Determination of leading coefficients in Sturm–Liouville operator from boundary measurements. I. A stripping algorithm

摘要

We present a stripping algorithm for determination of the unknown coefficient k=k(x) in the Sturm–Liouville operator Au≡(k(x)u′(x))′+q(x)u(x), x∈(a,b), form boundary measurements. Due to the only two physically possible measured data at the boundary, the problem is of strong unstable. The formulation of the problem based on the Tikhonov's quasisolution approach. The coefficient k(x)∈L2[a,b] is assumed to be a monotone and uniform bounded function. This class of functions Kc is compact in L2[a,b] and hence the inverse problem has at least one quasisolution in Kc. The stripping algorithm is implemented for the cases, when the unknown function k(x) is interpolated by the first- and second-order polynomials. Effectiveness of the method is demonstrated on concrete numerical examples with exact and noisy data.