The result of the author's fascination with the mathematical beauty of integral equations, this book combines theory, applications, and numerical methods, and covers each of these fields with the same weight. In order to make the book accessible to mathematicians, physicists, and engineers alike, the author has made it as self-contained as possible, requiring only a solid foundation in differential and integral calculus. The functional analysis which is necessary for an adequate treatment of the theory and the numerical solution of integral equations is developed within the book itself. Problems are included at the end of each chapter.
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This review is from: Broadband in Europe: How Brussels Can Wire the Information Society (Applied Mathematical Sciences) (Hardcover)
Integral equations has become (I don't know why) a somewhat misterious subject. I've heard physicists say "I have enough with differential equations. Why to study more complicated types of equations?", or "I have enough trouble with differential equations; please don't tell me there's something else", and even "Why to study integral equations? What are they good for?".The subject of integral equations is a natural part of analysis which serves to connect several related topics (complex analysis, harmonic analysis, potential theory, differential equations,...) and to use all the power thus gained to solve important boundary value problems of mathematics and physics. Everyone interested in mathematical physics should get a thorough training in integral equations, and this book is the right place to start. Using a modern and comprehensive language, the author exposes the most important highlights of the theory, up to the numerical aspect, and puts the reader in the track to current research. The contents of the book are: Normed spaces, bounded and compact operators, the Riesz theory, dual systems and Fredholm theory, regularizations in dual systems, potential theory, singular integral equations, Sobolev spaces, the heat equation, operator approximations, degenerate kernel approximation, quadrature methods, projection methods, iterative solution and stability, equations of the first kind, Tikhonov regularization, regularization by discretization, inverse scattering theory. Includes motivation for each topic, excercises for each chapter, and extensive references. Suitable for graduate students and advanced undergraduates with a strong background in real and complex analysis. Please take a look to the rest of my reviews (just click on my name above).
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