Error analysis of a multisingular inverse coefficient problem for the Sturm-Liouville operator based on boundary measurement


Hasanov A.

APPLIED MATHEMATICS AND COMPUTATION, cilt.150, ss.493-524, 2004 (SCI İndekslerine Giren Dergi) identifier identifier

  • Cilt numarası: 150 Konu: 2
  • Basım Tarihi: 2004
  • Doi Numarası: 10.1016/s0096-3003(03)00287-x
  • Dergi Adı: APPLIED MATHEMATICS AND COMPUTATION
  • Sayfa Sayısı: ss.493-524

Özet

In this paper we study the inverse problem of determining the unknown leading coefficient k = k(x) of the linear Sturm-Liouville operator Au = -(k(x)u(x))' + q(x)u(x), x is an element of (0, 1), from boundary measurements, when u'(x) or/and u"(x) vanishes at several, called singular, points of the interval (0, 1). As a result the considered inverse problem has simultaneously different types (moderate or severely) of ill-conditioned situations in different parts of the interval (0, 1). The presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial Cauchy data for the unknown coefficient. Error estimations for the polynomial approximations are presented for well-conditioned, as well as for ill-conditioned situations. The behaviour of the inverse problem solution with respect to both types (Dirichlet and Neumann) of noisy data is analyzed. The obtained results are illustrated by various numerical examples. (C) 2003 Elsevier Inc. All rights reserved.