OPTIMAL RECOVERY OF THE WAVE EQUATION SOLUTION BY INACCURATE INPUT DATA

Abstract. In this paper the problem of optimal recovery of the wave equation solution by inaccurate values of Fourier coefficients of a function defined the initial form of the string is considered. A solution of more general problem of recovery of operator defined on a weighted space of vectors from l2 by inaccurate values of its coordinates is given. 1. Statement of the problem Consider the wave equation with zero boundary conditions and the initial velocity equals zero (1) utt = uxx, u(0, t) = u(pi, t) = 0, u(x, 0) = f(x), ut(x, 0) = 0. It is known that the precise solution of this problem has the following form (2) u(x, t) = j=1 aj(f) cos jt sin jx, where aj(f) = 2 pi ∫ pi 0 f(x) sin jx dx are the Fourier coefficients of f(x). Assume that f(·) ∈W n2 ([0, pi]), where W n2 ([0, pi]) = { f(·) ∈ L2([0, pi]) : f (n−1)(·) abs. cont. on [0, pi]