On one direct method for the approximate solution of a periodic boundary-value problem

Abstract

We propose a direct method for the approximate solution of integral equations that arise in the course of approximate solution of a periodic boundary-value problem for linear differential equations by the method of boundary conditions. We show that the proposed direct method is optimal in order. Ly = y(r) + Z Pi(t)y(i) = f(t),where r > 1; the requirements concerning the smoothness of the coefficients Pi(t) and the right-hand side f(t) are formulated below. One of the methods for the approximate solution of the boundary-value problem (1), (2) is the method of a function of boundary conditions proposed by Atekseenko [I]. According to this method, a solution of (1), (2) ----Pi(t)Dr_i(t-'r,) u(~)dx + f(t). o (4)Let u N(t ) be an approximate solution of the integral equation (4) obtained by a direct method according to which the integral operator in (4) is replaced by an operator whose rank does not exceed N. As an approximate solution of problem (1), (2), we take the function 2~ _ C]UN(T,)dT,. YN(t) = I [ 1Dr(t '~)+ (5) 0If we assume that Eq. (4) and, consequently, problem (1), (2) are uniquely solvable for any function f e C, then (3) and (5) yield the following estimate for the error of the approximate solution:Institute of Mathematics, Ukrainian Academy of Sciences, Kiev.