The boundary function method for singular perturbation problems. (English) Zbl 0823.34059

Vasil’eva, Adelaida B.; Butuzov, Valentin F.; Kalachev, Leonid V.

SIAM Studies in Applied Mathematics. 14. Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics. xiii, 256 p. (1995).

The mathematical modeling of numerous phenomena in biology, chemistry and other fields leads to singularly perturbed differential equations, that is, to equations with a small parameter \(\varepsilon\) at the highest derivative. Asymptotic methods (with respect to \(\varepsilon)\) are a basic tool to investigate initial value problems and boundary value problems in finite time and in bounded regions for such equations. For \(\varepsilon = 0\), the corresponding problem is called degenerate. The aim of the asymptotic methods is to relate the solution of a given problem for small \(\varepsilon\) to the solution of the corresponding degenerate problem. The convergence of the exact solution to the “degenerate” one is not uniformly, especially near the boundary and interior layers. To get an approximate solution of some order with respect to \(\varepsilon\), a formal asymptotic expansion is constructed which consists of a power series in \(\varepsilon\) (regular expansion) and of power series in \((x-x_ 0)/ \varepsilon\) where \(x_ 0\) is a placeholder for a boundary point or an interior point. The latter describe the behavior of the solution in the corresponding transition layers and are called boundary layer functions. This monograph is devoted to the method of boundary layer functions which represents a corner stone of the asymptotic methods in studying singularly perturbed problems. This method has been developed in the Russian school of A. B. Vasil’eva and V. F. Butuzov. The book under review is the first monograph in English which is devoted exclusively to this method. Compared with the first monograph of A. B. Vasil’eva and V. F. Butuzov on asymptotic methods published in 1973, the main goal of this English version is to explain the main ideas, to outline the proofs for the existence of an exact solution near a formal one and for the asymptotic error estimate, to include recent developments concerning the existence and stability of dissipative structures in reaction-diffusion equations and related to critical cases (existence of families of solutions of the degenerate problem), and to illustrate the importance of this method by studying applications in microelectronics, chemical kinetics, combustion theory and exitable media. This book can be warmly recommended to all who want to apply or teach asymptotic methods in the field of singularly perturbed problems. Readers which are interested in the detailed proofs have to use the original Russian papers quoted in the context. The bibliographic contains 167 references. The headlines of the chapters and sections characterize very well the content of this monograph. 1. Basic Ideas (regular and singular perturbations, asymptotic approximations, examples). 2. Singularly Perturbed Ordinary Differential Equations (initial value problems, critical cases, boundary value problems, dissipative structures). 3. Singularly Perturbed Partial Differential Equations (method of Vishik-Lyusternik, corner boundary layer functions, smoothing procedure, critical cases, periodic solutions, hyperbolic systems). 4. Applied Problems (combustion process, heat conduction, semiconductor devices, FitzHugh-Nagumo system).

Reviewer: K.R.Schneider (Berlin)

MSC:

34E15	Singular perturbations for ordinary differential equations
35B25	Singular perturbations in context of PDEs
34-02	Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35-02	Research exposition (monographs, survey articles) pertaining to partial differential equations

Keywords:

method of Vishik-Lyusternik; combustion; boundary and interior layers; asymptotic expansion; transition layers; boundary layer functions; singularly perturbed problems; dissipative structures; reaction-diffusion equations; singular perturbations; periodic solutions; semiconductor devices; FitzHugh-Nagumo system