Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

AvYuri A. Mitropolsky,G. Khoma

Inbunden, Engelska, 1997

Del i serien Mathematics and Its Applications

539 kr

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Häftad

539 kr

Beskrivning

This volume is devoted to the further development of the asymptotic theory for analyzing solutions of a range of nonlinear periodic boundary value problems. It suggests a systematic approach to constructing asymptotic methods for solving wave equations, hyperbolic differential equations and partial differential equations with small parameters. The book should be of interest to researchers and postgraduate students whose work involves partial differential equations, mathematical physics, or approximations and expansions.

Produktinformation

Utgivningsdatum:1997-04-30
Mått:155 x 235 x 17 mm
Vikt:512 g
Format:Inbunden
Språk:Engelska
Serie:Mathematics and Its Applications
Antal sidor:214
Upplaga:1997
Förlag:Kluwer Academic Publishers
ISBN:9780792345299

Utforska kategorier

Beräkning och matematisk analys inom Naturvetenskap och teknik

Innehållsförteckning

1 Existence Theorems for Hyperbolic Equations.- 1.1 Preliminary remarks.- 1.2 Homogeneous mixed problem.- 1.3 Nonhomogeneous mixed problem.- 1.4 Reduction of the second order quasiwave equation to the first order systems.- 1.5 Reduction of the quasiwave equation to a system of integral equations.- 1.6 Quasilinear mixed problem.- 1.7 A property of solutions of quasilinear mixed problem.- 1.8 Justification of the asymptotic methods to be applied to the investigation of quasilinear mixed problems.- 1.9 A periodic boundary value problem.- 2 Periodic Solutions of The Wave Ordinary Diferential Equations of Second Order.- 2.1 Preliminary remarks.- 2.2 The existence of solutions periodic in time for wave equations.- 2.3 Periodic solutions of autonomous wave differential equations.- 3 Periodic Solutions of The First Class Systems.- 3.1 Linear systems.- 3.2 Nonlinear systems.- 4 Periodic Solutions of The Second Class Systems.- 4.1 Some preliminaries.- 4.2 The structure of generalized periodic solutions of the second order wave equation of the first kind.- 4.3 The structure of generalized periodic solutions of the second order wave equation of the second kind.- 4.4 The structure of continuous periodic solutions of systems.- 5 Periodic Solutions of The Second Order Integro-Diffrential Equations of Hyperbolic Type.- 5.1 Some preliminaries.- 5.2 Classical and smooth periodic solutions.- 5.3 The existence of generalized periodic solutions of hyperbolic integro-differential equations.- 5.4 Periodic solutions of nonlinear wave equations with small parameter.- 6 Hyperbolic Systems with Fast and Slow Variables and Asymptotic Methods For Solving Them.- 6.1 The first approximation of asymptotic solutions of the second order equations.- 6.2 Analytical dependence of solutions of hyperbolic equations on parameter.- 6.3 Bounded solutions of a linear hyperbolic system of first order.- 6.4 Almost periodic solutions of an almost linear hyperbolic system of first order.- 6.5 Mathematical justification of the Bogolyubov averaging method over the infinite time interval for hyperbolic systems of first order.- 6.6 The averaging methods for hyperbolic systems with fast and slow variables.- 6.7 Reduction of quasilinear equations to a countable system.- 6.8 Truncation of a countable system of partial differential equations. Problems of mathematical justification.- 6.9 Investigation into the multifrequency oscillation modes of the quasiwave equation.- 6.10 Asymptotic solution of nonlinear systems of first order partial differential equations.- 7 Asymptotic Methods For The Second Order Partial Differential Equations of Hyperbolic Type.- 7.1 The reduction of quasilinear equations of hyperbolic type to a countable system of ordinary differential equations in standard form.- 7.2 The reduction method in application to a countable system of differential equations.- 7.3 Summation of trigonometric Fourier series with coefficients given approximately.- 7.4 Shortening countable systems.- 7.5 Determination of the approximate solutions of truncated systems.- 7.6 Reduction of the nonlinear equations of hyperbolic type to countable systems.- 7.7 Investigation of solutions of the equation describing string transverse vibrations in a medium whose resistance is proportional to the velocity in first degree.- 7.8 A remark on shortening countable systems obtained when solving nonlinear hyperbolic equations.- 7.9 Construction of asymptotic approximations to solutions of linear mixed problems appearing when investigating multi-frequency modes of oscillations.- 7.10 Investigation of single-frequency oscillations for the equation utt-a2uxx = eu2.- 7.11 Construction of asymptotic approximations to solutions of nonlinear mixed problems used for investigating single-frequency modes of oscillations with fast and slow variables.- 7.12 A method for constructing asymptotic approximations to solutions of partial differential equations with application to multi-frequency modes of oscillations.