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: 6209

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.

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: 1475730691

Category: Mathematics

Page: 593

View: 2377

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. Aimed at teaching the most useful insights in approaching new problems, the text avoids special methods and tricks that only work for particular problems. Intended for graduates and advanced undergraduates, it assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations, then develops local asymptotic methods for such equations, and explains perturbation and summation theory before concluding with an exposition of global asymptotic methods. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach readers how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions, over 600 problems of varying levels of difficulty, and an appendix summarizing the properties of special functions.

Introduction to Perturbation Methods

Author: Mark H. Holmes

Publisher: Springer Science & Business Media

ISBN: 1461253470

Category: Mathematics

Page: 356

View: 3749

This introductory graduate text is based on a graduate course the author has taught repeatedly over the last ten years to students in applied mathematics, engineering sciences, and physics. Each chapter begins with an introductory development involving ordinary differential equations, and goes on to cover such traditional topics as boundary layers and multiple scales. However, it also contains material arising from current research interest, including homogenisation, slender body theory, symbolic computing, and discrete equations. Many of the excellent exercises are derived from problems of up-to-date research and are drawn from a wide range of application areas.

Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd

Publisher: Springer Science & Business Media

ISBN: 1475720637

Category: Mathematics

Page: 520

View: 5891

This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.

Perturbation Methods for Engineers and Scientists

Author: Alan W. Bush

Publisher: CRC Press

ISBN: 9780849386145

Category: Mathematics

Page: 320

View: 3611

The subject of perturbation expansions is a powerful analytical technique which can be applied to problems which are too complex to have an exact solution, for example, calculating the drag of an aircraft in flight. These techniques can be used in place of complicated numerical solutions.

Numerical Partial Differential Equations

Conservation Laws and Elliptic Equations

Author: J.W. Thomas

Publisher: Springer Science & Business Media

ISBN: N.A

Category: Mathematics

Page: 556

View: 5186

Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student in applied mathematics and engineering, this text offers a means of coming out of a course with a large number of methods that provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation. Prerequisites suggested for using this book in a course might include at least one semester of partial differential equations and some programming capability. The author stresses the use of technology throughout the text, allowing the student to utilize it as much as possible. The use of graphics for both illustration and analysis is emphasized, and algebraic manipulators are used when convenient. This is the second volume of a two-part book.

Singular Perturbation in the Physical Sciences

Author: John C. Neu

Publisher: American Mathematical Soc.

ISBN: 1470425556

Category: Asymptotic expansions

Page: 336

View: 6714

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.

A Guided Tour of Mathematical Methods for the Physical Sciences

Author: Roel Snieder,Kasper van Wijk,Matthew M. Haney

Publisher: Cambridge University Press

ISBN: 1107084962

Category: Mathematics

Page: 584

View: 2141

This completely revised edition provides a tour of the mathematical knowledge and techniques needed by students across the physical sciences. There are new chapters on probability and statistics and on inverse problems. It serves as a stand-alone text or as a source of exercises and examples to complement other textbooks.

Ordinary Differential Equations

Author: George F. Carrier,Carl E. Pearson

Publisher: SIAM

ISBN: 9781611971293

Category: Differential equations

Page: 220

View: 1356

Offers an alternative to the "rote" approach of presenting standard categories of differential equations accompanied by routine problem sets. The exercises presented amplify and provide perspective for the material, often giving readers opportunity for ingenuity. Little or no previous acquaintance with the subject is required to learn usage of techniques for constructing solutions of differential equations in this reprint volume.

The Optimal Homotopy Asymptotic Method

Engineering Applications

Author: Vasile Marinca,Nicolae Herisanu

Publisher: Springer

ISBN: 3319153749

Category: Technology & Engineering

Page: 465

View: 7615

This book emphasizes in detail the applicability of the Optimal Homotopy Asymptotic Method to various engineering problems. It is a continuation of the book “Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches”, published at Springer in 2011 and it contains a great amount of practical models from various fields of engineering such as classical and fluid mechanics, thermodynamics, nonlinear oscillations, electrical machines and so on. The main structure of the book consists of 5 chapters. The first chapter is introductory while the second chapter is devoted to a short history of the development of homotopy methods, including the basic ideas of the Optimal Homotopy Asymptotic Method. The last three chapters, from Chapter 3 to Chapter 5, are introducing three distinct alternatives of the Optimal Homotopy Asymptotic Method with illustrative applications to nonlinear dynamical systems. The third chapter deals with the first alternative of our approach with two iterations. Five applications are presented from fluid mechanics and nonlinear oscillations. The Chapter 4 presents the Optimal Homotopy Asymptotic Method with a single iteration and solving the linear equation on the first approximation. Here are treated 32 models from different fields of engineering such as fluid mechanics, thermodynamics, nonlinear damped and undamped oscillations, electrical machines and even from physics and biology. The last chapter is devoted to the Optimal Homotopy Asymptotic Method with a single iteration but without solving the equation in the first approximation.

Advanced Mathematical Methods

Author: Adam Ostaszewski

Publisher: Cambridge University Press

ISBN: 9780521289641

Category: Mathematics

Page: 545

View: 2312

Written in an appealing and informal style, this text is a self-contained second course on mathematical methods dealing with topics in linear algebra and multivariate calculus that can be applied to statistics, operations research, computer science, econometrics and mathematical economics. The prerequisites are elementary courses in linear algebra and calculus, but the author has maintained a balance between a rigorous theoretical and a cookbook approach, giving concrete and geometric explanations, so that the material will be accessible to students who have not studied mathematics in depth. Indeed, as much of the material is normally available only in technical textbooks, this book will have wide appeal to students whose interest is in application rather than theory. The book is amply supplied with examples and exercises: complete solutions to a large proportion of these are provided.

Statistical Physics

Statics, Dynamics and Renormalization

Author: Leo P Kadanoff

Publisher: World Scientific Publishing Company

ISBN: 981310290X

Category: Science

Page: 500

View: 6396

The material presented in this invaluable textbook has been tested in two courses. One of these is a graduate-level survey of statistical physics; the other, a rather personal perspective on critical behavior. Thus, this book defines a progression starting at the book-learning part of graduate education and ending in the midst of topics at the research level. To supplement the research-level side the book includes some research papers. Several of these are classics in the field, including a suite of six works on self-organized criticality and complexity, a pair on diffusion-limited aggregation, some papers on correlations near critical points, a few of the basic sources on the development of the real-space renormalization group, and several papers on magnetic behavior in a plain geometry. In addition, the author has included a few of his own papers.

Applied Asymptotic Analysis

Author: Peter David Miller

Publisher: American Mathematical Soc.

ISBN: 0821840789

Category: Mathematics

Page: 467

View: 7866

"The book is intended for a beginning graduate course on asymptotic analysis in applied mathematics and is aimed at students of pure and applied mathematics as well as science and engineering. The basic prerequisite is a background in differential equations, linear algebra, advanced calculus, and complex variables at the level of introductory undergraduate courses on these subjects."--BOOK JACKET.

Mathematical Methods and Models in Composites

Author: Vladislav Mantič

Publisher: World Scientific

ISBN: 178326411X

Category: Technology & Engineering

Page: 520

View: 3466

This book provides a representative selection of the most relevant, innovative, and useful mathematical methods and models applied to the analysis and characterization of composites and their behaviour on micro-, meso-, and macroscale. It establishes the fundamentals for meaningful and accurate theoretical and computer modelling of these materials in the future. Although the book is primarily concerned with fibre-reinforced composites, which have ever-increasing applications in fields such as aerospace, many of the results presented can be applied to other kinds of composites. The topics covered include: scaling and homogenization procedures in composite structures, thin plate and wave solutions in anisotropic materials, laminated structures, instabilities, fracture and damage analysis of composites, and highly efficient methods for simulation of composites manufacturing. The results presented are useful in the design, fabrication, testing, and industrial applications of composite components and structures. The book is written by well-known experts in different areas of applied mathematics, physics, and composite engineering and is an essential source of reference for graduate and doctoral students, as well as researchers. It is also suitable for non-experts in composites who wish to have an overview of both the mathematical methods and models used in this area and the related open problems requiring further research. Contents:Asymptotic Homogenization Method and Micromechanical Models for Composite Materials and Thin-Walled Composite Structures (Alexander L Kalamkarov)Scaling and Homogenization in Spatially Random Composites (Martin Ostoja-Starzewski and Shivakumar I Ranganathan)Stroh-Like Formalism for General Thin Laminated Plates and Its Applications (Chyanbin Hwu)Classical, Refined, Zig-Zag and Layer-Wise Models for Laminated Structures (Erasmo Carrera and Maria Cinefra)Bifurcation of Elastic Multilayers (Davide Bigoni, Massimiliano Gei and Sara Roccabianca)Propagation of Rayleigh Waves in Anisotropic Media and an Inverse Problem in the Characterization of Initial Stress (Kazumi Tanuma and Chi-Sing Man)Advanced Model Order Reduction for Simulating Composite-Forming Processes (Francisco Chinesta, Adrien Leygue and Arnaud Poitou)Modeling Fracture and Complex Crack Networks in Laminated Composites (Carlos G Dávila, Cheryl A Rose and Endel V Iarve)Delamination and Adhesive Contact Models and Their Mathematical Analysis and Numerical Treatment (Tomáš Roubíček, Martin Kružík and Jan Zeman)Crack Nucleation at Stress Concentration Points in Composite Materials — Application to Crack Deflection by an Interface (Dominique Leguillon and Eric Martin)Singular Elastic Solutions in Anisotropic Multimaterial Corners. Applications to Composites (Vladislav Mantič, Alberto Barroso and Federico París) Readership: Graduate students and researchers in composite engineering. Key Features:Competing titles are much more specialized - providing in-depth treatments but of only one area of research related to compositesContributing authors are worldwide prominent experts in the very different areas of applied mathematics, physics and engineering related to compositesEspecially suitable for young researchers showing a great variety of different approaches available today to model composites manufacturing, behavior and damage mechanisms. This unique volume allows them to find different approaches to related problems, to realize possible connections between them, and to finally take advantage of the already existing in-depth results related to composites in sometimes apparently very distant scientific fieldsKeywords:Composite Material;Laminate;Plate;Thin-Walled;Elastic Multilayer;Fiber-Reinforced;Homogenization;Anisotropic Material;Stroh Formalism;Nonlinear Elasticity;Wave;Inverse Problem;Fabrication;Manufacturing;Damage Mechanism;Crack;Delamination;Adhesive Contact;Fracture Criteria;Instability;Stress Singularity;X-FEM;SGBEM;Zig-Zag;Matched Asymptotic Expansion

Green's Functions and Boundary Value Problems

Author: Ivar Stakgold,Michael J. Holst

Publisher: John Wiley & Sons

ISBN: 0470906529

Category: Mathematics

Page: 736

View: 8167

Praise for the Second Edition "This book is an excellent introduction to the wide field of boundary value problems."—Journal of Engineering Mathematics "No doubt this textbook will be useful for both students and research workers."—Mathematical Reviews A new edition of the highly-acclaimed guide to boundary value problems, now featuring modern computational methods and approximation theory Green's Functions and Boundary Value Problems, Third Edition continues the tradition of the two prior editions by providing mathematical techniques for the use of differential and integral equations to tackle important problems in applied mathematics, the physical sciences, and engineering. This new edition presents mathematical concepts and quantitative tools that are essential for effective use of modern computational methods that play a key role in the practical solution of boundary value problems. With a careful blend of theory and applications, the authors successfully bridge the gap between real analysis, functional analysis, nonlinear analysis, nonlinear partial differential equations, integral equations, approximation theory, and numerical analysis to provide a comprehensive foundation for understanding and analyzing core mathematical and computational modeling problems. Thoroughly updated and revised to reflect recent developments, the book includes an extensive new chapter on the modern tools of computational mathematics for boundary value problems. The Third Edition features numerous new topics, including: Nonlinear analysis tools for Banach spaces Finite element and related discretizations Best and near-best approximation in Banach spaces Iterative methods for discretized equations Overview of Sobolev and Besov space linear Methods for nonlinear equations Applications to nonlinear elliptic equations In addition, various topics have been substantially expanded, and new material on weak derivatives and Sobolev spaces, the Hahn-Banach theorem, reflexive Banach spaces, the Banach Schauder and Banach-Steinhaus theorems, and the Lax-Milgram theorem has been incorporated into the book. New and revised exercises found throughout allow readers to develop their own problem-solving skills, and the updated bibliographies in each chapter provide an extensive resource for new and emerging research and applications. With its careful balance of mathematics and meaningful applications, Green's Functions and Boundary Value Problems, Third Edition is an excellent book for courses on applied analysis and boundary value problems in partial differential equations at the graduate level. It is also a valuable reference for mathematicians, physicists, engineers, and scientists who use applied mathematics in their everyday work.

Asymptotic Expansions of Integrals

Author: Norman Bleistein,Richard A. Handelsman

Publisher: Courier Corporation

ISBN: 0486650820

Category: Mathematics

Page: 425

View: 1793

Excellent introductory text, written by two experts, presents a coherent and systematic view of principles and methods. Topics include integration by parts, Watson's lemma, LaPlace's method, stationary phase, and steepest descents. Additional subjects include the Mellin transform method and less elementary aspects of the method of steepest descents. 1975 edition.

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: 4431

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.