Determination of leading coefficients in Sturm–Liouville operator from boundary measurements. II. Unicity and an engineering approach

摘要

We study the problem of determining the leading coefficient k=k(x) of the Sturm–Liouville operator Au≡−(k(x)u′(x))′+q(x)u(x),x∈(a,b), from a measured data given in the form of Dirichlet or/and Neumann type (additional) boundary conditions. The contribution of the each measured data to the unicity of the inverse problem solution is analysed. Based on the analysis the problem is treated as a Cauchy problem for the first-order ordinary differential equation with respect to k(x). For the case when the direct problem solution u=u(x) belongs to the class of the second-order polynomials and u′(x)≠0, we present an analytical formula for an approximate solution of the inverse problem. Numerical examples related to accuracy of the solution and to essential features of the inverse problem are demonstrated.