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Linear integral equations. 2nd ed. (English) Zbl 0920.45001
Applied Mathematical Sciences. 82. New York, NY: Springer. xiv, 365 p. (1999).
[For a review of the first edition (1989) see Zbl 0671.45001).]
The book adequately combines the theory, applications and numerical methods for solving integral equations. It consists of 18 chapters, references and an index. Besides, well selected problems are given at the end of each chapter. The first two chapters are introductory and deal with basic concepts of normed spaces, bounded and compact operators. The next seven chapters are devoted to the theory and applications of the integral equations. They deal with basic Riesz-Fredholm theory; dual systems and Fredholm alternative; regularization in dual systems; potential theory; singular integral equations; Sobolev spaces and the heat equation. In Chapters 10 and 11 operator approximations and degenerate kernel approximation are studied. In the next two chapters numerical solution of integral equations by quadrature and projection methods are taken up. Chapter 14 deals with iterative solution and stability. In the next three chapters ill-posed problems are considered. In Chapter 15, regularization methods for the stable solution of equations of the first kind in a Hilbert space setting are introduced. Again in Chapters 16 and 17 Tikhonov regularization and regularization by discretization are discussed. In the last chapter, the application of ill-posed integral equations of the first kind and regularization techniques to an inverse boundary problem for Laplace’s equation is briefly considered.
In the present edition of the book, the author has made corrections and adjustments throughout the text of the first edition of the book without altering the basic framework of its contents. He has also updated the references. In addition, he has made certain extensions and changes in sections 4.3, 5.3, 6.3, 7.6 and 14.4 as detailed in the preface of the book. Also the numerical analysis of the boundary integral equations in Sobolev space settings is extended for both integral equations of the first and second kind at appropriate places in the book.
This is a good introductory text book on linear integral equations. It contains almost all the topics necessary for a student. The presentation of the subject matter is lucid, clear and in the proper modern framework without being too abstract. It is suitable for students interested in studying integral equations. The problems given at the end of each chapters will be quite useful for them. The book will also attract mathematicians, scientists and engineers who want to learn the theory of integral equations, some of its recent developments, their applications and the basic ideas for the numerical solution of integral equations.
Reviewer: K.C.Gupta (Jaipur)

45-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations
65R20 Numerical methods for integral equations
65R30 Numerical methods for ill-posed problems for integral equations
45A05 Linear integral equations
45B05 Fredholm integral equations
45E05 Integral equations with kernels of Cauchy type
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
45L05 Theoretical approximation of solutions to integral equations
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
65N38 Boundary element methods for boundary value problems involving PDEs