Green’s Functions 2nd Edition by G. F. Roach (PDF)

Description

Green’s functions are an important tool used in solving boundary value problems associated with ordinary and partial differential equations. This self-contained and systematic introduction to Green’s functions has been written with applications in mind. The material is presented in an unsophisticated and rather more practical manner than usual. Consequently advanced undergraduates and beginning postgraduate students in mathematics and the applied sciences will find this account particularly attractive. Many exercises and examples have been supplied throughout to reinforce comprehension and to increase familiarity with the technique.

User’s Reviews

Editorial Reviews: Review ‘It is well planned and splendidly motivated and the computational material is covered without obscuring the conceptual development.’ Bulletin of the Institute of Mathematics and its Applications’The exposition is generally clear and well motivated mathematically… Generally this book should fill the need of those who want an introduction to the theory of Green’s Functions but lack the mathematical background to understand more advanced accounts.’ Mathematical Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I guess this book shows up when you search for Green functions because that is the title. I would say this book is not so great however. It is not very practical. There is a lot of time spend on theoretical mathematical analysis. This book won’t help you solve actual problems. I found the other books on this topic (Duffy and Barton) to be excellent books. If you are a math person only then you might like this book more. If you are physics person wanting to learn how to actually solve and use Green functions, you can just pass on this book.

⭐I purchased this book used. The book and content itself is more-or-less what it is claimed to be, though it is not the definitive work on Green’s Function. It is recommended by some college/university profs because they used it before, and remember good experiences using it?I began buying more books used to minimize the costs. I bought this copy from a merchant using Amazon to handle the transaction for $50. Notice that is cheaper than any of prices on this site, so I got the lowest price for a book in good condition.When I received the book, I found the price listed on the fly sheet — $16.50. That shows that the price this book sold more than 200 percent its purchased value. Also, looking at the date of reprinting (1992), it is obvious — this book is a collector’s item. Buy it but don’t use it…

⭐Preface: “our main purpose in this book is to provide a self-contained and systematic introduction…” My review concentrates upon that goal. I like this text quite a bit, due to the clarity of mathematical exposition. True it is, physical applications are not the dominant theme, but the background acquired here will serve the needs of the reader well into the future. I also point out that this book makes an excellent precursor to Barton or Stakgold. (Or, concurrently with study of those textbooks). Survey of content:(1) We start with an elementary synopsis– setting the stage via simple ordinary differential equations. So, refresh your memory with the separation-of-variables technique, trigonometric identities, integration-by-parts–these are utilized in the initial ten pages. Let me add that the synopsis here will be encountered in much greater detail throughout the text, therefore do not let the first chapter frighten you off. Much will be revisited. This material is not as forbidding as first appearances would suggest. Now, the fun begins:(2) Linear Vector Spaces: twenty pages, the skeleton which supports the remainder of the monograph. Pay attention, as the material here segues into the third chapter–matrices of finite dimension. Study examples #3.2 and # 3.5 (pages 44 and 49). The toughest portion of third chapter is saved for last: upper bounds for sequences (page 59). The material should be within grasp at this juncture. The initial three chapters are background.(3) Next, continuous functions, will be a bit more theoretical. My recommendation: if any part of this chapter poses a problem, peruse Taylor’s Advanced Calculus–then, return to Roach. Because, what is here–fourth chapter–segues into the fifth chapter (arguably the most important chapter, Integral Operators). The exercises are straightforward. In fact, glance at page 78, problems #4, 5, and, 6–they should be most assailable. If not, something has been missed.(4) Integral Operators. Here we travel from finite to infinite dimensional considerations. Read: “as may be anticipated, integral transformations, defined on an infinite-dimensional vector space, have many properties similar to linear transformations defined on a finite-dimensional vector space (page 81).Thus,the preceding will be utilized (via analogy) for this chapter. Symmetric and separable kernels elucidated. Recall sequences, upper bounds, inequalities (that is, elementary theory of chapter four, see Taylor’s book). The entire panoply utilized: . You will learn of Fourier series (page 100) encounter Bessel’s inequality (page 102), uniform convergence and Cauchy criteria utilized utmost. I reiterate, if those concepts and definitions have been ignored, this chapter (the most important chapter) will make no sense. The first four chapters were preparation for the fifth chapter. The fifth will be of utmost importance for the remainder of the text, being a deeper look into…(5) Fourier expansions. Schwarz inequality (page 113), Parseval’s inequality (page 115). Now, stop at page 123, study the example. If it makes sense, continue. Next up, we will meet Hilbert space (page 128). All presented simply, lucidly.(6) Boundary value problems, this is chapter seven and beyond. Notice, how they arrived on the scene later than is fashionable in many applied texts. I believe its placement further down the pipeline provides for excellent pedagogic strategy. Prime focus: Dirac-delta function. Linear functionals and distributions introduced. Study the example #7.3, (page 149), if all has been assimilated, then this example will flow effortlessly. This chapter (seven, differential operators) is as lucid as one can find elsewhere. In fact, I found it to be exceptional in execution. I make a stronger statement: This chapter on “Differential Operators” should be required study for any student intending to proceed forward. By “forward,” I imply “to more advanced and applied offerings” (and, I believe Barton and Stakgold are really more advanced and applied !).(7) Integral Equations–you get an introduction to them in chapter eight. Fredholm and Volterra. Here again, perusal of this chapter will serve one in good stead when attacking books such as Kanwal (1971) or Porter and Stirling (1990); Although, my favorite introduction to Integral Equations remains Petrovskii (Lectures, 1957). Recall, Fubini’s Theorem (page 204 ). Recall, too, uniform convergence. The remainder of this text is more advanced:(8) Higher Dimensions. Green’s theorem and Identities will cement the foundation. The great efficacy of this chapter is its help in confronting Jackson’s Classical Electrodynamics. Study pages 220-224, potentials and point charge. Reading: ” the most powerful analytical tools available for the study of linear second-order differential partial differential equations are the Green’s Identities.” ( page 225). The chapter begins with elliptic (Laplace), then goes to parabolic (heat) then, hyperbolic (wave) equations. The two chapters–nine and ten–form somewhat of a set. Together they provide plenty of calculated, solved examples. Reiterating, these chapters provide an excellent precursor to Jackson’s Classical Electrodynamics (Barton, too.). Tenth chapter, Roach begins with the so-called method-of-images, in a plane (Jackson, page 55, second edition–he begins with three dimensions), then we are off to three dimensions.(9) Approximation Methods concludes the text. It is a brief summary and can provide only an entree. There you have my take on this textbook. I believe it is an excellent “self-contained and systematic” introduction. The author has delivered on his goal. It is well-written. It presents a progressive build-up in technicality (excellent pedagogy).It is a tad theoretically oriented, less applied–however, that which is here will serve one well for the much-more-applied textbooks. By the way, the prime reference for this book is Apostol’s Mathematical Analysis.Sound advice, to keep Apostol at hand (I love that book), but, the student will find the book of Angus Taylor–Advanced Calculus– easier in that regard (Taylor being verbose in contrast to the terseness of Apostol).Roach is highly recommended for the enterprising senior-undergraduate student.

⭐This book is an outstanding overview. While the preface claims the book is written for a senior undergraduate or a junior graduate student, this book would be a little high level for all but the most motivated undergrads. Surprisingly, this book also contains an excellent review of linear vector spaces and transformations.

⭐The author gave a classical discussion to the classical topic. I was first impressed by the Chapter 1. The concept of green’s function is well explained by the simple example.Then, dear reader, I ask you read it by yourself. It will be an interesting journey!

⭐Very useful indeed!

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