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: 6602

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: 1903

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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Introduction to Linear Control Systems

Author: Yazdan Bavafa-Toosi

Publisher: Academic Press

ISBN: 012812749X

Category: Technology & Engineering

Page: 1008

View: 4670

Introduction to Linear Control Systems is designed as a standard introduction to linear control systems for all those who one way or another deal with control systems. It can be used as a comprehensive up-to-date textbook for a one-semester 3-credit undergraduate course on linear control systems as the first course on this topic at university. This includes the faculties of electrical engineering, mechanical engineering, aerospace engineering, chemical and petroleum engineering, industrial engineering, civil engineering, bio-engineering, economics, mathematics, physics, management and social sciences, etc. The book covers foundations of linear control systems, their raison detre, different types, modelling, representations, computations, stability concepts, tools for time-domain and frequency-domain analysis and synthesis, and fundamental limitations, with an emphasis on frequency-domain methods. Every chapter includes a part on further readings where more advanced topics and pertinent references are introduced for further studies. The presentation is theoretically firm, contemporary, and self-contained. Appendices cover Laplace transform and differential equations, dynamics, MATLAB and SIMULINK, treatise on stability concepts and tools, treatise on Routh-Hurwitz method, random optimization techniques as well as convex and non-convex problems, and sample midterm and endterm exams. The book is divided to the sequel 3 parts plus appendices. PART I: In this part of the book, chapters 1-5, we present foundations of linear control systems. This includes: the introduction to control systems, their raison detre, their different types, modelling of control systems, different methods for their representation and fundamental computations, basic stability concepts and tools for both analysis and design, basic time domain analysis and design details, and the root locus as a stability analysis and synthesis tool. PART II: In this part of the book, Chapters 6-9, we present what is generally referred to as the frequency domain methods. This refers to the experiment of applying a sinusoidal input to the system and studying its output. There are basically three different methods for representation and studying of the data of the aforementioned frequency response experiment: these are the Nyquist plot, the Bode diagram, and the Krohn-Manger-Nichols chart. We study these methods in details. We learn that the output is also a sinusoid with the same frequency but generally with different phase and magnitude. By dividing the output by the input we obtain the so-called sinusoidal or frequency transfer function of the system which is the same as the transfer function when the Laplace variable s is substituted with . Finally we use the Bode diagram for the design process. PART III: In this part, Chapter 10, we introduce some miscellaneous advanced topics under the theme fundamental limitations which should be included in this undergraduate course at least in an introductory level. We make bridges between some seemingly disparate aspects of a control system and theoretically complement the previously studied subjects. Appendices: The book contains seven appendices. Appendix A is on the Laplace transform and differential equations. Appendix B is an introduction to dynamics. Appendix C is an introduction to MATLAB, including SIMULINK. Appendix D is a survey on stability concepts and tools. A glossary and road map of the available stability concepts and tests is provided which is missing even in the research literature. Appendix E is a survey on the Routh-Hurwitz method, also missing in the literature. Appendix F is an introduction to random optimization techniques and convex and non-convex problems. Finally, appendix G presents sample midterm and endterm exams, which are class-tested several times.
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Optimal Trajectory Tracking of Nonlinear Dynamical Systems

Author: Jakob Lober

Publisher: Springer

ISBN: 3319465740

Category: Science

Page: 243

View: 9619

By establishing an alternative foundation of control theory, this thesis represents a significant advance in the theory of control systems, of interest to a broad range of scientists and engineers. While common control strategies for dynamical systems center on the system state as the object to be controlled, the approach developed here focuses on the state trajectory. The concept of precisely realizable trajectories identifies those trajectories that can be accurately achieved by applying appropriate control signals. The resulting simple expressions for the control signal lend themselves to immediate application in science and technology. The approach permits the generalization of many well-known results from the control theory of linear systems, e.g. the Kalman rank condition to nonlinear systems. The relationship between controllability, optimal control and trajectory tracking are clarified. Furthermore, the existence of linear structures underlying nonlinear optimal control is revealed, enabling the derivation of exact analytical solutions to an entire class of nonlinear optimal trajectory tracking problems. The clear and self-contained presentation focuses on a general and mathematically rigorous analysis of controlled dynamical systems. The concepts developed are visualized with the help of particular dynamical systems motivated by physics and chemistry.
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DCDS-A

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 2016

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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: 759

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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Perturbation Methods in Applied Mathematics

Author: Jirair Kevorkian,J.D. Cole

Publisher: Springer Science & Business Media

ISBN: 1475742134

Category: Mathematics

Page: 560

View: 9990

This book is a revised and updated version, including a substantial portion of new material, of J. D. Cole's text Perturbation Methods in Applied Mathe matics, Ginn-Blaisdell, 1968. We present the material at a level which assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate level course on the subject. The applied mathematician, attempting to understand or solve a physical problem, very often uses a perturbation procedure. In doing this, he usually draws on a backlog of experience gained from the solution of similar examples rather than on some general theory of perturbations. The aim of this book is to survey these perturbation methods, especially in connection with differ ential equations, in order to illustrate certain general features common to many examples. The basic ideas, however, are also applicable to integral equations, integrodifferential equations, and even to_difference equations. In essence, a perturbation procedure consists of constructing the solution for a problem involving a small parameter B, either in the differential equation or the boundary conditions or both, when the solution for the limiting case B = 0 is known. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of B.
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Singular Perturbation in the Physical Sciences

Author: John C. Neu

Publisher: American Mathematical Soc.

ISBN: 1470425556

Category: Asymptotic expansions

Page: 336

View: 3002

This book is the testimony of a physical scientist whose language is singular perturbation analysis. Classical mathematical notions, such as matched asymptotic expansions, projections of large dynamical systems onto small center manifolds, and modulation theory of oscillations based either on multiple scales or on averaging/transformation theory, are included. The narratives of these topics are carried by physical examples: Let's say that the moment when we "see" how a mathematical pattern fits a physical problem is like "hitting the ball." Yes, we want to hit the ball. But a powerful stroke includes the follow-through. One intention of this book is to discern in the structure and/or solutions of the equations their geometric and physical content. Through analysis, we come to sense directly the shape and feel of phenomena. The book is structured into a main text of fundamental ideas and a subtext of problems with detailed solutions. Roughly speaking, the former is the initial contact between mathematics and phenomena, and the latter emphasizes geometric and physical insight. It will be useful for mathematicians and physicists learning singular perturbation analysis of ODE and PDE boundary value problems as well as the full range of related examples and problems. Prerequisites are basic skills in analysis and a good junior/senior level undergraduate course of mathematical physics.
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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: 7486

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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Multiple Scale and Singular Perturbation Methods

Author: J.K. Kevorkian,J.D. Cole

Publisher: Springer Science & Business Media

ISBN: 1461239680

Category: Mathematics

Page: 634

View: 6738

This book is a revised and updated version, including a substantial portion of new material, of our text Perturbation Methods in Applied Mathematics (Springer Verlag, 1981). We present the material at a level that assumes some familiarity with the basics of ordinary and partial differential equations. Some of the more advanced ideas are reviewed as needed; therefore this book can serve as a text in either an advanced undergraduate course or a graduate-level course on the subject. Perturbation methods, first used by astronomers to predict the effects of small disturbances on the nominal motions of celestial bodies, have now become widely used analytical tools in virtually all branches of science. A problem lends itself to perturbation analysis if it is "close" to a simpler problem that can be solved exactly. Typically, this closeness is measured by the occurrence of a small dimensionless parameter, E, in the governing system (consisting of differential equations and boundary conditions) so that for E = 0 the resulting system is exactly solvable. The main mathematical tool used is asymptotic expansion with respect to a suitable asymptotic sequence of functions of E. In a regular perturbation problem, a straightforward procedure leads to a system of differential equations and boundary conditions for each term in the asymptotic expansion. This system can be solved recursively, and the accuracy of the result improves as E gets smaller, for all values of the independent variables throughout the domain of interest. We discuss regular perturbation problems in the first chapter.
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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: 9624

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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Singular-Perturbation Theory

An Introduction with Applications

Author: Donald R. Smith

Publisher: Cambridge University Press

ISBN: 9780521300421

Category: Mathematics

Page: 500

View: 4843

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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Introduction to Asymptotic Methods

Author: David Y. Gao,Vadim A. Krysko

Publisher: CRC Press

ISBN: 9781420011739

Category: Mathematics

Page: 272

View: 744

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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Engineering Mathematics II

Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization

Author: Sergei Silvestrov,Milica Rančić

Publisher: Springer

ISBN: 3319421050

Category: Computers

Page: 436

View: 5341

This book highlights the latest advances in engineering mathematics with a main focus on the mathematical models, structures, concepts, problems and computational methods and algorithms most relevant for applications in modern technologies and engineering. It addresses mathematical methods of algebra, applied matrix analysis, operator analysis, probability theory and stochastic processes, geometry and computational methods in network analysis, data classification, ranking and optimisation. The individual chapters cover both theory and applications, and include a wealth of figures, schemes, algorithms, tables and results of data analysis and simulation. Presenting new methods and results, reviews of cutting-edge research, and open problems for future research, they equip readers to develop new mathematical methods and concepts of their own, and to further compare and analyse the methods and results discussed. The book consists of contributed chapters covering research developed as a result of a focused international seminar series on mathematics and applied mathematics and a series of three focused international research workshops on engineering mathematics organised by the Research Environment in Mathematics and Applied Mathematics at Mälardalen University from autumn 2014 to autumn 2015: the International Workshop on Engineering Mathematics for Electromagnetics and Health Technology; the International Workshop on Engineering Mathematics, Algebra, Analysis and Electromagnetics; and the 1st Swedish-Estonian International Workshop on Engineering Mathematics, Algebra, Analysis and Applications. It serves as a source of inspiration for a broad spectrum of researchers and research students in applied mathematics, as well as in the areas of applications of mathematics considered in the book.
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Applied Mathematics And Modeling For Chemical Engineers

Author: Richard G. Rice,Duong D. Do

Publisher: John Wiley & Sons

ISBN: 1118343026

Category: Technology & Engineering

Page: 396

View: 5827

This Second Edition of the go-to reference combines the classical analysis and modern applications of applied mathematics for chemical engineers. The book introduces traditional techniques for solving ordinary differential equations (ODEs), adding new material on approximate solution methods such as perturbation techniques and elementary numerical solutions. It also includes analytical methods to deal with important classes of finite-difference equations. The last half discusses numerical solution techniques and partial differential equations (PDEs). The reader will then be equipped to apply mathematics in the formulation of problems in chemical engineering. Like the first edition, there are many examples provided as homework and worked examples.
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Applied Mathematics in Aerospace Science and Engineering

Author: Angelo Miele,Attilio Salvetti

Publisher: Springer Science & Business Media

ISBN: 147579259X

Category: Technology & Engineering

Page: 514

View: 6472

This book contains the proceedings ofthe meeting on "Applied Mathematics in the Aerospace Field," held in Erice, Sicily, Italy from September 3 to September 10, 1991. The occasion of the meeting was the 12th Course of the School of Mathematics "Guido Stampacchia," directed by Professor Franco Giannessi of the University of Pisa. The school is affiliated with the International Center for Scientific Culture "Ettore Majorana," which is directed by Professor Antonino Zichichi of the University of Bologna. The objective of the course was to give a perspective on the state-of the-art and research trends concerning the application of mathematics to aerospace science and engineering. The course was structured with invited lectures and seminars concerning fundamental aspects of differential equa tions, mathematical programming, optimal control, numerical methods, per turbation methods, and variational methods occurring in flight mechanics, astrodynamics, guidance, control, aircraft design, fluid mechanics, rarefied gas dynamics, and solid mechanics. The book includes 20 chapters by 23 contributors from the United States, Germany, and Italy and is intended to be an important reference work on the application of mathematics to the aerospace field. It reflects the belief of the course directors that strong interaction between mathematics and engineering is beneficial, indeed essential, to progresses in both areas.
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Analytical Techniques in Electromagnetics

Author: Matthew N. O. Sadiku,Sudarshan R. Nelatury

Publisher: CRC Press

ISBN: 1498709028

Category: Technology & Engineering

Page: 246

View: 1585

Analytical Techniques in Electromagnetics is designed for researchers, scientists, and engineers seeking analytical solutions to electromagnetic (EM) problems. The techniques presented provide exact solutions that can be used to validate the accuracy of approximate solutions, offer better insight into actual physical processes, and can be utilized in finding precise quantities of interest over a wide range of parameter values. Beginning with a review of basic EMs, the text: Describes the use of the separation of variables technique in Laplace, heat, and wave equations, covering rectangular, cylindrical, and spherical coordinate systems Explains the series expansion method, providing the solution of Poisson's equation in a cube and in a cylinder, and scattering by cylinders and spheres, as examples Addresses the conformal transformation technique, offering a visual display of conformal mapping and a brief introduction to the Schwarz–Christoffel transformation Employs worked-out problems to demonstrate various applications of Fourier sine and cosine, two-sided Fourier, Laplace, Hankel, and Mellin transform techniques Discusses perturbation techniques, supplying examples of perturbed results degenerating to their unperturbed versions as the perturbation parameters tend to zero Analytical Techniques in Electromagnetics maintains a balanced view of techniques for solving EM problems, refusing to overemphasize the importance of analytical methods at the expense of numerical techniques. Carefully selected topics give readers an appreciation of the kinds of EM problems that can be solved exactly.
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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: 9713

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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Methods for Constructing Exact Solutions of Partial Differential Equations

Mathematical and Analytical Techniques with Applications to Engineering

Author: Sergey V. Meleshko

Publisher: Springer Science & Business Media

ISBN: 0387252657

Category: Mathematics

Page: 352

View: 8289

Differential equations, especially nonlinear, present the most effective way for describing complex physical processes. Methods for constructing exact solutions of differential equations play an important role in applied mathematics and mechanics. This book aims to provide scientists, engineers and students with an easy-to-follow, but comprehensive, description of the methods for constructing exact solutions of differential equations.
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