The determination of the leading coefficient in the monotone potential Sturm-Liouville operator from boundary measurements

This paper deals with the problem of determining the leading coefficient k 1⁄4 kððu0ÞÞ of the nonlinear (monotone potential) Sturm–Liouville operator Au1⁄4 ðkððu0ÞÞu0ðxÞÞþ qðxÞuðxÞ, x2 ða;bÞ. As an additional condition only two measured data at the boundary (x1⁄4 a, x1⁄4 b) are used. Solvability and linearization of the corresponding nonlinear direct problem are given. An existence of a quasi-solution of the inverse problem is obtained in a suitable compact class of admissible coefficients. In the second part of the paper an approximate analytical solution for the inverse problem is derived. The approach presented permits to analyze well-posed, as well as, all ill-posed situations for the inverse coefficient problem. Numerical examples corresponding to the all considered situations are presented. 2003 Elsevier Inc. All rights reserved.