Аpproximation of the solutions to Cauchy problem for wave equation in uniform metric by inaccurate information

Abstract

On the article is considered problem of discretization of solutions to the Cauchy problem for the wave equation with initial information from the Korobov classes in a uniform metric. In the studied formulation, the Cauchy problem has an explicit representation of the solution as the sum of an absolutely convergent trigonometric Fourier series, which is generally completely determined by an infinite set of Fourier coefficients of the initial information. This raises the problem of approximating the solution—an infinite object—using finite numerical information of a given volume $N$, obtained from values of the Fourier coefficients of the initial information. The study is conducted within the framework of the Computational (Numerical) Diameter problem, the purpose of which is to construct optimal computational aggregate under conditions of inaccurate information. Given previously obtained orders of limiting errors by accurate information, unimprovable orders of inaccurate information are established, preserving the orders of decreasing errors for accurate information, with the optimal computational aggregates specified. It is shown that the found limiting error is the largest possible for all computational aggregates that provide the optimal order of approximation based on precise information and constructed based on an arbitrary finite spectrum of trigonometric Fourier coefficients.