Akmaral Arystangalikyzy

Partial differential equations along with a function, derivative, and integral are basic mathematical models. Therefore, the problem of their approximation by accurate and inaccurate information with the construction of optimal aggregates (methods) of approximation is relevant and many articles are devoted to this issue. The article considers the approximation of solutions of the Cauchy problem for the Poisson equation with the right-hand side from the Nikol'skii classes $H_2^r(0,1)^s$ in the Lebesgue metric $L^2(0,1)^s.$ The orders of error of approximation of solutions of the Poisson equation by accurate and inaccurate information in the form of trigonometric Fourier coefficients are obtained (Problem C(N)D-1). Namely, a lower bound for the approximation error based on accurate information is found for all possible computational agregates using a finite set of trigonometric Fourier coefficients. A computational agregate (approximation method) is constructed that confirms this lower bound. The boundaries of $\tilde{\varepsilon}$ of inaccurate information are determined -- the limiting error that preserves the order of limiting error based on accurate information (Problem C(N)D-2). Further, the massiveness of the limiting error $\tilde{\varepsilon}$ from C(N)D-2 is indicated -- a set of approximation agregates based on trigonometric Fourier coefficients is constructed, the limiting error of which does not exceed the limiting error from C(N)D-2.

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.