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

In this paper we study the inverse problem of determining the unknown leading coefficient k 1⁄4 kðxÞ of the linear Sturm–Liouville operator Au 1⁄4 ðkðxÞu0ðxÞÞþ qðxÞuðxÞ, x 2 ð0; 1Þ, from boundary measurements, when u0ðxÞ or/and u00ðxÞ vanishes at several, called singular, points of the interval ð0; 1Þ. As a result the considered inverse problem has simultaneously different types (moderate or severely) of ill-conditioned situations in different parts of the interval ð0; 1Þ. 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. 2003 Elsevier Inc. All rights reserved.