Advertisement
| perturbation methods in applied mathematics: Perturbation Methods in Applied Mathematics J. Kevorkian, J.D. Cole, 2013-03-09 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. |
| perturbation methods in applied mathematics: Perturbation Methods Ali H. Nayfeh, 2008-09-26 The Wiley Classics Library consists of selected books that have become recognized classics in their respective fields. With these new unabridged and inexpensive editions, Wiley hopes to extend the life of these important works by making them available to future generations of mathematicians and scientists. Currently available in the Series: T. W. Anderson The Statistical Analysis of Time Series T. S. Arthanari & Yadolah Dodge Mathematical Programming in Statistics Emil Artin Geometric Algebra Norman T. J. Bailey The Elements of Stochastic Processes with Applications to the Natural Sciences Robert G. Bartle The Elements of Integration and Lebesgue Measure George E. P. Box & Norman R. Draper Evolutionary Operation: A Statistical Method for Process Improvement George E. P. Box & George C. Tiao Bayesian Inference in Statistical Analysis R. W. Carter Finite Groups of Lie Type: Conjugacy Classes and Complex Characters R. W. Carter Simple Groups of Lie Type William G. Cochran & Gertrude M. Cox Experimental Designs, Second Edition Richard Courant Differential and Integral Calculus, Volume I RIchard Courant Differential and Integral Calculus, Volume II Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume I Richard Courant & D. Hilbert Methods of Mathematical Physics, Volume II D. R. Cox Planning of Experiments Harold S. M. Coxeter Introduction to Geometry, Second Edition Charles W. Curtis & Irving Reiner Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume I Charles W. Curtis & Irving Reiner Methods of Representation Theory with Applications to Finite Groups and Orders, Volume II Cuthbert Daniel Fitting Equations to Data: Computer Analysis of Multifactor Data, Second Edition Bruno de Finetti Theory of Probability, Volume I Bruno de Finetti Theory of Probability, Volume 2 W. Edwards Deming Sample Design in Business Research |
| perturbation methods in applied mathematics: Introduction to Perturbation Methods Mark H. Holmes, 2013-12-01 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. |
| perturbation methods in applied mathematics: Perturbation Methods E. J. Hinch, 1991-10-25 A textbook presenting the theory and underlying techniques of perturbation methods in a manner suitable for senior undergraduates from a broad range of disciplines. |
| perturbation methods in applied mathematics: Random Perturbation Methods with Applications in Science and Engineering Anatoli V. Skorokhod, Frank C. Hoppensteadt, Habib D. Salehi, 2007-06-21 This book develops methods for describing random dynamical systems, and it illustrats how the methods can be used in a variety of applications. Appeals to researchers and graduate students who require tools to investigate stochastic systems. |
| perturbation methods in applied mathematics: Multiple Scale and Singular Perturbation Methods J.K. Kevorkian, J.D. Cole, 1996-05-15 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. |
| perturbation methods in applied mathematics: Perturbations James A. Murdock, 1999-01-01 Perturbations: Theory and Methods gives a thorough introduction to both regular and singular perturbation methods for algebraic and differential equations. Unlike most introductory books on the subject, this one distinguishes between formal and rigorous asymptotic validity, which are commonly confused in books that treat perturbation theory as a bag of heuristic tricks with no foundation. The meaning of uniformity is carefully explained in a variety of contexts. All standard methods, such as rescaling, multiple scales, averaging, matching, and the WKB method are covered, and the asymptotic validity (in the rigorous sense) of each method is carefully proved. First published in 1991, this book is still useful today because it is an introduction. It combines perturbation results with those known through other methods. Sometimes a geometrical result (such as the existence of a periodic solution) is rigorously deduced from a perturbation result, and at other times a knowledge of the geometry of the solutions is used to aid in the selection of an effective perturbation method. Dr. Murdock's approach differs from other introductory texts because he attempts to present perturbation theory as a natural part of a larger whole, the mathematical theory of differential equations. He explores the meaning of the results and their connections to other ways of studying the same problems. |
| perturbation methods in applied mathematics: Singular Perturbation Methods in Control Petar Kokotovic, Hassan K. Khalil, John O'Reilly, 1999-01-01 Singular perturbations and time-scale techniques were introduced to control engineering in the late 1960s and have since become common tools for the modeling, analysis, and design of control systems. In this SIAM Classics edition of the 1986 book, the original text is reprinted in its entirety (along with a new preface), providing once again the theoretical foundation for representative control applications. This book continues to be essential in many ways. It lays down the foundation of singular perturbation theory for linear and nonlinear systems, it presents the methodology in a pedagogical way that is not available anywhere else, and it illustrates the theory with many solved examples, including various physical examples and applications. So while new developments may go beyond the topics covered in this book, they are still based on the methodology described here, which continues to be their common starting point. |
| perturbation methods in applied mathematics: Perturbation Methods in Applied Mathematics Julian D. Cole, 1968 |
| perturbation methods in applied mathematics: Perturbations James A. Murdock, 1999-01-01 This book gives a thorough introduction to both regular and singular perturbation methods for algebraic and differential equations. |
| perturbation methods in applied mathematics: Perturbation Methods in Science and Engineering Reza N. Jazar, 2021-07-12 Perturbation Methods in Science and Engineering provides the fundamental and advanced topics in perturbation methods in science and engineering, from an application viewpoint. This book bridges the gap between theory and applications, in new as well as classical problems. The engineers and graduate students who read this book will be able to apply their knowledge to a wide range of applications in different engineering disciplines. The book begins with a clear description on limits of mathematics in providing exact solutions and goes on to show how pioneers attempted to search for approximate solutions of unsolvable problems. Through examination of special applications and highlighting many different aspects of science, this text provides an excellent insight into perturbation methods without restricting itself to a particular method. This book is ideal for graduate students in engineering, mathematics, and physical sciences, as well as researchers in dynamic systems. |
| perturbation methods in applied mathematics: Advanced Mathematical Methods for Scientists and Engineers I Carl M. Bender, Steven A. Orszag, 2013-03-09 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. |
| perturbation methods in applied mathematics: Methods and Applications of Singular Perturbations Ferdinand Verhulst, 2006-06-04 Contains well-chosen examples and exercises A student-friendly introduction that follows a workbook type approach |
| perturbation methods in applied mathematics: Singular Perturbation Methods for Ordinary Differential Equations Robert E., Jr. O'Malley, 2012-12-06 This book results from various lectures given in recent years. Early drafts were used for several single semester courses on singular perturbation meth ods given at Rensselaer, and a more complete version was used for a one year course at the Technische Universitat Wien. Some portions have been used for short lecture series at Universidad Central de Venezuela, West Vir ginia University, the University of Southern California, the University of California at Davis, East China Normal University, the University of Texas at Arlington, Universita di Padova, and the University of New Hampshire, among other places. As a result, I've obtained lots of valuable feedback from students and listeners, for which I am grateful. This writing continues a pattern. Earlier lectures at Bell Laboratories, at the University of Edin burgh and New York University, and at the Australian National University led to my earlier works (1968, 1974, and 1978). All seem to have been useful for the study of singular perturbations, and I hope the same will be true of this monograph. I've personally learned much from reading and analyzing the works of others, so I would especially encourage readers to treat this book as an introduction to a diverse and exciting literature. The topic coverage selected is personal and reflects my current opin ions. An attempt has been made to encourage a consistent method of ap proaching problems, largely through correcting outer limits in regions of rapid change. Formal proofs of correctness are not emphasized. |
| perturbation methods in applied mathematics: Perturbation Techniques in Mathematics, Engineering and Physics Richard Ernest Bellman, 2003-01-01 Graduate students receive a stimulating introduction to analytical approximation techniques for solving differential equations in this text, which introduces scientifically significant problems and indicates useful solutions. 1966 edition. |
| perturbation methods in applied mathematics: Perturbation Methods for Differential Equations Bhimsen Shivamoggi, 2012-12-06 Perturbation methods are widely used in the study of physically significant differential equations, which arise in Applied Mathematics, Physics and Engineering.; Background material is provided in each chapter along with illustrative examples, problems, and solutions.; A comprehensive bibliography and index complete the work.; Covers an important field of solutions for engineering and the physical sciences.; To allow an interdisciplinary readership, the book focuses almost exclusively on the procedures and the underlying ideas and soft pedal the proofs; Dr. Bhimsen K. Shivamoggi has authored seven successful books for various publishers like John Wiley & Sons and Kluwer Academic Publishers. |
| perturbation methods in applied mathematics: Algebraic Methods in Nonlinear Perturbation Theory V.N. Bogaevski, A. Povzner, 2012-12-06 Of interest to everybody working on perturbation theory in differential equations, this book requires only a standard mathematical background in engineering and does not require reference to the special literature. Topics covered include: matrix perturbation theory; systems of ordinary differential equations with small parameters; reconstruction and equations in partial derivatives. While boundary problems are not discussed, the book is clearly illustrated by numerous examples. |
| perturbation methods in applied mathematics: Difference Methods for Singular Perturbation Problems Grigory I. Shishkin, Lidia P. Shishkina, 2008-09-22 Difference Methods for Singular Perturbation Problems focuses on the development of robust difference schemes for wide classes of boundary value problems. It justifies the ε-uniform convergence of these schemes and surveys the latest approaches important for further progress in numerical methods. The first part of the book e |
| perturbation methods in applied mathematics: Perturbation Methods in Fluid Mechanics Milton Van Dyke, 1975 |
| perturbation methods in applied mathematics: Perturbation Methods in Non-Linear Systems Georgio Eugenio Oscare Giacaglia, 2012-12-06 This volume is intended to provide a comprehensive treatment of recent developments in methods of perturbation for nonlinear systems of ordinary differ ential equations. In this respect, it appears to be a unique work. The main goal is to describe perturbation techniques, discuss their ad vantages and limitations and give some examples. The approach is founded on analytical and numerical methods of nonlinear mechanics. Attention has been given to the extension of methods to high orders of approximation, required now by the increased accuracy of measurements in all fields of science and technology. The main theorems relevant to each perturbation technique are outlined, but they only provide a foundation and are not the objective of these notes. Each chapter concludes with a detailed survey of the pertinent literature, supplemental information and more examples to complement the text, when necessary, for better comprehension. The references are intended to provide a guide for background information and for the reader who wishes to analyze any particular point in more detail. The main sources referenced are in the fields of differential equations, nonlinear oscillations and celestial mechanics. Thanks are due to Katherine MacDougall and Sandra Spinacci for their patience and competence in typing these notes. Partial support from the Mathematics Program of the Office of Naval Research is gratefully acknowledged. |
| perturbation methods in applied mathematics: Introduction to Perturbation Methods Mark H. Holmes, 2012-12-05 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. One hundred new pages added including new material on transcedentally small terms, Kummer's function, weakly coupled oscillators and wave interactions. |
| perturbation methods in applied mathematics: Perturbation Methods in Matrix Analysis and Control Mihail M. Konstantinov, Petko Hr Petkov, 2020-04-06 Notation and preliminaries -- Perturbation problems -- Splitting operators and Lyapunov majorants -- Schur decomposition -- Hamiltonian matrices : basic relations -- Hamiltonian matrices : asymptotic analysis -- Hamiltonian matrices : non-local analysis -- Orthogonal canonical forms -- Feedback synthesis problem. |
| perturbation methods in applied mathematics: A First Look at Perturbation Theory James G. Simmonds, James E. Mann, 2013-07-04 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. |
| perturbation methods in applied mathematics: Singular Perturbation Theory R.S. Johnson, 2005-12-28 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. |
| perturbation methods in applied mathematics: Asymptotic Analysis and Perturbation Theory William Paulsen, 2013-07-18 Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge o |
| perturbation methods in applied mathematics: Perturbation theory for linear operators Tosio Kato, 2013-06-29 |
| perturbation methods in applied mathematics: Multiple Scale and Singular Perturbation Methods J.K. Kevorkian, J.D. Cole, 2012-12-06 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. |
| perturbation methods in applied mathematics: Mathematics Applied to Deterministic Problems in the Natural Sciences C. C. Lin, L. A. Segel, 1988-12-01 This book addresses the construction, analysis, and intepretation of mathematical models that shed light on significant problems in the physical sciences, with exercises that reinforce, test and extend the reader's understanding. It may be used as an upper level undergraduate or graduate textbook as well as a reference for researchers. |
| perturbation methods in applied mathematics: Introduction to Asymptotic Methods David Y. Gao, Vadim A. Krysko, 2006-05-03 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 m |
| perturbation methods in applied mathematics: Perturbation Methods for Engineers and Scientists Alan W. Bush, 1992-02-03 Perturbation Methods for Engineers and Scientists examines the main techniques of perturbation expansions applied to both differential equations and integral expressions. It describes several fluid dynamics applications, including aerofoils, boundary layers in momentum heat, and mass transfer. In addition, it applies the multiple scale technique to the description of surface roughness effects in lubrication. The book's intuitive, rather than formal, approach enables these advanced techniques to be used by scientists and engineers as well as by students. |
| perturbation methods in applied mathematics: Solving Transcendental Equations John P. Boyd, 2014-09-23 Transcendental equations arise in every branch of science and engineering. While most of these equations are easy to solve, some are not, and that is where this book serves as the mathematical equivalent of a skydiver's reserve parachute--not always needed, but indispensible when it is. The author's goal is to teach the art of finding the root of a single algebraic equation or a pair of such equations. |
| perturbation methods in applied mathematics: Multiscale Methods Grigoris Pavliotis, Andrew Stuart, 2008-01-18 This introduction to multiscale methods gives you a broad overview of the methods’ many uses and applications. The book begins by setting the theoretical foundations of the methods and then moves on to develop models and prove theorems. Extensive use of examples shows how to apply multiscale methods to solving a variety of problems. Exercises then enable you to build your own skills and put them into practice. Extensions and generalizations of the results presented in the book, as well as references to the literature, are provided in the Discussion and Bibliography section at the end of each chapter.With the exception of Chapter One, all chapters are supplemented with exercises. |
| perturbation methods in applied mathematics: Finite Difference Methods for Ordinary and Partial Differential Equations Randall J. LeVeque, 2007-01-01 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. |
| perturbation methods in applied mathematics: Fitted Numerical Methods For Singular Perturbation Problems: Error Estimates In The Maximum Norm For Linear Problems In One And Two Dimensions (Revised Edition) John J H Miller, Eugene O'riordan, G I Shishkin, 2012-02-29 Since the first edition of this book, the literature on fitted mesh methods for singularly perturbed problems has expanded significantly. Over the intervening years, fitted meshes have been shown to be effective for an extensive set of singularly perturbed partial differential equations. In the revised version of this book, the reader will find an introduction to the basic theory associated with fitted numerical methods for singularly perturbed differential equations. Fitted mesh methods focus on the appropriate distribution of the mesh points for singularly perturbed problems. The global errors in the numerical approximations are measured in the pointwise maximum norm. The fitted mesh algorithm is particularly simple to implement in practice, but the theory of why these numerical methods work is far from simple. This book can be used as an introductory text to the theory underpinning fitted mesh methods. |
| perturbation methods in applied mathematics: Introduction to Perturbation Techniques Ali H. Nayfeh, 2011-04-08 Similarities, differences, advantages and limitations of perturbation techniques are pointed out concisely. The techniques are described by means of examples that consist mainly of algebraic and ordinary differential equations. Each chapter contains a number of exercises. |
| perturbation methods in applied mathematics: Applied Mathematics J. David Logan, 2013-06-18 Praise for the Third Edition “Future mathematicians, scientists, and engineers should find the book to be an excellent introductory text for coursework or self-study as well as worth its shelf space for reference.” —MAA Reviews Applied Mathematics, Fourth Edition is a thoroughly updated and revised edition on the applications of modeling and analyzing natural, social, and technological processes. The book covers a wide range of key topics in mathematical methods and modeling and highlights the connections between mathematics and the applied and natural sciences. The Fourth Edition covers both standard and modern topics, including scaling and dimensional analysis; regular and singular perturbation; calculus of variations; Green’s functions and integral equations; nonlinear wave propagation; and stability and bifurcation. The book provides extended coverage of mathematical biology, including biochemical kinetics, epidemiology, viral dynamics, and parasitic disease. In addition, the new edition features: Expanded coverage on orthogonality, boundary value problems, and distributions, all of which are motivated by solvability and eigenvalue problems in elementary linear algebra Additional MATLAB® applications for computer algebra system calculations Over 300 exercises and 100 illustrations that demonstrate important concepts New examples of dimensional analysis and scaling along with new tables of dimensions and units for easy reference Review material, theory, and examples of ordinary differential equations New material on applications to quantum mechanics, chemical kinetics, and modeling diseases and viruses Written at an accessible level for readers in a wide range of scientific fields, Applied Mathematics, Fourth Edition is an ideal text for introducing modern and advanced techniques of applied mathematics to upper-undergraduate and graduate-level students in mathematics, science, and engineering. The book is also a valuable reference for engineers and scientists in government and industry. |
| perturbation methods in applied mathematics: Perturbation Methods in Optimal Control Alain Bensoussan, 1988-06-23 Describes, analyzes, and generalizes the principal results concerning perturbation methods in optimal control for systems governed by deterministic or stochastic differential equations. Covers the most important theorems in deterministic and stochastic optimal control, the theory of ergodic control, and the use of control, including regular perturbations and singular perturbations. |
| perturbation methods in applied mathematics: Beyond Perturbation Shijun Liao, 2003-10-27 Solving nonlinear problems is inherently difficult, and the stronger the nonlinearity, the more intractable solutions become. Analytic approximations often break down as nonlinearity becomes strong, and even perturbation approximations are valid only for problems with weak nonlinearity. This book introduces a powerful new analytic method for nonlinear problems-homotopy analysis-that remains valid even with strong nonlinearity. In Part I, the author starts with a very simple example, then presents the basic ideas, detailed procedures, and the advantages (and limitations) of homotopy analysis. Part II illustrates the application of homotopy analysis to many interesting nonlinear problems. These range from simple bifurcations of a nonlinear boundary-value problem to the Thomas-Fermi atom model, Volterra's population model, Von Karman swirling viscous flow, and nonlinear progressive waves in deep water. Although the homotopy analysis method has been verified in a number of prestigious journals, it has yet to be fully detailed in book form. Written by a pioneer in its development, Beyond Pertubation: Introduction to the Homotopy Analysis Method is your first opportunity to explore the details of this valuable new approach, add it to your analytic toolbox, and perhaps make contributions to some of the questions that remain open. |
| perturbation methods in applied mathematics: Partial Differential Equations of Applied Mathematics Erich Zauderer, 1998-08-04 The only comprehensive guide to modeling, characterizing, and solving partial differential equations This classic text by Erich Zauderer provides a comprehensive account of partial differential equations and their applications. Dr. Zauderer develops mathematical models that give rise to partial differential equations and describes classical and modern solution techniques. With an emphasis on practical applications, he makes liberal use of real-world examples, explores both linear and nonlinear problems, and provides approximate as well as exact solutions. He also describes approximation methods for simplifying complicated solutions and for solving linear and nonlinear problems not readily solved by standard methods. The book begins with a demonstration of how the three basic types of equations (parabolic, hyperbolic, and elliptic) can be derived from random walk models. It continues in a less statistical vein to cover an exceptionally broad range of topics, including stabilities, singularities, transform methods, the use of Green's functions, and perturbation and asymptotic treatments. Features that set Partial Differential Equations of Applied Mathematics, Second Edition above all other texts in the field include: Coverage of random walk problems, discontinuous and singular solutions, and perturbation and asymptotic methods More than 800 practice exercises, many of which are fully worked out Numerous up-to-date examples from engineering and the physical sciences Partial Differential Equations of Applied Mathematics, Second Edition is a superior advanced-undergraduate to graduate-level text for students in engineering, the sciences, and applied mathematics. The title is also a valuable working resource for professionals in these fields. Dr. Zauderer received his doctorate in mathematics from the New York University-Courant Institute. Prior to joining the staff of Polytechnic University, he was a Senior Weitzmann Fellow of the Weitzmann Institute of Science in Rehovot, Israel. |
| perturbation methods in applied mathematics: Advanced Numerical and Semi-Analytical Methods for Differential Equations Snehashish Chakraverty, Nisha Mahato, Perumandla Karunakar, Tharasi Dilleswar Rao, 2019-04-16 Examines numerical and semi-analytical methods for differential equations that can be used for solving practical ODEs and PDEs This student-friendly book deals with various approaches for solving differential equations numerically or semi-analytically depending on the type of equations and offers simple example problems to help readers along. Featuring both traditional and recent methods, Advanced Numerical and Semi Analytical Methods for Differential Equations begins with a review of basic numerical methods. It then looks at Laplace, Fourier, and weighted residual methods for solving differential equations. A new challenging method of Boundary Characteristics Orthogonal Polynomials (BCOPs) is introduced next. The book then discusses Finite Difference Method (FDM), Finite Element Method (FEM), Finite Volume Method (FVM), and Boundary Element Method (BEM). Following that, analytical/semi analytic methods like Akbari Ganji's Method (AGM) and Exp-function are used to solve nonlinear differential equations. Nonlinear differential equations using semi-analytical methods are also addressed, namely Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Variational Iteration Method (VIM), and Homotopy Analysis Method (HAM). Other topics covered include: emerging areas of research related to the solution of differential equations based on differential quadrature and wavelet approach; combined and hybrid methods for solving differential equations; as well as an overview of fractal differential equations. Further, uncertainty in term of intervals and fuzzy numbers have also been included, along with the interval finite element method. This book: Discusses various methods for solving linear and nonlinear ODEs and PDEs Covers basic numerical techniques for solving differential equations along with various discretization methods Investigates nonlinear differential equations using semi-analytical methods Examines differential equations in an uncertain environment Includes a new scenario in which uncertainty (in term of intervals and fuzzy numbers) has been included in differential equations Contains solved example problems, as well as some unsolved problems for self-validation of the topics covered Advanced Numerical and Semi Analytical Methods for Differential Equations is an excellent text for graduate as well as post graduate students and researchers studying various methods for solving differential equations, numerically and semi-analytically. |
Perturbation Methods (M16) - University of Cambridge
Perturbation Methods (M16) S.J. Cowley. This course will deal with the asymptotic solution to problems in applied mathematics in gen-eral when some parameter or coordinate in the …
Perturbation Methods GM01 Dr. Helen J. Wilson Autumn Term …
perturbation methods are a cornerstone of applied mathematics. The second use of perturbation methods is coupled with numerical methods. Although computed solutions to a problem can be …
Perturbation Methods - api.pageplace.de
There are two methods for obtain ing precise approximations to the solutions of an equation, numerical methods and analytic methods, and this book is about the latter. The analytic …
2 Introduction to perturbation methods
Perturbation methods are methods which rely on there being a dimensionless parameter in the problem that is relatively small: " 1. The most common example you may have seen before is …
Perturbation methods, Physics 2400 - Mathematical methods for …
25 Apr 2017 · The approach of perturbation theory is to decompose a tough problem into an (infinite) number of relatively easy ones. The perturbation theory is most useful when the first …
Perturbation Methods - University of Pennsylvania
Regular perturbation: a small change in the problem induces a small change in the solution. Singular perturbation: a small change in the problem induces a large change in the solution. …
Perturbation Methods in Applied Mathematics
The goal of this course is to explore mathematical methods for obtaining approximate analytical solutions to differential equations that cannot be solved exactly and to develop insights and …
Perturbation Methods (M16)
Perturbation Methods (M16) Prof. I.D. Abrahams. This course will deal with the asymptotic solution to problems in applied mathematics when some parameter or coordinate assumes …
Introduction to Perturbation Methods - Springer
The methods covered in the text vary widely in their applicability. The first chapter introduces the fundamental ideas underlying asymptotic ap …
Multiple Scale and Singular Perturbation Methods
Multiple scale and singular perturbation methods/ J. Kevorkian, J.D. Cole. p. em. - (Applied mathematical sciences; v. 114) Includes bibliographical references and index. ISBN 0-387 …
Asymptotic analysis notes - Heriot-Watt University
10 2. PERTURBATION METHODS Expansion method. The idea behind this method is to formally expand the solution about one of the unperturbed roots, say x0 = +1, as a power series in : x( ) …
Prof. I.D. Abrahams
start with a brief review of classical asymptotic methods for the evaluation of integrals, but most of the lectures will be devoted to singular perturbation problems (including the methods of …
Perturbation Methods in Applied Mathematics - Arizona State …
The goal of this course is to explore mathematical methods for obtaining approximate analytical solutions to differential equations that cannot be solved exactly, and to develop insights and …
Math 537 - Ordinary Di erential Equations - San Diego State …
Approximations often use perturbation methods: The equations have a small term. One physical process is signi cantly less important than another dominant one. Often rescale problem with a …
Perturbation Methods for Solving Non-Linear Ordinary Differential …
Perturbation uses traditional perturbation methods and Homotopy techniques to reduce nonlinear problems to linear problems and generate rapid convergent series solutions in most cases. In …
Perturbation Theory and the WKB Method - Colorado State …
Pertubation Theory/Asymptotic Analysis is an important subject in applied mathematics, and there are many techniques used in this study, one of which is the WKB method.
MATH34011 - Asymptotic Expansions and Perturbation Methods
Perturbation methods represent a very powerful tool in modern mathematical physics and in particular, in fluid dynamics. This course unit will introduce students to a range of modern …
Math 551: Peturbation Methods for PDE’s in Applied Mathematics
The general concepts and methods are illustrated and developed for a wide variety of specific problems arising in math biology, fluid mechanics, materials science, and wave propagation. …
An Integral Equation Perturbation Technique in Applied …
An Integral Equation Perturbation Technique in Applied Mathematics R. P. KANWAL* Communicated by T. Y. Thomas 1. Introduction. The theory of diffraction and scattering has …
Perturbation Methods - Springer
this chapter, we give a brief account of some of the most successful …
Perturbation Methods (M16) - University of Cambridge
Perturbation Methods (M16) S.J. Cowley. This course will deal with …
Perturbation Methods GM01 Dr. Helen J. Wilson Autumn Te…
perturbation methods are a cornerstone of applied mathematics. …
Perturbation Methods - api.pageplace.de
There are two methods for obtain ing precise approximations to the …
2 Introduction to perturbation methods
Perturbation methods are methods which rely on there being a …
