Open AccessReconstruction of potential function in inverse Sturm-Liouville problem via partial data
Author(s) -
Mehmet Açil,
Ali Konuralp
Publication year - 2021
Publication title -
an international journal of optimization and control: theories and applications/e-an international journal of optimization and control: theories and applications
Language(s) - English
Resource type - Journals
eISSN - 2146-5703
pISSN - 2146-0957
DOI - 10.11121/ijocta.01.2021.001090
Subject(s) - eigenfunction , mathematics , eigenvalues and eigenvectors , computation , boundary value problem , function (biology) , inverse problem , sturm–liouville theory , uniqueness , spectrum (functional analysis) , mathematical analysis , boundary (topology) , algorithm , physics , quantum mechanics , evolutionary biology , biology
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 R\"{o}hrl's objective function, the steepest descent method is used in the computation of potential functions. To decrease the volume of computation, we propose a theorem to precalculate the minimization 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 boundary conditions are also eliminated. As partial data, we take two spectra, the set of the $j$th 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 continuous, and noncontinuous, respectively.