Partial Differential Equations of Mathematical Physics (Dover Books on Physics)

Description

The classical partial differential equations of mathematical physics, formulated by the great mathematicians of the 19th century, remain today the basis of investigation into waves, heat conduction, hydrodynamics, and other physical problems. In this comprehensive treatment by a well-known Soviet mathematician, the equations are presented and explained in a manner especially designed to be accessible to the novice in the field. The reader is assumed to have no previous knowledge other than elementary analysis. From there, more advanced concepts are developed in detail and with great precision; moreover, theorems are often approached through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems. After deriving the fundamental equations, the author provides illuminating expositions of such topics as Riemann's method, Lebesgue integration of multiple integrals, the equation of heat conduction, Laplace's equation and Poisson's equation, the theory of integral equations, Green's function, Fourier's method, harmonic polynomials and spherical functions, and much more. For this third edition, various improvements in style and clarifications of the presentations were made, including a simplification of the theory of multiple Lebesgue integrals and greater precision in the proof of the Fourier method. Finally, the translation is both idiomatic as well as accurate, making the vast amount of information in this book more readily accessible to the English reader. Review: Prerequisite knowledge of ODEs and intro PDEs needed. - Prerequisite knowledge of ODEs and intro PDEs needed. Also some potential theory. It's an excellent book. and agrees w/ other books on PDEs an Potential Theory. Recommend (all on Dover) simultaneous study of: 1) Potential Theory, O. Kellog 2) Intro to PDEs w/ Applications, E.C. Zachmanoglou 3) Intro to ODEs, E. Coddington Will be continuing my study of this book. Mr. Sobolev has very interesting perspectives on this PDEs subject. Review: Sobolev: "concentrating upon classical pde's of wave, Laplace and heat conduction" - Quite frankly, I had almost forgotten about this book. However, while working my way through Gabriel Barton's Book, Elements Of Green's Functions And Propagation, then his references, I was struck by what he wrote regarding this text: "accessibly written...a truly exceptional text, head and shoulders above most others with similar titles." Thus, I retrieved the book from my bookshelf. I confess to having ignored it for some time. The reason is obvious, as it is decidedly much more advanced than others "with similar titles." (1) Sobolev is advanced, this implies that a course in analysis has already been assimilated. Happily, if you have gone through much of Weinberger's text, A First Course in Partial Differential Equations, Sobolev will be easier to assail. Onward, then: (2) Of particular note is chapter six: multiple integrals and lebesgue integration (this is a "measure theory first," introduction). Also, if a sentence such as "any closed set can always be taken to be the intersection of a nest of open sets" is unfamiliar, this will be a tough chapter. On the positive side, the exposition is exceptionally lucid, and the initial five chapters (referred to by Sobolev as "lectures") are, as a whole, easier than the sixth lecture. (3) Surprisingly few intermediate steps are left for the reader. Two examples: glance at pages,134 and 135. Here you find the computations pedestrian and spelled-out for the reader (this is lecture eight, heat conduction). If you have glanced at Weinberger, then lectures eight and nine will not pose too much of an issue. (4) You meet interesting approaches in Lecture fifteen, properties of potentials. A glance at this chapter affords an opportunity to observe interplay between inequalities and differential geometric analysis. A most interesting analysis. (5) Next up, integral equations. Read: "the properties of potentials which we established in the last lecture enable us to solve the Dirichlet and Neumann problems by reducing the problems to the form of integral equations." (page 225). Lectures 18 and 19 revisit this theme. That is, lecture 18 will present theory, lecture 19 presents applications. (6) Green's Functions, next: "we will begin with a very simple case." (page 265). Green will occupy two chapters. These two chapters are ancillary to Barton's textbook. (7) Lecture twenty-two: The lecture has a major theme, inequalities: Of Minkowski and Schwarz. A fine exposition. (8) Fourier and separation-of-variables. We read: "the reader who has mastered the arguments set out in the previous lectures will have no difficulty in understanding immediately how and in what circumstances the Fourier method enables the solution of a problem to be found." (page 327). (9) Next, examining more general integral equations. So you will for three chapters following. After that, more Fourier methods, culminating in an exposition of spherical functions and harmonics--the final two lectures. (10) Concluding: Given adequate preparation (a first course in analysis) Sobolev is a fine text (no student exercises). In conjunction with Barton's text (with its copious supply of student exercises) a sound foundation is achieved. The exposition is exceptional, most intermediate steps are made explicit, the analysis (the proofs) fascinating. As preliminary, read Sobolev's fine forty-five page exposition of partial differential equations in Mathematics: Its Content, Methods and Meaning (volume two, MIT Press). Highly recommended for advanced students.