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Wave propagation. An invariant imbedding approach. (English) Zbl 0645.65054

The book covers many ideas of previous papers and books of the first author in particular, devoted to invariant imbedding and dynamic programming approaches to different problems. As the title indicates, the principles of both techniques are explained and successfully applied to wave equations (one-dimensional models). In addition to some of the currently used “classical” methods, the authors’ attention is focussed on invariant imbedding approaches, explained using a simple transport process model, and on the dynamic programming approach. The latter point of view is also used to analyze the classical variational principles. Such problems as scattering, the Lippmann-Schwinger equations and Bremmer series are also considered. The book gives good insight into the main principles and techniques. It could serve as a very good initiation into ideas of invariant imbedding for anyone intending to apply this approach. The numerical power of the technique, which is of good robustness, is not neglected, though the reader is often referred to the annotated bibliography. The final part of the book contains some exercises for every chapter of the book.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
35L05 Wave equation
90C39 Dynamic programming
49L20 Dynamic programming in optimal control and differential games