Mathematics and Computers in Simulation 61 (2002) 47–52 Short communication Simulation of ill-conditioned situations in inverse coefficient problem for the Sturm–Liouville operator based on boundary measurements Alemder Hasanov ∗ , Burhan Pektas Department of Mathematics, Applied Mathematical Sciences Research Center, University of Kocaeli, Ataturk Bulvari, 41300 Izmit, Turkey Accepted 1 May 2002 Abstract The problem of determining the unknown coefficient k = k(x) of the Sturm–Liouville operator Au ≡-(k(x)u ′ (x)) ′ + q(x)u(x) from the measured data at the boundary x = 0; 1 is considered. It is assumed that the function u = u(x) has several singular points in (0, 1) of different types. As a result different types of ill-conditioned situations (mild, moderate and severe) in (0, 1) arise. We analyze all the ill-conditioned situations and then based on the analysis construct computational method for the solution of the inverse problem. © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. ITB: 34A55; 34A30 Keywords: Inverse coefficient problem; Ill-conditioned situations; Polynomial approximation 1. Well- and ill-conditioned situations Consider the inverse problem (the problem (ICP)) of determining the leading coefficient k = k(x) of the Sturm–Liouville operator, from the boundary measurements (k(x)u ′ (x ; k))| x=0 = φ, u(x ; k)| x=1 = β, (1) where u = u(x ; k) is the solution to the boundary value problem -(k(x)u ′ (x)) ′ + q(x)u(x) = f (x), x ∈ (0, 1), (2) u(0) = 0, (k(x)u ′ (x))| x=1 = ϕ, ϕ ∈ R 1 , (3) and the values ϕ = 0, φ = 0 and β = 0 are given measured data. ∗ Corresponding author. E-mail address: ahasanov@kou.edu.tr (A. Hasanov). 0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII:S0378-4754(02)00134-9