Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics.

*(English)*Zbl 1148.35006
Texts in Applied Mathematics 50. New York, NY: Springer (ISBN 0-387-22966-3/hbk). xvi, 328 p. (2005).

The introduction (Chapters 1,2) of the reviewed monograph contains a historical introduction and auxiliary material on asymptotic convergence and asymptotic expansions, boundary layers and one example about the boundary of a laser-sustained plasma. Ch. 1.1 presents the techniques of expanding integrals useful in various applications. It illuminates ideas of such methods as partial integration, Laplace and Fourier integrals, stationary phase. Ch. 4 is devoted to boundary layer phenomena in many examples and situations. Ch. 5 contains their appearance at the solving of two-point boundary value problems for ODEs at one or two endpoints of the segment when small parameter belongs to higher differential expressions, the expansion and matching techniques in WKBJ method, the problem of suspension bridge. Further nonlinear boundary value problems follow (Ch. 6), where the theory has many complications and interesting phenomena and therefore remains far from complete.

Ch. 7 considers nonautonomous elliptic Dirichlet boundary value problems with small parameter at the Laplace operator, on the circle, rectangle and nonconvex domains.

Further nonstationary problems with boundary layers in time are considered, at first in ODE (A.N. Tikhonov theorem in p. 8.2, attraction of the outer expansion, 8.3. The O’Malley-Vasil’eva expansion for nonlinear initial value problem, 8.4 The two-body problem with variable mass and 8.5-6 Existence of slow manifold and behavior near it, 8.7 Periodic solutions and oscillations), then Evolution Equations with boundary layer (Ch. 9), i.e. singular perturbation in PDEs, where analysis of expansions for parabolic and hyperbolic equations is given. Here results of Kevorkian and Cole, De Jager and J. Furu, for linear parabolic (Slow diffusion with heat production and Slow diffusion on a semi-definite domain) and hyperbolic (9.5 A wave equation with singular perturbation at the main part of the second order) equations, and results of Vasil’eva, Butusov and Kalachev (1995) for singularly perturbed parabolic equations (9.3 A Chemical reaction with diffusion, 9.4 Periodic solutions of parabolic equations) are presented. Applications to signalling or radiation problem is given.

The subsequent chapters are devoted to continuation and averaging methods in singular perturbation theory. In Ch. 10 these are Poincaré expansion theorem with application to Poincaré-Lindstedt method for autonomous equation \(\dot{x}=f(x, \varepsilon)\), \(x\in \mathbb R^n\); \(T\)-periodic solutions (\(T\) is near to \(2\pi\)) of the periodic nonautonomous equation \(\ddot{x}+x=\varepsilon f(x,\dot{x},t,\varepsilon)\) with applications to forced Van der Pol and damped forced Duffing equations, Mathieu equation, where sometimes bifurcation phenomena arise. In n.10.4 the instability phenomenon of stable periodic solution of an unperturbed subsystem is investigated, so-called autoparametric resonance, which is interesting in engineering problems on behavior of flexible structures such as vibrations of overhead transmission lines, connecting cables or chimney pipes. A short description of works on the radius of convergence on small parameter is given.

In the Ch. 11 various schemes of averaging with approximation on timescales are considered (basic periodic averaging with application to Van der Pol and damped forced Duffing equations, transformation to slowly varying system, asymptotic character of averaging; nonperiodic averaging; the multiple-timescales method). Ch. 12 Advanced averaging is looking a number of useful and important extensions of averaging together with discussion of timescales. These are averaging over an angle and over more angles in connection with the investigation of resonance phenomena. Here at the usage of averaging techniques it is investigated also the question about persistence of invariant manifolds, such as tori or cylinders in nonlinear equation under perturbation (tori in the dissipative case, Hamiltonian case, the Neimark-Sacker bifurcation). Averaging of higher orders and the questions of timescales identification are considered.

Analysis of weakly nonlinear PDEs with evolution in time containes in Ch. 13. It concerns with conservative systems and nearly integrable systems of KAM theory in finite-dimensional Hamiltonian systems (KAM theorems for infinite-dimensional conservative systems, developed by S. B. Kuksin [Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics. 1556. Berlin: Springer-Verlag. (1993; Zbl 0784.58028)]). Operators with continuous and discrete spectrum are considered separately.

The concluding Ch. 14 is devoted to singular perturbation in wave equation on unbounded domains such as weakly nonlinear Klein-Gordon and perturbed Korteweg-de Vries equations. Here some results are discussed without strong mathematical justification.

The appendices of Ch. 15 contains auxiliary material such as du Bois-Reymond theorem about asymptotic sequences, approximation of integrals, perturbation of constant matrices, application of maximum principles, almost-periodic functions etc.

Detailed illustrations, stimulating examples in every chapter make this monograph very useful for applied mathematicians in science and engineering fields.

Ch. 7 considers nonautonomous elliptic Dirichlet boundary value problems with small parameter at the Laplace operator, on the circle, rectangle and nonconvex domains.

Further nonstationary problems with boundary layers in time are considered, at first in ODE (A.N. Tikhonov theorem in p. 8.2, attraction of the outer expansion, 8.3. The O’Malley-Vasil’eva expansion for nonlinear initial value problem, 8.4 The two-body problem with variable mass and 8.5-6 Existence of slow manifold and behavior near it, 8.7 Periodic solutions and oscillations), then Evolution Equations with boundary layer (Ch. 9), i.e. singular perturbation in PDEs, where analysis of expansions for parabolic and hyperbolic equations is given. Here results of Kevorkian and Cole, De Jager and J. Furu, for linear parabolic (Slow diffusion with heat production and Slow diffusion on a semi-definite domain) and hyperbolic (9.5 A wave equation with singular perturbation at the main part of the second order) equations, and results of Vasil’eva, Butusov and Kalachev (1995) for singularly perturbed parabolic equations (9.3 A Chemical reaction with diffusion, 9.4 Periodic solutions of parabolic equations) are presented. Applications to signalling or radiation problem is given.

The subsequent chapters are devoted to continuation and averaging methods in singular perturbation theory. In Ch. 10 these are Poincaré expansion theorem with application to Poincaré-Lindstedt method for autonomous equation \(\dot{x}=f(x, \varepsilon)\), \(x\in \mathbb R^n\); \(T\)-periodic solutions (\(T\) is near to \(2\pi\)) of the periodic nonautonomous equation \(\ddot{x}+x=\varepsilon f(x,\dot{x},t,\varepsilon)\) with applications to forced Van der Pol and damped forced Duffing equations, Mathieu equation, where sometimes bifurcation phenomena arise. In n.10.4 the instability phenomenon of stable periodic solution of an unperturbed subsystem is investigated, so-called autoparametric resonance, which is interesting in engineering problems on behavior of flexible structures such as vibrations of overhead transmission lines, connecting cables or chimney pipes. A short description of works on the radius of convergence on small parameter is given.

In the Ch. 11 various schemes of averaging with approximation on timescales are considered (basic periodic averaging with application to Van der Pol and damped forced Duffing equations, transformation to slowly varying system, asymptotic character of averaging; nonperiodic averaging; the multiple-timescales method). Ch. 12 Advanced averaging is looking a number of useful and important extensions of averaging together with discussion of timescales. These are averaging over an angle and over more angles in connection with the investigation of resonance phenomena. Here at the usage of averaging techniques it is investigated also the question about persistence of invariant manifolds, such as tori or cylinders in nonlinear equation under perturbation (tori in the dissipative case, Hamiltonian case, the Neimark-Sacker bifurcation). Averaging of higher orders and the questions of timescales identification are considered.

Analysis of weakly nonlinear PDEs with evolution in time containes in Ch. 13. It concerns with conservative systems and nearly integrable systems of KAM theory in finite-dimensional Hamiltonian systems (KAM theorems for infinite-dimensional conservative systems, developed by S. B. Kuksin [Nearly integrable infinite-dimensional Hamiltonian systems. Lecture Notes in Mathematics. 1556. Berlin: Springer-Verlag. (1993; Zbl 0784.58028)]). Operators with continuous and discrete spectrum are considered separately.

The concluding Ch. 14 is devoted to singular perturbation in wave equation on unbounded domains such as weakly nonlinear Klein-Gordon and perturbed Korteweg-de Vries equations. Here some results are discussed without strong mathematical justification.

The appendices of Ch. 15 contains auxiliary material such as du Bois-Reymond theorem about asymptotic sequences, approximation of integrals, perturbation of constant matrices, application of maximum principles, almost-periodic functions etc.

Detailed illustrations, stimulating examples in every chapter make this monograph very useful for applied mathematicians in science and engineering fields.

Reviewer: Boris V. Loginov (Ul’yanovsk)

##### MSC:

35B25 | Singular perturbations in context of PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |