Galerkin and generalized Galerkin), as wellas ofcollocationmethods, is devel oped for the spatial discretization. This theory is then specified to two numer ical subspace realizations of remarkable interest: the finite element method (conforming, non-conforming, mixed, hybrid) and the spectral method (Leg endre and Chebyshev expansion).
Numerical Approximation Of Ordinary Differential Problems
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Author : Raffaele D'Ambrosio
language : en
Publisher: Springer Nature
Release Date : 2023-09-26
Numerical Approximation Of Ordinary Differential Problems written by Raffaele D'Ambrosio and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-26 with Mathematics categories.
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.
Numerical Approximation Of Partial Differential Equations
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Author : Sören Bartels
language : en
Publisher: Springer
Release Date : 2018-05-30
Numerical Approximation Of Partial Differential Equations written by Sören Bartels and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-05-30 with Mathematics categories.
Finite element methods for approximating partial differential equations have reached a high degree of maturity, and are an indispensible tool in science and technology. This textbook aims at providing a thorough introduction to the construction, analysis, and implementation of finite element methods for model problems arising in continuum mechanics. The first part of the book discusses elementary properties of linear partial differential equations along with their basic numerical approximation, the functional-analytical framework for rigorously establishing existence of solutions, and the construction and analysis of basic finite element methods. The second part is devoted to the optimal adaptive approximation of singularities and the fast iterative solution of linear systems of equations arising from finite element discretizations. In the third part, the mathematical framework for analyzing and discretizing saddle-point problems is formulated, corresponding finte element methods are analyzed, and particular applications including incompressible elasticity, thin elastic objects, electromagnetism, and fluid mechanics are addressed. The book includes theoretical problems and practical projects for all chapters, and an introduction to the implementation of finite element methods.
Finite Difference Methods For Ordinary And Partial Differential Equations
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Author : Randall J. LeVeque
language : en
Publisher: SIAM
Release Date : 2007-01-01
Finite Difference Methods For Ordinary And Partial Differential Equations written by Randall J. LeVeque and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-01-01 with Mathematics categories.
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.
Analysis Of Approximation Methods For Differential And Integral Equations
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Author : Hans-Jürgen Reinhardt
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Analysis Of Approximation Methods For Differential And Integral Equations written by Hans-Jürgen Reinhardt 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 2012-12-06 with Mathematics categories.
This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite difference methods and of projection methods for approximating their variational formulations.
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
Numerical Solution Of Ordinary Differential Equations
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Author :
language : en
Publisher: Academic Press
Release Date : 1971-03-31
Numerical Solution Of Ordinary Differential Equations written by and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1971-03-31 with Mathematics categories.
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
The Numerical Solution Of Ordinary And Partial Differential Equations
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Author : Granville Sewell
language : en
Publisher: Academic Press
Release Date : 2014-05-10
The Numerical Solution Of Ordinary And Partial Differential Equations written by Granville Sewell 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.
The Numerical Solution of Ordinary and Partial Differential Equations is an introduction to the numerical solution of ordinary and partial differential equations. Finite difference methods for solving partial differential equations are mostly classical low order formulas, easy to program but not ideal for problems with poorly behaved solutions or (especially) for problems in irregular multidimensional regions. FORTRAN77 programs are used to implement many of the methods studied. Comprised of six chapters, this book begins with a review of direct methods for the solution of linear systems, with emphasis on the special features of the linear systems that arise when differential equations are solved. The next four chapters deal with the more commonly used finite difference methods for solving a variety of problems, including both ordinary differential equations and partial differential equations, and both initial value and boundary value problems. The final chapter is an overview of the basic ideas behind the finite element method and covers the Galerkin method for boundary value problems. Examples using piecewise linear trial functions, cubic hermite trial functions, and triangular elements are presented. This monograph is appropriate for senior-level undergraduate or first-year graduate students of mathematics.
Numerical Approximation Methods
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Author : Harold Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-12-10
Numerical Approximation Methods written by Harold Cohen 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 2011-12-10 with Mathematics categories.
This book presents numerical and other approximation techniques for solving various types of mathematical problems that cannot be solved analytically. In addition to well known methods, it contains some non-standard approximation techniques that are now formally collected as well as original methods developed by the author that do not appear in the literature. This book contains an extensive treatment of approximate solutions to various types of integral equations, a topic that is not often discussed in detail. There are detailed analyses of ordinary and partial differential equations and descriptions of methods for estimating the values of integrals that are presented in a level of detail that will suggest techniques that will be useful for developing methods for approximating solutions to problems outside of this text. The book is intended for researchers who must approximate solutions to problems that cannot be solved analytically. It is also appropriate for students taking courses in numerical approximation techniques.
Numerical Methods For Nonlinear Partial Differential Equations
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Author : Sören Bartels
language : en
Publisher: Springer
Release Date : 2015-01-19
Numerical Methods For Nonlinear Partial Differential Equations written by Sören Bartels and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-19 with Mathematics categories.
The description of many interesting phenomena in science and engineering leads to infinite-dimensional minimization or evolution problems that define nonlinear partial differential equations. While the development and analysis of numerical methods for linear partial differential equations is nearly complete, only few results are available in the case of nonlinear equations. This monograph devises numerical methods for nonlinear model problems arising in the mathematical description of phase transitions, large bending problems, image processing, and inelastic material behavior. For each of these problems the underlying mathematical model is discussed, the essential analytical properties are explained, and the proposed numerical method is rigorously analyzed. The practicality of the algorithms is illustrated by means of short implementations.