Product Description
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.

The book is organized into two main sections and a set of appendices. Part I addresses steady-state boundary value problems, starting with two-point boundary value problems in one dimension, followed by coverage of elliptic problems in two and three dimensions. It concludes with a chapter on iterative methods for large sparse linear systems that emphasizes systems arising from difference approximations. Part II addresses time-dependent problems, starting with the initial value problem for ODEs, moving on to initial boundary value problems for parabolic and hyperbolic PDEs, and concluding with a chapter on mixed equations combining features of ODEs, parabolic equations, and hyperbolic equations. The appendices cover concepts pertinent to Parts I and II. Exercises and student projects, developed in conjunction with this book, are available on the book s webpage along with numerous MATLAB m-files.

Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics and further explore the theory and/or use of finite difference methods according to their interests and needs.

Audience: This book is designed as an introductory graduate-level textbook on finite difference methods and their analysis. It is also appropriate for researchers who desire an introduction to the use of these methods.

Contents: Preface; Part I: Boundary Value Problems and Iterative Methods. Chapter 1: Finite Difference Approximations; Chapter 2: Steady States and Boundary Value Problems; Chapter 3: Elliptic Equations; Chapter 4: Iterative Methods for Sparse Linear Systems; Part II: Initial Value Problems. Chapter 5: The Initial Value Problem for Ordinary Differential Equations; Chapter 6: Zero-Stability and Convergence for Initial Value Problems; Chapter 7: Absolute Stability for Ordinary Differential Equations; Chapter 8: Stiff Ordinary Differential Equations; Chapter 9: Diffusion Equations and Parabolic Problems; Chapter 10: Advection Equations and Hyperbolic Systems; Chapter 11: Mixed Equations; Appendix A: Measuring Errors; Appendix B: Polynomial Interpolation and Orthogonal Polynomials; Appendix C: Eigenvalues and Inner-Product Norms; Appendix D: Matrix Powers and Exponentials; Appendix E: Partial Differential Equations; Bibliography; Index. ... Read more

Customer Reviews (1)

Good book
This is a great book for numerical analysis and finite differences. The author makes it simple to understand(well mostly) without sacrificing rigor.
... Read more

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Customer Reviews (1)

Excellent text for abstract differential equations
"A Second Course in Elementary Differential Equations" by Waltman combines an eloquence in proofs and an ease in explanation that makes the important proofs readable.This book should be in every mathematics library. ... Read more

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A concise yet solid introduction to advanced numerical methods,
this textbook introduces several major numerical methods for solving various partial differential equations (PDEs) in science and engineering, including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques that include the classic finite difference method and the finite element method as well as state-of-the-art numerical methods, such as the high-order compact difference method and the radial basis function meshless method.

Helps Students Better Understand Numerical Methods through Use of MATLAB^®
The authors uniquely emphasize both theoretical numerical analysis and practical implementation of the algorithms in MATLAB, making the book useful for students in computational science and engineering. They provide students with simple, clear implementations instead of sophisticated usages of MATLAB functions.

All the Material Needed for a Numerical Analysis Course
Based on the authors’ own courses, the text only requires some knowledge of computer programming, advanced calculus, and difference equations. It includes practical examples, exercises, references, and problems, along with a solutions manual for qualifying instructors. Readers can download MATLAB source code from crcpress.com, which will allow them to easily modify or improve the codes to solve their own problems.

Customer Reviews (2)

One of the best book on numerical analysis
I bought this book from Amazon.com for a graduate course in numerical solution of PDE. It comes with a CD that contain all MATLAB codes. This book is kind of middle of engneer and mathematician's perspective on numerical analysis. MATLAB codes runs great, exercise problems are also good to understand many difficult relavent concept. Overall, I think, this book is one the best numerical analysis book. Amazon did a little delay to deliver this book to me.

A valuale numerical PDE book
This book is one of the best written on the subject and is suitable for readers in a wide variety of fields, including mathematics, computational sciences and engineering. It is certainly well-suited for classroom use, and it includes many stand alone MATLAB source codes.

The book not only cover classic finite difference methods, but also finite element methods, and meshless methods. The book also include some advanced topics such as high-order compact difference methods, radial basis meshless methods and Maxwell's equaations in dispersive media.

I found those codes are very helpful for me to learn the algorithms (many books just talk the algorithms in the air), and I can even extend some codes in the book immediately for my research, since the authors kindly released their most recent work (such as compact scheme and meshless methods, which are currently quite active research areas) in the book.

Very useful!
... Read more

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Customer Reviews (4)

rigorous but understandable
I got this book after finishing Farlow's Partial Differential Equations for Scientists and Engineers. This is a much more mathematically rigorous and sophisticated book, and it took me a lot longer to go through it. (I still haven't done the last two sections on the Laplace transform and approximation methods). My original goal was to learn complex analysis, and since Farlow recommended this book for that topic, I started in the middle with Section VIII, Analytic Functions of a Complex Variable, then did IX and X, then I through VII. This caused a few difficulties solving some of the problems in the later sections due to unfamiliarity with material presented in the earlier sections. E.g. problem 71.4 presumes a knowledge of Green's function, which is covered in section V. Nonetheless, this worked out OK. I now feel as though I have a pretty solid grounding in the material (by which I mean not that I can remember it all, but I can recognize when I've seen it before, and know where to look it up again). That is my operational definition of having "learned" the material.

Frankly, I'm quite surprised by the second reviewer's comment that this book would not be good for self-instruction. I thought it was great! But I had to be very patient and resist the urge to move ahead before I really understood what I had just read. Doing the problems is an excellent reality check! That's one of the features of this book that I liked the most: lots of problems, with ALL the answers in the back, so you can't fool yourself. Most of the answers can be reached with persistence, although I was stumped by a few. But don't buy this book if you don't have a background in calculus and ordinary differential equations! It would also be helpful to review the topic of partial differential equations in an easier format, like Farlow, before progressing to this text.

The material is interesting and varied. There are sections heavy on abstract math and proofs of theorems, other sections mostly practical in orientation. The section on complex analysis lives up to its billing in Farlow. In general, I found that I could skim through the really abstract material without losing the thread of the argument. I'm not good at proofs so I skipped over most of the problems that called for proving something. But at least now I know what is meant by existence and uniqueness proofs, etc.

The book has very few errors. I've listed the ones I think are errors below. Maybe they are and maybe they aren't!

p. 182 second paragraph R(r) sin(n*theta) etc. should be R(r) sin(m*theta) etc.

p. 192 fifth line from bottom "degree n in x,y, and z" should be "degree n in x,y, and r".

p. 223 second equation from bottom should be du2/de*du1/dx - dv2/de*dv1/dx + i(dv2/de*du1/dx + du2/de*dv1/dx)

p. 265 last equation right hand side integrand should be e^(-i*theta), not e^(i*theta).

Solutions to exercises:

1.4: rho subzero du1/dx + d rho subone/dt = 0

5.4: answer can be reduced further to 157/4096, which is how my calculator displayed it

7.5: last line: dA/dt greater than or equal to 0

10.2: second line: epsilon 3 = z*sqrt(alpha/D)

14.1e: e^(-n^2*pi^2*t-t-x)*sin(n*pi*(x-1))

18.4: right side of equation should be 1/2 (e^pi + e^(-pi)) i.e. cosh(pi)

31.4: one times first sum, not two times first sum, and difference between first sum and second (double) sum, not sum of the two

32.3: second line, numerator, sinh term should be sinh (sqrt((n-1/2)^2 + (2m-1)^2 +1) * (1-z))

37.1: substituting xbar = x+1 in formula for c sub-n I get c sub-n twice what answer gives.

37.4: formula for c sub-n: integrand should include (1+x)^-1, not ^-2.

44.4: the exponential t-term is e^(-(mu sub-k supra-(n+1/2))*t), not e^(-mu sub-k supra-((n+1/2)*t))

50.7: log term is log(2 plus or minus sqrt(3))

53.4: the part of the disk to the LEFT of the line....

58.2: ? pi*i*(e-2)

58.7: ? -pi/3

Happy self-instructing!

Book teaches specifics while ignoring big picture
This is not so much a book as it is a published version of lecture notes: it might make a useful supplement to a textbook in a course, or a useful review text for someone who already knows the subject, but it is useless as a standalone textbook for a course or for someone studying the material on their own.The book dives into specific applications and problems without giving much theoretical background.Even for a student with a strong background in ODE's and some exposure to PDE's, this book will be difficult to follow and understand.

The book is essentially a walkthrough of solutions to specific problems.There is little discussion of the theoretical foundations behind the material, little motivation for any of the solutions, little discussion of why a particular line of attack was chosen for a specific problem.

The reader is left with the ability to carry out specific techniques but without any global picture of how to know which techniques are appropriate for which situations.

very thorough book, a bit dated but sophisticated
I used this book for a yearlong course in PDE'swhen I was an undergraduate. I really wish I hadhad that course before quantum mechanics, as manyof the difficult mathematics there would have been cakeafter weinberger. This book is fairly terse but quite complete. The textcan be hard to follow by onesself and I often would refer to easier bookson PDE's to get over certain humps. However, weinberger covers most of thematerial that anyundergraduate would want and does it at a relativelyhigh level. The problems are simply posed but are quite solvable and allhave answers in the back of the book. My recollection is that the book hasvery few typos. The discussion ismore mathematically rigourous than manymore popular books but not so rigourous that it ispainful for someone whoviews the mathematics asa tool. The book isorganized into about 90sections, each corresponding roughly to a 50 minute lecture there areproblem sets of approximately 10 problems for each of these. More recenttopics touching on nonlinear PDEs and the like are not here as the book waswritten in 1966. If you want to get a very solid background in PDEs thatwill leave you in goodstead for conquering books like Jackson'sElectrodynamics or Schiff's quantum mechanics, this is a good book. If youwant a simple introduction, for limited use go somewhere elselikechurchill or powers.

Temario
No entiendo completamente lo que dicen, por ello no se como adquirir el libro(necesitoadquirir varios ejemplares).Manden mayor información. ... Read more

Product Description
Covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities. Illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Softcover. ... Read more

Customer Reviews (1)

A Classic ODE Text
This classic text on ordinary differential equations has withstood the test of time and after 30 years is still considered to be one of the finest books in its field.

After reading Hartman's book the mathematician iswell equiped to contribute to research in the area of differentialequations. ... Read more

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Product Description
Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems. ... Read more

Customer Reviews (3)

Payback in longrun; not a Cookbook.
In the graduate school, a Math professor used this book, as it talked about a bit of everything. Everything as in initial value problems (IVP) in ODE, 2 point boundary value problems ( BVP ) for ODE, finite difference schemes for boundary value elliptic PDE's and initial-boundary type hypebolic PDE's.

However, the book is written for a mathematician in mind,as the author clearly mentions in the preface. It is not for the light hearted. Book would serve as a
starting point for rigourous foundations in numerical analysis methods for solving
ODE/PDE. Excellent book over all, with numerical examples, watertight arguments,
and crisp prose, without being boring.

Informal, nice text
A very informal style of writing with lots of explanation.He doesn't skip large steps like in the old-fashioned terse style of math texts, which makes it very readable, though some readers may not like it.Not very rigorous, but he's upfront about it.

The original version from 1996 has quite a few errors, and the author maintains information on errata on his website.The most recent reprinting has corrected most of these errors.So, even though there is only a single edition, some versions have errors and some don't.So, BEWARE BUYING USED EDITIONS because they will most likely be from an earlier printing and thus have more errors.I assume the new version on amazon is the corrected version.

Excellent for a graduate course on numerical DE
This is an excellent reference and textbook for someone hoping to go beyond the introduction to numerical DE found in any of the standard numerical analysis textbooks.It is not a research monograph, but is also not easy reading.It has already become a fairly standard reference in the literature because of its complete coverage and further references to more specialized sources.I have used it as the textbook for a graduate courseon numerical differential equations.I highly recommend it for that purpose and as a reference for someone doing independent reading. ... Read more

Product Description
Based on a one-year course taught by the author to graduates at the University of Missouri, this book provides a student-friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations. In a second semester, these ideas can be expanded by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem, while the latter sections of the book introduce the basic ideas of perturbation theory as applications of this Theorem. The book also contains material differing from standard treatments, for example, the Fiber Contraction Principle is used to prove the smoothness of functions that are obtained as fixed points of contractions. The ideas introduced in this section can be extended to infinite dimensions. ... Read more

Product Description
Excellent undergraduate/graduate-level introduction presents full introduction to the subject and to the Fourier series as related to applied mathematics, considers principal method of solving partial differential equations, examines first-order systems, computation methods, and much more. Over 600 problems and exercises, with answers for many. Ideal for a one-semester or full-year course.
... Read more

Customer Reviews (3)

Views from the student
Having just taken this course from Professor Gustafson, I can say, with complete support from everyone in every PDEs class he's taught as of yet, that this book leaves a lot to be desired.There is absolutely no mention of separation of variables, the section on greens functions forgets to mention at all what they're used for, and further how to do anything with them, and numerous other grave omissions.I feel a more apt title would be: "An overview of PDE's for the advanced student, and interesting applications."So perhaps, for the right teacher this will be a fine book, but beware, you are going to have a lot of work filling in the large gaps the book leaves.

Also, thought this has no bearing on the quality of the book, he has a strange obsession with the number 3.

review 1
This is an excellent book for the beginning engineer/scientist as well as the more experienced technical person. I will use this as a reference in the class I teach on Mathematical Methods for Electromagnetic Theory.

Unique Organization
I recently taught a one-semester course out of this text, having chosen Gustafson's book after a careful review of most of the standard introductory PDE texts.The feature which distinguishes this text from its competitors is its organization, which is based upon the author's belief in the pedagogical style of reinforcement through repetition.Within the first 50 pages, the reader has already seen a first treatment of (i) separation of variables and Fourier techniques, (ii) Green's functions, and (iii) variational (or energy) methods. One then repeatedly studies each of these standard solution techniques in greater depth at later points in the text.By contrast, with many alternative texts one can read 300 pages and still know nothing about Green's functions or variational techniques.Additionally, Gustafson writes so clearly that the text could be used for independent study.His selection of problems (3 Problems and 3 Exercises at the end of each section)reflects careful, deliberate choices.One is not overwhelmed with endless pages of repetitious "drill" exercises. Instead, each problem has a definite purpose and illustrates an important point.This text is a masterpiece, and Dover is to be congratulated for keeping it in print. ... Read more

Product Description

This self-contained text offers an elementary introduction to partial differential equations (pdes), primarily focusing on linear equations, but also providing some perspective on nonlinear equations. The classical treatment is mathematically rigorous with a generally theoretical layout, though indications to some of the physical origins of pdes are made throughout in references to potential theory, similarity solutions for the porous medium equation, generalized Riemann problems, and others.

The material begins with a focus on the Cauchy–Kowalewski theorem, discussing the notion of characteristic surfaces to classify pdes. Next, the Laplace equation and connected elliptic theory are treated, as well as integral equations and solutions to eigenvalue problems. The heat equation and related parabolic theory are then presented, followed by the wave equation in its basic aspects. An introduction to conservation laws, the uniqueness theorem, viscosity solutions, ill-posed problems, and nonlinear equations of first order round out the key subject matter.

Large parts of this revised second edition have been streamlined and rewritten to incorporate years of classroom feedback, correct errors, and improve clarity. Most of the necessary background material has been incorporated into the complements and certain nonessential topics have been given reduced attention (noticeably, numerical methods) to improve the flow of presentation.

The exposition is replete with examples, problems and solutions that complement the material to enhance understanding and solidify comprehension. The only prerequisites are advanced differential calculus and some basic L^p theory. The work can serve as a text for advanced undergraduates and graduate students in mathematics, physics, engineering, and the natural sciences, as well as an excellent reference for applied mathematicians and mathematical physicists.

From Reviews of the First Edition:

"The author's intent is to present an elementary introduction to pdes... In contrast to other elementary textbooks on pdes...much more material is presented on the three basic equations: Laplace's equation, the heat and wave equations...The presentation is clear and well organized...The text is complemented by numerous exercises and hints to proofs."

–Mathematical Reviews

"This is a well-written, self-contained, elementary introduction to linear, partial differential equations."

–ZentrallblattMATH

"This book certainly can be recommended as an introduction to PDEs in mathematical faculties and technical universities."

–Applications of Mathematics

Product Description
Most mathematicians, engineers, and many other scientists are well-acquainted with theory and application of ordinary differential equations.This book seeks to present Volterra integral and functional differential equations in that same framwork, allowing the readers to parlay their knowledge of ordinary differential equations into theory and application of the more general problems.Thus, the presentation starts slowly with very familiar concepts and shows how these are generalized in a natural way to problems involving a memory.Liapunov's direct method is gently introduced and applied to many particular examples in ordinary differential equations, Volterra integro-differential equations, and functional differential equations.

By Chapter 7 the momentum has built until we are looking at problems on the frontier.Chapter 7 is entirely new, dealing with fundamental problems of the resolvent, Floquet theory, and total stability.Chapter 8 presents a solid foundation for the theory of functional differential equations.Many recent results on stability and periodic solutions of functional differential equations are given and unsolved problems are stated.

Key Features:

- Smooth transition from ordinary differential equations to integral and functional differential equations.

- Unification of the theories, methods, and applications of ordinary and functional differential equations.

- Large collection of examples of Liapunov functions.

- Description of the history of stability theory leading up to unsolved problems.

- Applications of the resolvent to stability and periodic problems.

1. Smooth transition from ordinary differential equations to integral and functional differential equations.
2. Unification of the theories, methods, and applications of ordinary and functional differential equations.
3. Large collection of examples of Liapunov functions.
4. Description of the history of stability theory leading up to unsolved problems.
5. Applications of the resolvent to stability and periodic problems. ... Read more

Product Description
This is a two volume introduction to the computational solution of differential equations using a unified approach organized around the adaptive finite element method. It presents a synthesis of mathematical modeling, analysis, and computation. The goal is to provide the student with theoretical and practical tools useful for addressing the basic questions of computational mathematical modeling in science and engineering: How can we model physical phenomena using differential equations? What are the properties of solutions of differential equations?How do we compute solutions in practice? How do we estimate and control the accuracy of computed solutions? The first volume begins by developing the basic issues at an elementary level in the context of a set of model problems in ordinary differential equations. The authors then widen the scope to cover the basic classes of linear partial differential equations modeling elasticity, heat flow, wave propagation and convection-diffusion-absorption problems. The book concludes with a chapter on the abstract framework of the finite element method for differential equations.Volume 2, to be published in early 1997, extends the scope to nonlinear differential equations and systems of equations modeling a variety of phenomena such as reaction-diffusion, fluid flow, many-body dynamics and reaches the frontiers of research. It also addresses practical implementation issues in detail. These volumes are ideal for undergraduates studying numerical analysis or differential equations. This is a new edition of a 1988 text of 275 pages by C. Johnson. ... Read more

Customer Reviews (4)

Not even anywhere near as good as Johnson's first FEM book.
I honestely don't understand the point of this book. It spends a great deal of time developing FEM for ODEs. Do we really need FEM for ODEs?????????????? There are already many robust methods for numerically solving ODEs; FEM has no great advantage for ODEs. The real power of FEM is for solving PDEs. I thought this book would be a sequel to Johnson's FEM for PDEs, but instead we get this very long development of FEM for ODEs.

I would recommend that you save a lot of money and buy Johnson's FEM for PDEs book instead. There isn't much useful to be found in this one.

Best of EDP in !!!
This book is excelent because there's reviews of calculus, linear algebra and all sort of classical numerical methods needed to understand the finite element method. Moreover, the authors try and succed in giving a clear explanation of many applications of the method.

Well, a must of numerical analysis that every scientific searchers and engineers must own !!!

An excellent book
Great book to learn the fundamentals of finite element methods (FEM). The authors combine both rigor and simplicity in their presentation of the FEM as a means of numerically solving a wide variety of partial differentialequations. Highly recommended for engineers--especially those who have beentaught FEM using the traditional "structural mechanics" approach.

Amazing
Excellent approach for numerical and computational issues. Perfect for creating solid programs that work and it also explains every detail in a great way. ... Read more

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Customer Reviews (1)

Not for Kindergarten level
First, regardless of what mathematicians may believe or say, this is not a book for beginners. Second, if you understand stochastic processes at the level of Gnedenko, Stratonovich, and Wax and want to understand what is possible beyond that, then this is a valuable reference. From a modern standpoint the index is terrible, there's far more in the book than the index and table of contents indicate. I review the book to the chapter before the Cameron-Martin-Girsanov theorem, for that topic I recommend Durrett's 1984 book. Let me indicate briefly what you can find in Friedman.

The claim (without proof) that the Chapman-Kolmogorov eqn. defines a semi-group even in the absence of time translational invariance (pg. 23). Time translational invariance means that the drift R(x,t) and diffusion D(x,t) coefficients are t-independent, but the process is not stationary unless the 1-point density f(x,t) approaches a statistical equilibrium density f(x), and this is typically impossible when R and D depend on x (see L. Arnold, e.g.). Proving a semi-group for the case of time translational invariance is trivial. Would have been nice to see the proof for the general case.

The condition for a Wiener process (pg. 36) emphasizes time-translational invariance. This is misleading. The condition for a Wiener process is stationary increments combined with variance linear in time. This yields time translational invariance. Generalizing time translational invariance leads to stationary processes, this includes the Zwanzig-Mori memory processes (these are not Ito processes)) popular in physics in the 1970s. Combining stationary increments with variance nonlinear in time leads to processes with long time increment autocorrlelations like fractional Brownian motion (fBm), also not an Ito process. But, one can easily generalize from Wiener processes to Martingales, processes with uncorrelated and generally nonstationary increments x(t,T)=x(t+T)-x(t), and from there to general Ito processes with drift. Increments and increment autocorrelations are not discussed in Friedman, but see our 2005-2007 papers posted on xxx.lanl.gov.

Hölder continuity, nondifferentiability and infinite length of Brownian paths are discussed on pp. 39-40.

On pg. 46, eqns. (5.1,2), "Levy's Theorem", a word of warning: for an arbitrary martingale x(t), x^2(t)-t is not a martingale. An arbitrary martingale is topologically inequivalent to a Wiener process. The assumption that martingales are generally equivalent to Wiener processes is the source of much too confusion in financial math texts.

The derivation of both Kolmogorov's backward time pde (K1) and the Fokker-Planck pde (K2) from Ito's lemma (pp. 139-150). The proof that the Green functions (transition densities) of K1 and K2 are adjoints of each other implies the Chapman-Kolmogorov eqn. (assigned as problem 11, pg. 151)! This is the reverse of the usual derivation, where one derives K2 from a Chapman-Kolmogorov eqn., and is extremely enlightening. Friedman's assumption is that we have a Markov process, and that is guaranteed iff. the drift and diffusion coefficients R and D are history-independent, depend on (x,t) alone and on no earlier states. Because Friedman's emphasis is on solving elliptic and parabolic pdes, rather than on transition densities for Markov processes, he first writes the Fokker-Planck pde (Kolmogorov's second pde, or K2) in standard elliptic form (pg. 142). For the same reason, he could have started with K1 directly on pg. 139 but instead delays introducing K1 until pg. 142. I.e., he first discusses how to solve elliptic and parabolic pdes by 'running an Ito process'.

The 'Feynman-Katz' formula is found on pp. 147-8, and is applied to Black-Scholes type pdes over 20 yeas before Duffie published and popularized the same solution in the financial economics literature.

Friedman's very general treatment shows that not only Kolmogorov's two pdes but a more general class of Black-Scholes like pdes and adjoints satisfy the Chapman-Kolmogorov eqn. Feller's example of a discrete nonMarkov system satisfying the C-K eqn. was known in that era, maybe not to Friedman. Friedman provides us with the general formalism of nonMarkov systems continuous in both x and t that satisfy C-K.

A word of warning: Friedman (mis-)defines a Markov process by the Chapman-Kolmogorov eqn.As Feller showed, systems containing memory also satisfy that eqn. Friedman apparently did not realize this. Friedman's theoretical development is very general and applies to Ito processes with finite memory, which is a big plus. See cond-mat/0702517 on xxx.lanl.gov. ... Read more

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Product Description
Symmetry methods have long been recognized to be of great importance for the study of the differential equations arising in mathematics, physics, engineering, and many other disciplines. The purpose of this book is to provide a solid introduction to those applications of Lie groups to differential equations that have proved to be useful in practice, including determination of symmetry groups, integration of ordinary differential equations, construction of group-invariant solutions to partial differential equations, symmetries and conservation laws, generalized symmetries, and symmetry methods in Hamiltonian systems. The computational methods are presented so that graduate students and researchers in other fields can readily learn to use them.

Following an exposition of the applications, the book develops the underlying theory. Many of the topics are presented in a novel way, with an emphasis on explicit examples and computations. Further examples, as well as new theoretical developments, appear in the exercises at the end of each chapter.

This second edition contains a new section on formal symmetries and the calculus of pseudodifferential operators, simpler proofs of some theorems, new exercises, and a substantially updated bibliography. ... Read more

Customer Reviews (4)

The kindle edition is riddled with typos
This book is excellent. Olver presents a vast amount of material in full detail, and does it in an accessible way. Unfortunately the Kindle edition has many typos which are not present in the original book. In one proof (proposition 3.37), several lines are cut out of the middle (mid-sentence) and pasted back at the end of the proof. If you are familiar with this book and the subject matter then you can make sense of things. Otherwise, I would advise you to get a hard copy.

Why I like this book
First, let me preface this by saying my review is based on the FIRST EDITION of the book. Also, I have not read the entire thing, but much of it.I had no idea what a Lie Group was before picking this book up and found it to be an excellent introduction to a very fascinating subject.

The autor gives a fairly rigorous explication of the fundamentals of manifolds and groups in the first chapter, skipping proofs of harder facts.He then spends the rest of the book focusing on how to find symmetry groups of differential equations and their interpretation.He goes through detailed calculations and provides many helpful examples, without which I would have no chance of understanding the book.He gives very readable and easily applicable formulas for prolongation of group actions and vector fields, and supplies the heavy-handed theorems relating subvarieties of the prolonged group actions to symmetry groups of the DE's.

Algebraists will find the book lacking in details and probably fairly myopic in scope.Applied people such as myself will find it indispensible as a resource for actual computation.The focus of the book is consistent with the original formulations by Lie and Noether and is still relevant and largely untaught in standard courses.Reading this book, I have learned some very helpful TECHNIQUES, and I suspect if that's what you're looking for this book will be a Godsend.

GOOD
Not proper for first contact with the subject ,but as bibliographical resource, this is a realy good one.

This book fulfils a great lack
Application of symmetries and Lie groups to differential equations (primarily PDE's) is a hot issue in contemporary Mathematics and Physics. Unfortunately, only few textbooks are available on this area.

The bookis one of the best attempts to put this topic into an ordered, easy tostudying form, despite of its being a rapidly developing and thus hard toteaching issue. It survived two editions, and I am sure this is not thelast.

I am also sure that anyone who is involved in the area, will needto read this book, or have already read it. ... Read more

Product Description
Thirty years in the making, this revised text by three of the world's leading mathematicians covers the dynamical aspects of ordinary differential equations. it explores the relations between dynamical systems and certain fields outside pure mathematics, and has become the standard textbook for graduate courses in this area. The Second Edition now brings students to the brink of contemporary research, starting from a background that includes only calculus and elementary linear algebra.

The authors are tops in the field of advanced mathematics, including Steve Smale who is a recipient of the Field's Medal for his work in dynamical systems.

* Developed by award-winning researchers and authors
* Provides a rigorous yet accessible introduction to differential equations and dynamical systems
* Includes bifurcation theory throughout
* Contains numerous explorations for students to embark upon

NEW IN THIS EDITION
* New contemporary material and updated applications
* Revisions throughout the text, including simplification of many theorem hypotheses
* Many new figures and illustrations
* Simplified treatment of linear algebra
* Detailed discussion of the chaotic behavior in the Lorenz attractor, the Shil'nikov systems, and the double scroll attractor
* Increased coverage of discrete dynamical systems
... Read more

Customer Reviews (4)

A new version of a classic book
I bought a copy of this new book and I have its old version with Hirsch and Smale as its only authors. Main differences between these books are some new chapters covering chaos and the exercises. Old version has better chapters dealing with linear algebra.I find this new version hard to read and it leaves many details to be filled by the reader. I would say that the new version is still a good choice for a second course in ODE or supplementary text for a graduate course. I gave it four stars.

Excellent Book
This is a great introduction to the next stage of differential equations after a first course.Devaney is a master of presenation, and makes everything seem easy.It is not as encyclopedic as some other books on this material, such as Arnold and Perko, but it is easier to read and still covers the most important advanced material.

good, not ideal
the two books by hirsch smale, one with devaney, seem like good books, but I am not crazy about either, at least from the few pages one can search online here.

the latter book with devaney just seems a dumbed down version of the earlier book by the two more famous authors.i expected that earlier book to be far better, but found to my regret that the two books actually share almost the same first page, and the main difference noticeable in the early going is that the 2 author work is poorly written, and the 3 author one is not written much better.

it is clearer but seems to be talking down to the reader in an annoying way.so neither is the absolute pleasure to read that the wonderfully written text of arnol'd is, or the classic of hurewicz.i would skip these books and get arnold and hurewicz instead.

New Edition
You should be aware that there are two similar books with similar titles by the same authors. The old edition is a hardcover all green book by Hirsch and Smale called:

"Differential Equations, Dynamical Systems and Linear Algebra"

The second with the lorenz attractors in yellow on the cover is by Hirsch, Smale and Devaney and is called:

"Differential Equations, Dynamical Systems and an Introduction to Chaos"

Now, that may be obvious to you, but it is important to note that because those are VERY different books (which I have both of right here). The 'old' one is a more theoretical text that mainly addresses linear systems and is organized more like a math monograph than a contemporary (i.e. with pictures and examples) textbook. It is difficult for most people. The newer version is COMPLETELY different and is written for a more diverse audience. It starts with linear systems but then goes into nonlinear systems and discrete systems. It is somewhat similar in character to Strogatz's Nonlinear Dynamics and Chaos. If you do not have a very strong abstract theoretical type of math background I would not recommend you start learning about differential equations from the "old" edition. You will find it very difficult. If you are used to a general abstract presentation of results you should be fine. For the NEW edition the level is very different. I would guess that courses in multi-variable calc, elementary diff eq, and linear algebra (if you understood them) would be sufficient preparation. Both books are excellent, just be clear on what you are looking for. ... Read more

Product Description
Like all of Vladimir Arnold's books, this book is full of geometric insight. Arnold illustrates every principle with a figure. This book aims to cover the most basic parts of the subject and confines itself largely to the Cauchy and Neumann problems for the classical linear equations of mathematical physics, especially Laplace's equation and the wave equation, although the heat equation and the Korteweg-de Vries equation are also discussed. Physical intuition is emphasized. A large number of problems are sprinkled throughout the book, and a full set of problems from examinations given in Moscow are included at the end. Some of these problems are quite challenging!

What makes the book unique is Arnold's particular talent at holding a topic up for examination from a new and fresh perspective. He likes to blow away the fog of generality that obscures so much mathematical writing and reveal the essentially simple intuitive ideas underlying the subject. No other mathematical writer does this quite so well as Arnold. ... Read more

Customer Reviews (2)

Lectures on PDEs
Arnold's geometric point of view of differential equations is really intriguing. These lectures treat the subject of PDEs by considering specific examples and studying them thoroughly. By this approach the author aims at presenting fundamental ideas in a clear way. However, the interetested reader should be acquainted with some geometric ideas of differential equations.

buy it
OK, I did not read the book, just browsed through the pages. My first impression is that this is an excellent book, like most of Arnold's books. It is not trying to cover the theory of Partial Differential Equations, and this cannot be done in a single book anyway - just selected topics on a very accessible level. The geometric insight is excellent and even a specialist can benefit from that. I highly recommend this book to undergraduates (and higher) that are passionate about analysis.

I will update my review after I actually buy the book, I understand that I am not very helpful at the moment.

Update: I read parts of the book and I still like it a lot. I however found to be aimed at more advanced readers. It is an excellent book to complement any of the standard PDE texts. ... Read more