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

摘要

This paper deals with the problem of determining the leading coefficient k=k((u′)2) of the nonlinear (monotone potential) Sturm–Liouville operator Au=−(k((u′)2)u′(x))′+q(x)u(x), x∈(a,b). As an additional condition only two measured data at the boundary (x=a, x=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.