Algebraic methods in nonlinear perturbation theory / V. N. Bogaevski, A. Povzner

Auteur: Bogaevski, Vladimir Nikolaevich - AuteurCo-auteur: Povzner, Aleksandr Yakovlevich (1915-2008) - AuteurType de document: Livre numériqueCollection: Applied mathematical sciences ; 88Langue: anglaisÉditeur: Berlin : Springer, 1991 ISBN: 9781461287704 Note: he perturbation theory of the book concerns nonlinear systems of ordinary differential equations of first order containing a small parameter. The authors turn over from such a nonlinear system to the corresponding partial differential equation, for which the given system is the characteristic one. Using for the linear equation the familiar transformations to matrix normal forms, they obtain a formalism for calculating the asymptotic expansion of the wanted solutions. The asymptotic problems are classified into regular ones, where the asymptotics are valid in the whole domain of the variables, and singular ones, where different asymptotics must be constructed in various subdomains and a matching procedure must be applied in the overlapping domain. The book begins with the matrix perturbation theory and ends with an extension of the formalism to some partial differential equations. Emphasis is laid on a series of examples, which both illustrate the theory in detail and show the usefulness of the method for the solution of concrete applications in different parts of physics. (zbMath) Sujets MSC: 34E10 Ordinary differential equations -- Asymptotic theory -- Perturbations, asymptotics
34C20 Ordinary differential equations -- Qualitative theory -- Transformation and reduction of equations and systems, normal forms
34B20 Ordinary differential equations -- Boundary value problems -- Weyl theory and its generalizations
37J40 Dynamical systems and ergodic theory -- Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems -- Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
34D10 Ordinary differential equations -- Stability theory -- Perturbations
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he perturbation theory of the book concerns nonlinear systems of ordinary differential equations of first order containing a small parameter. The authors turn over from such a nonlinear system to the corresponding partial differential equation, for which the given system is the characteristic one. Using for the linear equation the familiar transformations to matrix normal forms, they obtain a formalism for calculating the asymptotic expansion of the wanted solutions. The asymptotic problems are classified into regular ones, where the asymptotics are valid in the whole domain of the variables, and singular ones, where different asymptotics must be constructed in various subdomains and a matching procedure must be applied in the overlapping domain. The book begins with the matrix perturbation theory and ends with an extension of the formalism to some partial differential equations. Emphasis is laid on a series of examples, which both illustrate the theory in detail and show the usefulness of the method for the solution of concrete applications in different parts of physics. (zbMath)

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