Singular Perturbation Theory

Mathematical and Analytical Techniques with Applications to Engineering

Author: R.S. Johnson

Publisher: Springer Science & Business Media

ISBN: 9780387232171

Category: Mathematics

Page: 292

View: 1804

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, is well established. The purpose of the set of volumes to which the present one belongs is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume provides a detailed introduction to a specific subject area of current importance that is summarized below, and then goes beyond this by reviewing recent contributions, and so serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems would otherwise have been impossible.
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Approximate Analytical Methods for Solving Ordinary Differential Equations

Author: T.S.L Radhika,T. K.V. Iyengar,T. Raja Rani

Publisher: CRC Press

ISBN: 1466588160

Category: Mathematics

Page: 200

View: 7331

Approximate Analytical Methods for Solving Ordinary Differential Equations (ODEs) is the first book to present all of the available approximate methods for solving ODEs, eliminating the need to wade through multiple books and articles. It covers both well-established techniques and recently developed procedures, including the classical series solution method, diverse perturbation methods, pioneering asymptotic methods, and the latest homotopy methods. The book is suitable not only for mathematicians and engineers but also for biologists, physicists, and economists. It gives a complete description of the methods without going deep into rigorous mathematical aspects. Detailed examples illustrate the application of the methods to solve real-world problems. The authors introduce the classical power series method for solving differential equations before moving on to asymptotic methods. They next show how perturbation methods are used to understand physical phenomena whose mathematical formulation involves a perturbation parameter and explain how the multiple-scale technique solves problems whose solution cannot be completely described on a single timescale. They then describe the Wentzel, Kramers, and Brillown (WKB) method that helps solve both problems that oscillate rapidly and problems that have a sudden change in the behavior of the solution function at a point in the interval. The book concludes with recent nonperturbation methods that provide solutions to a much wider class of problems and recent analytical methods based on the concept of homotopy of topology.
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Theory of Stochastic Differential Equations with Jumps and Applications

Mathematical and Analytical Techniques with Applications to Engineering

Author: Rong SITU

Publisher: Springer Science & Business Media

ISBN: 0387251758

Category: Mathematics

Page: 434

View: 2862

Stochastic differential equations (SDEs) are a powerful tool in science, mathematics, economics and finance. This book will help the reader to master the basic theory and learn some applications of SDEs. In particular, the reader will be provided with the backward SDE technique for use in research when considering financial problems in the market, and with the reflecting SDE technique to enable study of optimal stochastic population control problems. These two techniques are powerful and efficient, and can also be applied to research in many other problems in nature, science and elsewhere.
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DCDS-A

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 3989

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Inverse Problems

Mathematical and Analytical Techniques with Applications to Engineering

Author: Alexander G. Ramm

Publisher: Springer Science & Business Media

ISBN: 0387232184

Category: Mathematics

Page: 442

View: 7656

Inverse Problems is a monograph which contains a self-contained presentation of the theory of several major inverse problems and the closely related results from the theory of ill-posed problems. The book is aimed at a large audience which include graduate students and researchers in mathematical, physical, and engineering sciences and in the area of numerical analysis.
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Multicomponent and Multiscale Systems

Theory, Methods, and Applications in Engineering

Author: Juergen Geiser

Publisher: Springer

ISBN: 3319151177

Category: Mathematics

Page: 325

View: 6629

This book examines the latest research results from combined multi-component and multi-scale explorations. It provides theory, considers underlying numerical methods and presents brilliant computational experimentation. Engineering computations featured in this monograph further offer particular interest to many researchers, engineers and computational scientists working in frontier modeling and applications of multicomponent and multiscale problems. Professor Geiser gives specific attention to the aspects of decomposing and splitting delicate structures and controlling decomposition and the rationale behind many important applications of multi-component and multi-scale analysis. Multicomponent and Multiscale Systems: Theory, Methods and Applications in Engineering also considers the question of why iterative methods can be powerful and more appropriate for well-balanced multiscale and multicomponent coupled nonlinear problems. The book is ideal for engineers and scientists working in theoretical and applied areas.
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Advanced Mathematical Methods for Scientists and Engineers I

Asymptotic Methods and Perturbation Theory

Author: Carl M. Bender,Steven A. Orszag

Publisher: Springer Science & Business Media

ISBN: 9780387989310

Category: Mathematics

Page: 593

View: 3674

This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. These methods allow one to analyze physics and engineering problems that may not be solvable in closed form. The presentation provides insights that will be useful in approaching new problems.
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Singular-Perturbation Theory

An Introduction with Applications

Author: Donald R. Smith

Publisher: Cambridge University Press

ISBN: 9780521300421

Category: Mathematics

Page: 500

View: 8544

Introduction to singular perturbation problems. Since the nature of the nonuniformity can vary from case to case, the author considers and solves a variety of problems, mostly for ordinary differential equations.
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Quasi-static State Analysis of Differential, Difference, Integral, and Gradient Systems

Author: F. C. Hoppensteadt

Publisher: American Mathematical Soc.

ISBN: 0821852698

Category: Mathematics

Page: 163

View: 6667

This book is based on a course on advanced topics in differential equations given in Spring 2010 at the Courant Institute of Mathematical Sciences. It describes aspects of mathematical modeling, analysis, computer simulation, and visualization in the mathematical sciences and engineering that involve singular perturbations. There is a large literature devoted to singular perturbation methods for ordinary and partial differential equations, but there are not many studies that deal with difference equations, Volterra integral equations, and purely nonlinear gradient systems where there is no dominant linear part. Designed for a one-semester course for students in applied mathematics, it is the purpose of this book to present sufficient rigorous methods and examples to position the reader to investigate singular perturbation problems in such equations. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.|This book is based on a course on advanced topics in differential equations given in Spring 2010 at the Courant Institute of Mathematical Sciences. It describes aspects of mathematical modeling, analysis, computer simulation, and visualization in the mathematical sciences and engineering that involve singular perturbations. There is a large literature devoted to singular perturbation methods for ordinary and partial differential equations, but there are not many studies that deal with difference equations, Volterra integral equations, and purely nonlinear gradient systems where there is no dominant linear part. Designed for a one-semester course for students in applied mathematics, it is the purpose of this book to present sufficient rigorous methods and examples to position the reader to investigate singular perturbation problems in such equations. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.
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Introduction to Asymptotic Methods

Author: David Y. Gao,Vadim A. Krysko

Publisher: CRC Press

ISBN: 9781420011739

Category: Mathematics

Page: 272

View: 6078

Among the theoretical methods for solving many problems of applied mathematics, physics, and technology, asymptotic methods often provide results that lead to obtaining more effective algorithms of numerical evaluation. Presenting the mathematical methods of perturbation theory, Introduction to Asymptotic Methods reviews the most important methods of singular perturbations within the scope of application of differential equations. The authors take a challenging and original approach based on the integrated mathematical-analytical treatment of various objects taken from interdisciplinary fields of mechanics, physics, and applied mathematics. This new hybrid approach will lead to results that cannot be obtained by standard theories in the field. Emphasizing fundamental elements of the mathematical modeling process, the book provides comprehensive coverage of asymptotic approaches, regular and singular perturbations, one-dimensional non-stationary non-linear waves, Padé approximations, oscillators with negative Duffing type stiffness, and differential equations with discontinuous nonlinearities. The book also offers a method of construction for canonical variables transformation in parametric form along with a number of examples and applications. The book is applications oriented and features results and literature citations that have not been seen in the Western Scientific Community. The authors emphasize the dynamics of the development of perturbation methods and present the development of ideas associated with this wide field of research.
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A First Look at Perturbation Theory

Author: James G. Simmonds,James E. Mann, Jr.

Publisher: Courier Corporation

ISBN: 9780486675510

Category: Mathematics

Page: 139

View: 9560

Undergraduates in engineering and the physical sciences receive a thorough introduction to perturbation theory in this useful and accessible text. Students discover methods for obtaining an approximate solution of a mathematical problem by exploiting the presence of a small, dimensionless parameter — the smaller the parameter, the more accurate the approximate solution. Knowledge of perturbation theory offers a twofold benefit: approximate solutions often reveal the exact solution's essential dependence on specified parameters; also, some problems resistant to numerical solutions may yield to perturbation methods. In fact, numerical and perturbation methods can be combined in a complementary way. The text opens with a well-defined treatment of finding the roots of polynomials whose coefficients contain a small parameter. Proceeding to differential equations, the authors explain many techniques for handling perturbations that reorder the equations or involve an unbounded independent variable. Two disparate practical problems that can be solved efficiently with perturbation methods conclude the volume. Written in an informal style that moves from specific examples to general principles, this elementary text emphasizes the "why" along with the "how"; prerequisites include a knowledge of one-variable calculus and ordinary differential equations. This newly revised second edition features an additional appendix concerning the approximate evaluation of integrals.
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Theory and applications of stochastic differential equations

Author: Zeev Schuss

Publisher: John Wiley & Sons Inc

ISBN: N.A

Category: Mathematics

Page: 321

View: 8794

Presents theory, sources, and applications of stochastic differential equations of Ito's type; those containing white noise. Closely studies first passage problems by modern singular perturbation methods and their role in various fields of science. Introduces analytical methods to obtain information on probabilistic quantities. Demonstrates the role of partial differential equations in this context. Clarifies the relationship between the complex mathematical theories involved and sources of the problem for physicists, chemists, engineers, and other non-mathematical specialists.
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Nonlinear Time Scale Systems in Standard and Nonstandard Forms

Analysis and Control

Author: Anshu Narang-Siddarth,John Valasek

Publisher: SIAM

ISBN: 1611973341

Category: Mathematics

Page: 219

View: 9475

This book introduces key concepts for systematically controlling engineering systems that possess interacting phenomena occurring at widely different speeds. The aim is to present the reader with control techniques that extend the benefits of model reduction of singular perturbation theory to a larger class of nonlinear dynamical systems. New results and relevant background are presented through insightful examples that cover a wide range of applications from different branches of engineering. This book is unique because it: presents a new perspective on existing control methods and thus broadens their application to a larger class of nonlinear dynamical systems; discusses general rather than problem-specific developments to certain applications or disciplines in order to provide control engineers with useful analytical tools ; addresses new control problems using singular perturbation methods, including closed-form results for control of nonminimum phase systems.
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Modeling Uncertainty

An Examination of Stochastic Theory, Methods, and Applications

Author: Moshe Dror,Pierre L'Ecuyer,Ferenc Szidarovszky

Publisher: Springer Science & Business Media

ISBN: 9780792374633

Category: Business & Economics

Page: 770

View: 6325

Writing in honour of Sid Yakowitz, 50 internationally known scholars have collectively contributed 30 papers on modelling uncertainty to this volume. These include papers with a theoretical emphasis and others that focus on applications.
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Finite-Elemente-Methoden

Author: Klaus-Jürgen Bathe

Publisher: DrMaster Publications

ISBN: 9783540668060

Category: Technology & Engineering

Page: 1253

View: 4917

Dieses Lehr- und Handbuch behandelt sowohl die elementaren Konzepte als auch die fortgeschrittenen und zukunftsweisenden linearen und nichtlinearen FE-Methoden in Statik, Dynamik, Festkörper- und Fluidmechanik. Es wird sowohl der physikalische als auch der mathematische Hintergrund der Prozeduren ausführlich und verständlich beschrieben. Das Werk enthält eine Vielzahl von ausgearbeiteten Beispielen, Rechnerübungen und Programmlisten. Als Übersetzung eines erfolgreichen amerikanischen Lehrbuchs hat es sich in zwei Auflagen auch bei den deutschsprachigen Ingenieuren etabliert. Die umfangreichen Änderungen gegenüber der Vorauflage innerhalb aller Kapitel - vor allem aber der fortgeschrittenen - spiegeln die rasche Entwicklung innerhalb des letzten Jahrzehnts auf diesem Gebiet wieder. TOC:Eine Einführung in den Gebrauch von Finite-Elemente-Verfahren.-Vektoren, Matrizen und Tensoren.-Einige Grundbegriffe ingenieurwissenschaftlicher Berechnungen.-Formulierung der Methode der finiten Elemente.-Formulierung und Berechnung von isoparametrischen Finite-Elemente-Matrizen.-Nichtlineare Finite-Elemente-Berechnungen in der Festkörper- und Strukturmechanik.-Finite-Elemente-Berechnungen von Wärmeübertragungs- und Feldproblemen.-Lösung von Gleichgewichtsbeziehungen in statischen Berechnungen.-Lösung von Bewegungsgleichungen in kinetischen Berechnungen.-Vorbemerkungen zur Lösung von Eigenproblemen.-Lösungsverfahren für Eigenprobleme.-Implementierung der Finite-Elemente-Methode.
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Vorlesungen über Technische Akustik

Author: L. Cremer

Publisher: Springer-Verlag

ISBN: 3662108259

Category: Technology & Engineering

Page: 334

View: 3664

Obschon die Herausgabe von Vorlesungen in Buchform klassische Vorbilder hat, ist sie nicht ohne Problematik. Eine Vorlesung kann man nicht konservieren, sie soll viel mehr jeweils dem Wissensstand von Forschung und Hörern angepaßt werden. Das ge sprochene Wort verfügt über Nuancen, die dem gedruckten fehlen, und eine sich gleich als Ganzes präsentierende Formel oder Skizze ist etwas anderes als eine vor den Au gen der Zuhörer entwickelte. Kurzum, Bücher können Vorlesungen nicht ersetzen. Es gilt aber auch das Umgekehrte. Beim Buch bestimmt der Lesende das Tempo; er kann repetieren, unterbrechen und überschlagen. Er ist dabei mehr an einer systema tischen als an einer didaktischen Anordnung des Stoffes interessiert. Der Verfasser hofft, mit der vorliegenden Niederschrift seiner, an der Technischen Universität Berlin, im Rahmen der Fakultät für Elektrotechnik gehaltenen, zweisemestri gen "Vorlesung über Technische Akustik" - in der in den beiden letzten Semestern vor getragenen Form - beiden Gesichtspunkten gerecht geworden zu sein. Der systematische Aufbau der Vorlesung ist aus den Kapitelüberschriften ersichtlich: Elektroakustik, Entstehung der Wellen, Schallausbreitung, Schalldämmung und das Hören.
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Asymptotic Methods in Mechanics

Author: RŽmi Vaillancourt,Andrei L. Smirnov

Publisher: American Mathematical Soc.

ISBN: 9780821870266

Category: Technology & Engineering

Page: 282

View: 4770

Asymptotic methods constitute an important area of both pure and applied mathematics and have applications to a vast array of problems. This collection of papers is devoted to asymptotic methods applied to mechanical problems, primarily thin structure problems. The first section presents a survey of asymptotic methods and a review of the literature, including the considerable body of Russian works in this area. This part may be used as a reference book or as a textbook for advanced undergraduate or graduate students in mathematics or engineering. The second part presents original papers containing new results. Among the key features of the book are its analysis of the general theory of asymptotic integration with applications to the theory of thin shells and plates, and new results about the local forms of vibrations and buckling of thin shells which have not yet made their way into other monographs on this subject.
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Asymptotic Analysis and the Numerical Solution of Partial Differential Equations

Author: Hans G. Kaper,Marc Garbey

Publisher: CRC Press

ISBN: 9780585319674

Category: Mathematics

Page: 286

View: 8160

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
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