Reconstruction of Potential Function in Inverse Sturm-Liouville Problem Via Partial Data

Abstract

In this paper, three different uniqueness data are investigated to reconstruct the potential function in the Sturm-Liouville boundary value problem in the normal form. Taking account of Rohrl's objective function, the steepest de-scent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the mini-mization parameter that is required in the optimization. Further, we propose a novel time-saving algorithm in which the obligation of using the asymptotics of eigenvalues and eigenfunctions and the appropriateness of selected bound-ary conditions are also eliminated. As partial data, we take two spectra, the set of the jth elements of the infinite numbers of spectra obtained by changing boundary conditions in the problem, and one spectrum with the set of terminal velocities. In order to show the efficiency of the proposed method, numerical results are given for three test potentials which are smooth, nonsmooth con-tinuous, and noncontinuous, respectively.