Advanced Mathematical Methods for Scientists and Engineers: Asymptotic

Book Description

This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory and explains how to use these methods to obtain 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 and for which brute-force numerical methods may not converge to useful solutions. The objective of this book is to teaching the insights and problem-solving skills that are most useful in solving mathematical problems arising in the course of modern research. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with a an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader 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.

Customer Reviews:

Deeply insightful and utterly fascinating.......2006-12-20

I had the privilege to explore this guide to the universe of asymptotics under Prof. Bender. I used to think, and still do (about 2 decades on), that this book is like the Rosetta Stone, and enables us to solve anything in applied mathematics.

A must have for anyone looking to understand the incredible universe we find ourselves in!

The Classic Text on Asymptotics.......2005-12-09

I am a mathematics professor, and asymptotic methods is my area of expertise. Bender and Orszag was the standard text on this topic in the early 1980s, and it remains by far the most thorough book on the subject. I had two courses from it in graduate school and have taught from it 3 times now. I have never found a mistake. The book is very well written; its only weakness is that its graphics reflect the technology of the 1970s. Nevertheless, I am teaching from it again next year, because it is in a class by itself.

There is one thing that individual readers and faculty users should be aware of. Some of the exercises, including a few marked as "intermediate," are incredibly difficult. My instructor made the mistake of assigning exercises without working them first. I am careful not to assign any exercises until after I have worked them.

A Classic Resource.......2005-10-07

I'll echo the sentiments of the other reviewers here. I have a well-worn and well-used copy of the first (1978) edition. Believe it or not, I actually had a chance to use these methods as recently as last month (September 2005). I would recommend, though, if you want to get the most out of this book, pick up a symbolic calculation program or calculator capable of doing symbolic math. I use the open-source programs Maxima and Axiom for this, but if you want to actually spend money, Amazon has some calculators that will handle this sort of thing.

Extremely useful.......2003-12-13

Great for anyone taking Partial differential equations, mathematical physics, and the related courses. Carl Bender is a master of the field as well as a great book writer, as well as Orszag. The only complaint is the type setting--a bit too small. But not a big problem

Excellent.......2003-10-19

This is an essential textbook for all applied mathematicians, physicists and research oriented engineers. It is very well written, and explains an enormous lot. It explains it all very well too. I've used chapter 3 and 6 in class, and they were full of insights and information that you cannot find in any other textbook.

Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Encyclopaedia of Mathematical Sciences)

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Partial Differential Equations V: Asymptotic Methods for Partial Differential Equations (Encyclopaedia of Mathematical Sciences)

Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540533710

Book Description

The six articles in this EMS volume provide an overview of a number of contemporary techniques in the study of the asymptotic behavior of partial differential equations.These techniques include the Maslov canonical operator, semiclassical asymptotics of solutions and eigenfunctions, behavior of solutions near singular points of different kinds, matching of asymptotic expansions close to a boundary layer, and processes in inhomogeneous media. Asymptotic expansions are one of the most important areas in the theory of partial differential equations. Readers should find the wide variety of approaches of interest.

Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Cambridge Texts in Applied Mathematics)

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Brilliant book! By a brilliant man!!
Dimensional Scaling
Scaling , self-similarity, and intermediate asymptotics

Scaling, Self-similarity, and Intermediate Asymptotics: Dimensional Analysis and Intermediate Asymptotics (Cambridge Texts in Applied Mathematics)
Grigory Isaakovich Barenblatt
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0521435226

Book Description

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.

Customer Reviews:

Brilliant book! By a brilliant man!!.......2002-09-29

Last semester I took professor Barrenblatt's graduate course Math 275 at UC Berkeley: "Advanced topics in Applied Mathematics." The topics covered therein were more or less what is covered in this book. I am not a math major, but a civil engineering one, and the course a lot of times got way over my head. Nevertheless, it was a truly amazing experience. I learned a lot. But enough about the course...
This is a truly great book! The introduction (Chapter 0) is a little overwhelming because it attempts to present an overview of topics covered in the following chapters of the book, but the brevity and lack of rigor (it is a summary) may result in confusion. This was the one and only weak point in the book. So... what did I do? I skipped the intro chapter. You can go back to it after you have read the book (or a good part of it) and things will make a lot more sense. From chapter 1 forward, the book is excellent. The ideas are very interesting (this is an applied math book, and the author documents real world examples of where the ideas are applicable) and the concepts presented with sufficient rigor and lucidity that one expects from a mathematics book. Barenblatt is a truly brilliant mathematician and an excellent educator as well, and provides deep insight about dimensional analysis, scaling, similarity, and intermediate asymptotics in this book.
Buy it!

Dimensional Scaling.......2001-02-08

As a specialist in the topic of dimensional scaling I found the book one of the mot interesting in this field. The material is well explained.

Scaling , self-similarity, and intermediate asymptotics.......2000-04-05

Barenblatt's book, Scaling, self similarity and intermediate asymptotics, addresses the understanding of physical processes and the interpretation of calculations revealing these processes, two mental problems intertwined closely with the deeper more general issues raised by the recognition of patterns. The book contains excellent problems that are considered in detail and then followed by brilliant generalizations that inspire and provoke reflection.

This book contains many deep examples of analytic solutions to various problems, including propagation of heat from a source in linear and nonlinear cases, and energy propagation from a localized explosion, in which dimensions of the constants that characterize the medium and the dimensions of energy determine uniquely the exponents of the self-similar solutions. By introducing losses, however, the problems change, so that now the conservation of energy does not hold, but the self-similarity remains.

Problems of the non-linear propagation of waves on the surface of a heavy fluid, described by the Kortweg-de Vries equation, are excellent. This example is remarkable in that theorems exist proving the stability of solitons even after these solitons collide. The solutions giving the asymptotic behavior of generalized initial distributions are then transformed beautifully into a sequence of solitons.

In general problems included in this book are focused, cleverly presented and are exemplary. Many are non-linear, and their special solutions represent the asymptotics of a wider class of other more general solutions corresponding to many different initial conditions.

Book Description

The book deals with the symptotic analysis of relaxation oscillations, which are nonlinear oscillations characterized by rapid change of a variable within a short time interval of the cycle. The type of asymptotic approximation of the solution is known as the method of matched asymptotic expansions. In case of coupled oscillations it gives conditions for entrainment. For spatially distributed oscillators phase wave solutions can be constructed. The asymptotic theory also covers the chaotic dynamics of free and forced oscillations. The influence of stochastic perturbations upon the period of the oscillation is also covered. It is the first book on this subject which also provides a survey of the literature, reflecting historical developments in the field. Furthermore, relaxation oscillations are analyzed using the tools drawn from modern dynamical system theory. This book is intended for graduate students and researchers interested in the modelling of periodic phenomena in physics and biology and will provide a second knowledge of the application of the theory of nonlinear oscillations to a particular class of problems.

Maslov Classes, Metaplectic Representation and Lagrangian Quantization (Mathematical Research Vol. 5)

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Maslov Classes, Metaplectic Representation and Lagrangian Quantization (Mathematical Research Vol. 5)
Maurice de Gosson
Manufacturer: Wiley-VCH
ProductGroup: Book
Binding: Paperback

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ASIN: 3527400877

Book Description

The Maslov Classes have been playing an essential role in various parts of applied and pure mathematics, and physics, since the early 70's. Their correct definition is due to V. I. Arnold and J. Leray, in the transversal case, and to P. Dazord and the author in the general case. The aim of this book is to give a thorough treatment of the theory of the Maslov classes and of their relationship with the metaplectic group. It is (among other things) shown that these classes can be reconstructed, modulo 4, using only the analytic properties of the metaplectic group. In the last chapter the author sketches a scheme for geometric quantization by introducing two new concepts, that of metaplectic half-form and that of Lagrangian catalogue, the latter generalizes and simplifies the notion of "Lagrangian function" introduced by J. Leray. A Lagrangian catalogue is a collection of metaplectic half-forms which are themselves "cohomological wave functions", whose definition is made possible by using the combinatorial properties of the Maslov classes. The transformation of Lagrangian catalogues under the metaplectic group and of Hamiltonian flows is studied, and it is shown that one thus recovers very easily the so-called "quasi-classical approximation" to the solutions of Schrödinger equation if one introduces a natural concept, that of projection of a Lagrangian catalogue. An application to geometric phase shifts, including Berry's phase, is given.

Applied Asymptotic Methods in Nonlinear Oscillations (Solid Mechanics and Its Applications)

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Compact and well structured

Applied Asymptotic Methods in Nonlinear Oscillations (Solid Mechanics and Its Applications)
Yuri A. Mitropolsky , and Nguyen Van Dao
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 079234605X

Book Description

The present volume addresses the application of asymptotic methods in nonlinear oscillations. Such methods see a large variety of applications in physics, mechanics and engineering.
The advantages of using asymptotic methods in solving nonlinear problems are firstly simplicity, especially for computing higher approximations, and secondly their applicability to a large class of quasi-linear systems.
In contrast to the existing literature, this book is concerned with the applied aspects of the methods concerned and also contains problems relevant to the everyday practice of engineers, physicists and mathematicians.
Usually, dynamics systems are classified and examined by their degrees of freedom. This book is constructed from another point of view based upon the originating mechanism of the oscillations: free oscillation, self-excited oscillation, forced oscillation, and parametrically excited oscillation.
The text has been designed to cover material from the elementary to the more advanced, in increasing order of difficulty. It is of considerable interest to both students and researchers in applied mathematics, physical and mechanical sciences, and engineering.

Customer Reviews:

Compact and well structured.......1999-07-23

The book is a more compact version of the 'Asymptotic methods in the theory of nonlinear oscillations' by Mitropolskii and Bogoliobov published years back. Nonlinear equations close to linear ones are solved using the methods suggested by Krylov and Bogoliobov. The presentation is orderly dealing with free oscillations first, then self oscillations leading to forced oscillations. A neat article on the averaging method ends the book in a nice fashion bringing about some of the connections.

Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications (Springer Series in Synergetics)

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Asymptotic Methods for the Fokker-Planck Equation and the Exit Problem in Applications (Springer Series in Synergetics)
Johan Grasman , and Onno A., van Herwaarden
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540644350

Book Description

Asymptotic methods are of great importance for practical applications, especially in dealing with boundary value problems for small stochastic perturbations. This book deals with nonlinear dynamical systems perturbed by noise. It addresses problems in which noise leads to qualitative changes, escape from the attraction domain, or extinction in population dynamics. The most likely exit point and expected escape time are determined with singular perturbation methods for the corresponding Fokker-Planck equation. The authors indicate how their techniques relate to the Itô calculus applied to the Langevin equation. The book will be useful to researchers and graduate students.

Asymptotic Methods in Equations of Mathematical Physics

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Asymptotic Methods in Equations of Mathematical Physics
B Vainberg
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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Book Description

This book provides a single source for both students and advanced researchers on asymptotic methods employed in the linear problems of mathematical physics. It opens with a section based on material from special courses given by the author which gives detailed coverage of classical material on the equations of mathematical physics and their applications, and includes a simple explanation of the Maslov Canonical operator method. The book goes on to present more advanced material from the author's own research. Topics range from radiation conditions and the principle of limiting absorption for general exterior problems, to complete asymptotic expansion of spectral function of equations over all of space.

Asymptotic Methods in Mechanics (Crm Proceedings & Lecture Notes, Vol 3)

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Asymptotic Methods in Mechanics (Crm Proceedings & Lecture Notes, Vol 3)
Remi Vaillancourt
Manufacturer: American Mathematical Society
ProductGroup: Book
Binding: Paperback

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ASIN: 0821869930

Book Description

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

Asymptotics for Dissipative Nonlinear Equations (Lecture Notes in Mathematics)

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Asymptotics for Dissipative Nonlinear Equations (Lecture Notes in Mathematics)
N. Hayashi , E.I. Kaikina , P. Naumkin , and I.A. Shishmarev
Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 3540320598

Book Description

Many of problems of the natural sciences lead to nonlinear partial differential equations. However, only a few of them have succeeded in being solved explicitly. Therefore different methods of qualitative analysis such as the asymptotic methods play a very important role. This is the first book in the world literature giving a systematic development of a general asymptotic theory for nonlinear partial differential equations with dissipation. Many typical well-known equations are considered as examples, such as: nonlinear heat equation, KdVB equation, nonlinear damped wave equation, Landau-Ginzburg equation, Sobolev type equations, systems of equations of Boussinesq, Navier-Stokes and others.

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