The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have…mehr
The theory of partial differential equations is a wide and rapidly developing branch of contemporary mathematics. Problems related to partial differential equations of order higher than one are so diverse that a general theory can hardly be built up. There are several essentially different kinds of differential equations called elliptic, hyperbolic, and parabolic. Regarding the construction of solutions of Cauchy, mixed and boundary value problems, each kind of equation exhibits entirely different properties. Cauchy problems for hyperbolic equations and systems with variable coefficients have been studied in classical works of Petrovskii, Leret, Courant, Gording. Mixed problems for hyperbolic equations were considered by Vishik, Ladyzhenskaya, and that for general two dimensional equations were investigated by Bitsadze, Vishik, Gol'dberg, Ladyzhenskaya, Myshkis, and others. In last decade the theory of solvability on the whole of boundary value problems for nonlinear differentialequations has received intensive development. Significant results for nonlinear elliptic and parabolic equations of second order were obtained in works of Gvazava, Ladyzhenskaya, Nakhushev, Oleinik, Skripnik, and others. Concerning the solvability in general of nonlinear hyperbolic equations, which are connected to the theory of local and nonlocal boundary value problems for hyperbolic equations, there are only partial results obtained by Bronshtein, Pokhozhev, Nakhushev.
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.
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.
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