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Numerical Solution of Partial Differential Equations Theory Algorithms and Their Applications

Numerical Solution of Partial Differential Equations  Theory  Algorithms  and Their Applications Author Oleg P. Iliev
ISBN-10 9781461471721
Release 2013-06-04
Pages 327
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One of the current main challenges in the area of scientific computing​ is the design and implementation of accurate numerical models for complex physical systems which are described by time dependent coupled systems of nonlinear PDEs. This volume integrates the works of experts in computational mathematics and its applications, with a focus on modern algorithms which are at the heart of accurate modeling: adaptive finite element methods, conservative finite difference methods and finite volume methods, and multilevel solution techniques. Fundamental theoretical results are revisited in survey articles and new techniques in numerical analysis are introduced. Applications showcasing the efficiency, reliability and robustness of the algorithms in porous media, structural mechanics and electromagnetism are presented. Researchers and graduate students in numerical analysis and numerical solutions of PDEs and their scientific computing applications will find this book useful.

Numerical Approximation of Partial Differential Equations

Numerical Approximation of Partial Differential Equations Author Alfio Quarteroni
ISBN-10 9783540852681
Release 2009-02-11
Pages 544
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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).

Differential Equations with Mathematica

Differential Equations with Mathematica Author Martha L. Abell
ISBN-10 9780128047774
Release 2016-09-19
Pages 880
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Differential Equations with Mathematica, Fourth Edition is a supplementing reference which uses the fundamental concepts of the popular platform to solve (analytically, numerically, and/or graphically) differential equations of interest to students, instructors, and scientists. Mathematica’s diversity makes it particularly well suited to performing calculations encountered when solving many ordinary and partial differential equations. In some cases, Mathematica’s built-in functions can immediately solve a differential equation by providing an explicit, implicit, or numerical solution. In other cases, mathematica can be used to perform the calculations encountered when solving a differential equation. Because one goal of elementary differential equations courses is to introduce students to basic methods and algorithms so that they gain proficiency in them, nearly every topic covered this book introduces basic commands, also including typical examples of their application. A study of differential equations relies on concepts from calculus and linear algebra, so this text also includes discussions of relevant commands useful in those areas. In many cases, seeing a solution graphically is most meaningful, so the book relies heavily on Mathematica’s outstanding graphics capabilities. Demonstrates how to take advantage of the advanced features of Mathematica 10 Introduces the fundamental theory of ordinary and partial differential equations using Mathematica to solve typical problems of interest to students, instructors, scientists, and practitioners in many fields Showcases practical applications and case studies drawn from biology, physics, and engineering

Innovative Methods for Numerical Solutions of Partial Differential Equations

Innovative Methods for Numerical Solutions of Partial Differential Equations Author P. L. Roe
ISBN-10 9789810248109
Release 2002
Pages 382
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This book consists of 20 review articles dedicated to Prof. Philip Roe on the occasion of his 60th birthday and in appreciation of his original contributions to computational fluid dynamics. The articles, written by leading researchers in the field, cover many topics, including theory and applications, algorithm developments and modern computational techniques for industry.

Partial Differential Equations with Numerical Methods

Partial Differential Equations with Numerical Methods Author Stig Larsson
ISBN-10 9783540887058
Release 2008-12-05
Pages 262
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The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.

Numerical Solutions of Partial Differential Equations

Numerical Solutions of Partial Differential Equations Author Silvia Bertoluzza
ISBN-10 9783764389406
Release 2009-03-13
Pages 202
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This book presents some of the latest developments in numerical analysis and scientific computing. Specifically, it covers central schemes, error estimates for discontinuous Galerkin methods, and the use of wavelets in scientific computing.

Numerical Solution of Partial Differential Equations

Numerical Solution of Partial Differential Equations Author T. Meis
ISBN-10 9781461258858
Release 2012-12-06
Pages 556
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This book is the result of two courses of lectures given at the University of Cologne in Germany in 1974/75. The majority of the students were not familiar with partial differential equations and functional analysis. This explains why Sections 1, 2, 4 and 12 contain some basic material and results from these areas. The three parts of the book are largely independent of each other and can be read separately. Their topics are: initial value problems, boundary value problems, solutions of systems of equations. There is much emphasis on theoretical considerations and they are discussed as thoroughly as the algorithms which are presented in full detail and together with the programs. We believe that theoretical and practical applications are equally important for a genuine understa- ing of numerical mathematics. When writing this book, we had considerable help and many discussions with H. W. Branca, R. Esser, W. Hackbusch and H. Multhei. H. Lehmann, B. Muller, H. J. Niemeyer, U. Schulte and B. Thomas helped with the completion of the programs and with several numerical calculations. Springer-Verlag showed a lot of patience and under standing during the course of the production of the book. We would like to use the occasion of this preface to express our thanks to all those who assisted in our sometimes arduous task.

Optimal Control of Partial Differential Equations

Optimal Control of Partial Differential Equations Author Karl-Heinz Hoffmann
ISBN-10 9783034886918
Release 2012-12-06
Pages 326
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The application of PDE-based control theory and the corresponding numerical algorithms to industrial problems have become increasingly important in recent years. This volume offers a wide spectrum of aspects of the discipline, and is of interest to mathematicians and scientists working in the field.

Matrix Based Multigrid

Matrix Based Multigrid Author Yair Shapira
ISBN-10 9780387497648
Release 2008-07-02
Pages 318
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Matrix-Based Multigrid introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This book can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge only in basic linear algebra and calculus.

Decomposition Methods for Differential Equations

Decomposition Methods for Differential Equations Author Juergen Geiser
ISBN-10 1439810974
Release 2009-05-20
Pages 304
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Decomposition Methods for Differential Equations: Theory and Applications describes the analysis of numerical methods for evolution equations based on temporal and spatial decomposition methods. It covers real-life problems, the underlying decomposition and discretization, the stability and consistency analysis of the decomposition methods, and numerical results. The book focuses on the modeling of selected multi-physics problems, before introducing decomposition analysis. It presents time and space discretization, temporal decomposition, and the combination of time and spatial decomposition methods for parabolic and hyperbolic equations. The author then applies these methods to numerical problems, including test examples and real-world problems in physical and engineering applications. For the computational results, he uses various software tools, such as MATLAB®, R3T, WIAS-HiTNIHS, and OPERA-SPLITT. Exploring iterative operator-splitting methods, this book shows how to use higher-order discretization methods to solve differential equations. It discusses decomposition methods and their effectiveness, combination possibility with discretization methods, multi-scaling possibilities, and stability to initial and boundary values problems.

Fuzzy Partial Differential Equations and Relational Equations

Fuzzy Partial Differential Equations and Relational Equations Author Masoud Nikravesh
ISBN-10 3540203222
Release 2004-01-21
Pages 347
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This monograph presents the latest advances of fuzzy logic and soft computing in reservoir characterization and modeling. It proposes for the first time that future develoments require perception-based information processing. The book presents important steps in this direction by introducing fuzzy partial differential equations and relational equations. It provides a unique opportunity for soft computing researchers and oil industry practitioners to understand the significance of the changes in the fields by presenting recent accomplishments and new directions.

Numerical methods and scientific computing

Numerical methods and scientific computing Author Norbert Köckler
ISBN-10 0198596987
Release 1994
Pages 328
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This book covers the whole range of numerical mathematics--from linear equations to ordinary differential equations--and details the calculus of errors and partial differential equations. In attempting to give a unified approach of theory, algorithms, applications, and use of software, the book contains many helpful examples and applications. Topics include linear optimization, numerical integration, initial value problems, and nonlinear equations. The book is appearing simultaneously with the problem-solving environment PAN, a system that contains an enlarged hypertext version of the text together with all of the programs described in the book, help systems, and utility tools. (PAN is licensed public domain software.) The text is ideally suited as an introduction to numerical methods and programming for undergraduates in computer science, engineering, and mathematics. It will also be useful to software engineers using NAG libraries and numerical algorithms.

Group Theory and Numerical Analysis

Group Theory and Numerical Analysis Author Pavel Winternitz
ISBN-10 0821870343
Pages 298
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The Workshop on Group Theory and Numerical Analysis brought together scientists working in several different but related areas. The unifying theme was the application of group theory and geometrical methods to the solution of differential and difference equations. The emphasis was on the combination of analytical and numerical methods and also the use of symbolic computation. This meeting was organized under the auspices of the Centre de Recherches Mathematiques, Universite de Montreal (Canada). This volume has the character of a monograph and should represent a useful reference book for scientists working in this highly topical field.

Asymptotic Analysis and the Numerical Solution of Partial Differential Equations

Asymptotic Analysis and the Numerical Solution of Partial Differential Equations Author Hans G. Kaper
ISBN-10 0585319677
Release 1991-02-25
Pages 286
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Integrates two fields generally held to be incompatible, if not downright antithetical, in 16 lectures from a February 1990 workshop at the Argonne National Laboratory, Illinois. The topics, of interest to industrial and applied mathematicians, analysts, and computer scientists, include singular per

Finite Difference Methods for Ordinary and Partial Differential Equations

Finite Difference Methods for Ordinary and Partial Differential Equations Author Randall J. LeVeque
ISBN-10 0898717833
Release 2007
Pages 339
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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.

Computer Aided Analysis of Difference Schemes for Partial Differential Equations

Computer Aided Analysis of Difference Schemes for Partial Differential Equations Author Victor G. Ganzha
ISBN-10 0471129461
Release 1996-04-12
Pages 458
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Advances in computer technology have conveniently coincided with trends in numerical analysis toward increased complexity of computational algorithms based on finite difference methods. It is no longer feasible to perform stability investigation of these methods manually--and no longer necessary. As this book shows, modern computer algebra tools can be combined with methods from numerical analysis to generate programs that will do the job automatically. Comprehensive, timely, and accessible--this is the definitive reference on the application of computerized symbolic manipulations for analyzing the stability of a wide range of difference schemes. In particular, it deals with those schemes that are used to solve complex physical problems in areas such as gas dynamics, heat and mass transfer, catastrophe theory, elasticity, shallow water theory, and more. Introducing many new applications, methods, and concepts, Computer-Aided Analysis of Difference Schemes for Partial Differential Equations * Shows how computational algebra expedites the task of stability analysis--whatever the approach to stability investigation * Covers ten different approaches for each stability method * Deals with the specific characteristics of each method and its application to problems commonly encountered by numerical modelers * Describes all basic mathematical formulas that are necessary to implement each algorithm * Provides each formula in several global algebraic symbolic languages, such as MAPLE, MATHEMATICA, and REDUCE * Includes numerous illustrations and thought-provoking examples throughout the text For mathematicians, physicists, and engineers, as well as for postgraduate students, and for anyone involved with numeric solutions for real-world physical problems, this book provides a valuable resource, a helpful guide, and a head start on developments for the twenty-first century.


ISBN-10 9789491216275
Release 2010-09-01
Pages 715
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Boundary Element Methods have become a major numerical tool in scientific and engineering problem-solving, with particular applications to numerical computations and simulations of partial differential equations in engineering. Boundary Element Methods provides a rigorous and systematic account of the modern mathematical theory of Boundary Element Methods, including the requisite background on general partial, differential equation methods, Sobolev spaces, pseudo-differential and Fredholm operators and finite elements. It aims at the computation of many types of elliptic boundary value problems in potential theory, elasticity, wave propagation, and structural mechanics. Also presented are various methods and algorithms for nonlinear partial differential equations. This second edition has been fully revised and combines the mathematical rigour necessary for a full understanding of the subject, with extensive examples of applications illustrated with computer graphics. This book is intended as a textbook and reference for applied mathematicians, physical scientists and engineers at graduate and research level. It will be an invaluable sourcebook for all concerned with numerical modeling and the solution of partial differential equations.