Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type

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.