Theory of difference equations: Numerical methods and applications. (English) Zbl 0683.39001

Lakshmikantham, V.; Trigiante, D.

Mathematics in Science and Engineering, 181. Boston, MA etc.: Academic Press, Inc. x, 242 p. $ 49.00 (1988).

The book consists of seven chapters and three appendices. The first chapter presents some necessary preliminaries. The second chapter is devoted to the presentation of such important methods as variation of constants and generating functions for finding a solution of a linear difference equation. Here the authors also consider the stability and absolute stability of solutions.
Chapter three deals with linear systems of difference equations and an extension of previously described methods for one difference equation to the case of a system. Chapter four presents some more deep issues of stability theory such as linear equations with periodic coefficients, use of the comparison principle, Lyapunov functions, domain of asymptotic stability, total and practical stabilities.
Chapter five deals with applications of the previously presented results to numerical analysis, iterative methods, Miller’s, Olver’s, Clenshaw’s algorithms, and monotone iterative methods.
Chapter six presents some numerical methods for differential equations and their connections with the corresponding methods for difference equations. Chapter seven deals with some applications of the previous results to such models of real world phenomena as population dynamics, distillation of a binary liquid, economics, traffic in channels, etc. The appendices present useful facts on algebra and calculus of matrices, Schur criterium, Chebyshev polynomials.
All the chapters contain problems which help a lot the reader to understand the main part of the text.

Reviewer: Yu.V.Kostarchuk

MSC:

39Axx	Difference equations
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
39-02	Research exposition (monographs, survey articles) pertaining to difference and functional equations
65Q05	Numerical methods for functional equations (MSC2000)
65N22	Numerical solution of discretized equations for boundary value problems involving PDEs

Keywords:

Miller’s algorithm; Olver’s algorithm; textbook; variation of constants; generating functions; linear difference equation; absolute stability; linear systems; periodic coefficients; comparison principle; Lyapunov functions; asymptotic stability; Clenshaw’s algorithms; monotone iterative methods; population dynamics; economics; traffic in channels