Asymptotic Methods for Investigating Quasiwave Equations of Hyperbolic Type
Springer Science & Business Media, Apr 30, 1997 - Mathematics - 214 pages
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 differential equations 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.
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27r-periodic solutions 2n-periodic function a2uxx asymptotic methods averaged system Bogolyubov boundary condition boundary value problem bounded Cauchy problem Chapter classical solution condition of Theorem Consider const constant continuous with respect countable system defined eigenfunctions equality equation utt existence expanded fi(t following conditions Fourier coefficients Fourier series fulfilled function f(x function u(x,t holds hyperbolic equations hyperbolic type inequalities initial condition initial function integral equations investigation ISBN Let the condition linear Lipschitz condition mixed problem nonlinear nonperturbed equation obtain odd function operator ordinary differential equations oscillations partial differential equations periodic solutions proof prove region respect to variable right-hand side satisfies the condition series oo small parameter smooth functions smooth solution solution of equation solution u(x,t Suppose system of equations system of integral T-periodic T-system uniformly convergent uniformly with respect value problem 4.4 wave equation wave solution zero