This self-contained and well written book, which presents many original results of the authors, is devoted to the solvability theory and the numerical analysis of boundary integral equations on smooth curves. It will be very useful to anyone interested in the modern treatment of boundary value problems and integral equations.
The book starts by providing the necessary tools from functional analysis and the basic theory of boundary integral methods: Fredholm operators, Krylov subspace methods, single and double layer potentials, classical boundary integral equations for 2D Laplace and Helmholtz equations, singular integral equations, boundary integral operators in periodic Sobolev spaces. Then the authors present a systematic and elementary theory of (one-dimensional) periodic integral and pseudodifferential equations with full proofs, following an approach proposed by M. S. Agranovich.
The next chapters, which form the core of the book, are concerned with the stability and convergence analysis of numerical methods to solve periodic integral equations, including trigonometric Galerkin and collocation methods, their fully discrete versions, quadrature and spline based methods. Here the pseudodifferential structure is extensively used to construct fast solvers for integral equations. The authors’ approach also covers a class of integral equations on an open arc where the singularities of solutions can be “smoothed out” by using the cosine transform. Stability and convergence results for rather general classes of integral equations on intervals and curves with corners can be found in the monographs by S. Prößdorf and B. Silbermann [Numerical analysis for integral and related operator equations (1991; Zbl 0763.65102)] and by R. Hagen, S. Roch and B. Silbermann [\(C^*\)-algebras and numerical analysis (2001; Zbl 0964.65055)].
The book also contains many carefully chosen exercises. It can be strongly recommended as a course book for graduate and postgraduate students.