OPTIMAL APPROXIMATION OF SOLUTIONS OF POISSON EQUATIONS BY INITIAL DATA IN THE FORM OF ACCURATE AND INACCURATE INFORMATION OF TRIGONOMETRIC FOURIER COEFFICIENTS

Abstract

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.