Perturbation Methods

Author: E. J. Hinch

Publisher: Cambridge University Press

ISBN: 1139935844

Category: Mathematics

Page: N.A

View: 6155

Perturbation methods are one of the fundamental tools used by all applied mathematicians and theoretical physicists. In this book, the author has managed to present the theory and techniques underlying such methods in a manner which will give the text wide appeal to students from a broad range of disciplines. Asymptotic expansions, strained coordinates and multiple scales are illustrated by copious use of examples drawn from all areas of applied mathematics and theoretical physics. The philosophy adopted is that there is no single or best method for such problems, but that one may exploit the small parameter given some experience and understanding of similar perturbation problems. The author does not look to perturbation methods to give quantitative answers but rather to give a physical understanding of the subtle balances in a complex problem.

Introduction to Perturbation Methods

Author: Mark H. Holmes

Publisher: Springer Science & Business Media

ISBN: 1461253470

Category: Mathematics

Page: 356

View: 6329

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.

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: 2901

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.

Scaling, Self-similarity, and Intermediate Asymptotics

Dimensional Analysis and Intermediate Asymptotics

Author: G. I. Barenblatt

Publisher: Cambridge University Press

ISBN: 9780521435222

Category: Mathematics

Page: 386

View: 1777

Scaling (power-type) laws reveal the fundamental property of the phenomena--self similarity. Self-similar (scaling) phenomena repeat themselves in time and/or space. The property of self-similarity simplifies substantially the mathematical modeling of phenomena and its analysis--experimental, analytical and computational. The book begins from a non-traditional exposition of dimensional analysis, physical similarity theory and general theory of scaling phenomena. Classical examples of scaling phenomena are presented. It is demonstrated that scaling comes on a stage when the influence of fine details of initial and/or boundary conditions disappeared but the system is still far from ultimate equilibrium state (intermediate asymptotics). It is explained why the dimensional analysis as a rule is insufficient for establishing self-similarity and constructing scaling variables. Important examples of scaling phenomena for which the dimensional analysis is insufficient (self-similarities of the second kind) are presented and discussed. A close connection of intermediate asymptotics and self-similarities of the second kind with a fundamental concept of theoretical physics, the renormalization group, is explained and discussed. Numerous examples from various fields--from theoretical biology to fracture mechanics, turbulence, flame propagation, flow in porous strata, atmospheric and oceanic phenomena are presented for which the ideas of scaling, intermediate asymptotics, self-similarity and renormalization group were of decisive value in modeling.

Thinking about Ordinary Differential Equations

Author: Robert E. O'Malley

Publisher: Cambridge University Press

ISBN: 9780521557429

Category: Mathematics

Page: 247

View: 2183

While mastery of these equations is essential, adhering to any one method of solving them is not. This book stresses alternative examples and analyses by means of which students can understand a number of approaches to finding solutions and understanding their behavior. This book offers not only an applied perspective for the student learning to solve differential equations, but also the challenge to apply these analytical tools in the context of singular perturbations, which arises in many areas of application.

Stability, Instability and Chaos

An Introduction to the Theory of Nonlinear Differential Equations

Author: Paul Glendinning

Publisher: Cambridge University Press

ISBN: 9780521425667

Category: Mathematics

Page: 388

View: 2798

An introduction to nonlinear differential equations which equips undergraduate students with the know-how to appreciate stability theory and bifurcation.

Practical Applied Mathematics

Modelling, Analysis, Approximation

Author: Sam Howison

Publisher: Cambridge University Press

ISBN: 9780521842747

Category: Mathematics

Page: 326

View: 1296

Drawing from a wide variety of mathematical subjects, this book aims to show how mathematics is realised in practice in the everyday world. Dozens of applications are used to show that applied mathematics is much more than a series of academic calculations. Mathematical topics covered include distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling. The range of applications is similarly varied, from the modelling of hair to piano tuning, egg incubation and traffic flow. The style is informal but not superficial. In addition, the text is supplemented by a large number of exercises and sideline discussions, assisting the reader's grasp of the material. Used either in the classroom by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them.

Singular Perturbation in the Physical Sciences

Author: John C. Neu

Publisher: American Mathematical Soc.

ISBN: 1470425556

Category: Asymptotic expansions

Page: 336

View: 9083

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.

Applied Solid Mechanics

Author: Peter Howell,Gregory Kozyreff,John Ockendon

Publisher: Cambridge University Press

ISBN: 052185489X

Category: Mathematics

Page: 452

View: 1352

Emphasises the power of mathematics to provide quantitative insights across the whole area of solid mechanics; accessible and comprehensive.

Hydrodynamic Instabilities

Author: François Charru

Publisher: Cambridge University Press

ISBN: 1139500546

Category: Science

Page: N.A

View: 6138

The instability of fluid flows is a key topic in classical fluid mechanics because it has huge repercussions for applied disciplines such as chemical engineering, hydraulics, aeronautics, and geophysics. This modern introduction is written for any student, researcher, or practitioner working in the area, for whom an understanding of hydrodynamic instabilities is essential. Based on a decade's experience of teaching postgraduate students in fluid dynamics, this book brings the subject to life by emphasizing the physical mechanisms involved. The theory of dynamical systems provides the basic structure of the exposition, together with asymptotic methods. Wherever possible, Charru discusses the phenomena in terms of characteristic scales and dimensional analysis. The book includes numerous experimental studies, with references to videos and multimedia material, as well as over 150 exercises which introduce the reader to new problems.

Introduction to Numerical Methods in Differential Equations

Author: Mark H. Holmes

Publisher: Springer Science & Business Media

ISBN: 0387681213

Category: Mathematics

Page: 239

View: 7694

This book shows how to derive, test and analyze numerical methods for solving differential equations, including both ordinary and partial differential equations. The objective is that students learn to solve differential equations numerically and understand the mathematical and computational issues that arise when this is done. Includes an extensive collection of exercises, which develop both the analytical and computational aspects of the material. In addition to more than 100 illustrations, the book includes a large collection of supplemental material: exercise sets, MATLAB computer codes for both student and instructor, lecture slides and movies.

Rarefied Gas Dynamics

From Basic Concepts to Actual Calculations

Author: Carlo Cercignani

Publisher: Cambridge University Press

ISBN: 9780521659925

Category: Mathematics

Page: 320

View: 9100

The aim of this book is to present the concepts, methods and applications of kinetic theory to rarefied gas dynamics. After introducing the basic tools, problems in plane geometry are treated using approximation techniques (perturbation and numerical methods). These same techniques are later used to deal with two- and three-dimensional problems. The models include not only monatomic but also polyatomic gases, mixtures, chemical reactions. A special chapter is devoted to evaporation and condensation phenomena. Each section is accompanied by problems which are mainly intended to demonstrate the use of the material in the text and to outline additional subjects, results and equations. This will help ensure that the book can be used for a range of graduate courses in aerospace engineering or applied mathematics.

Nonlinear Dispersive Waves

Asymptotic Analysis and Solitons

Author: Mark J. Ablowitz

Publisher: Cambridge University Press

ISBN: 1139503480

Category: Mathematics

Page: N.A

View: 9722

The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg–de Vries (KdV) in the nineteenth century. In the 1960s, researchers developed effective asymptotic methods for deriving nonlinear wave equations, such as the KdV equation, governing a broad class of physical phenomena that admit special solutions including those commonly known as solitons. This book describes the underlying approximation techniques and methods for finding solutions to these and other equations. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. Most chapters feature exercise sets, making the book suitable for advanced courses or for self-directed learning. Graduate students and researchers will find this an excellent entry to a thriving area at the intersection of applied mathematics, engineering and physical science.

Mathematics Applied to Deterministic Problems in the Natural Sciences

Author: C. C. Lin,L. A. Segel

Publisher: SIAM

ISBN: 9781611971347

Category: Mathematics

Page: 609

View: 9126

Addresses the construction, analysis, and intepretation of mathematical models that shed light on significant problems in the physical sciences. The authors' case studies approach leads to excitement in teaching realistic problems. The many problems and exercises reinforce, test and extend the reader's understanding. This reprint volume may be used as an upper level undergraduate or graduate textbook as well as a reference for researchers working on fluid mechanics, elasticity, perturbation methods, dimensional analysis, numerical analysis, continuum mechanics and differential equations.

Introduction to Asymptotic Methods

Author: David Y. Gao,Vadim A. Krysko

Publisher: CRC Press

ISBN: 9781420011739

Category: Mathematics

Page: 272

View: 8389

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.