Finite difference methods for half inverse Sturm-Liouville problems
This paper extends the work of \textit{A. Kammanee} and \textit{C. Böckman} [Appl. Math. Comput. 214, No. 2, 342--352 (2009; Zbl 1170.65064)] in several ways. Only Dirichlet boundary conditions were considered in that paper, whereas this paper considers the boundary conditions \(y(0)=(\pi)=0, y(0)=y'(\pi)=0, y'(0)=y(\pi)=0, y'(0)=y'(\pi)=0\). This paper treats the generalized Hochstadt-Lieberman problem, using \(n\) eigenvalues and the values of \(q\) at \(m\) grid points with general \(m\) and \(n\), whereas that paper considered only the classical Hochstadt-Lieberman problem with \(m=n\). These methods use an asymptotic correction technique of Pain, deHoog and Anderson, and its extension to Numerov's method for various boundary conditions. In the classical case \(c=\dfrac{\pi}2\), Numerov's method is found to be particularly effective. Since eigenvalue data is scare in applications, the author also examines stability problems associated with the use of the extra information on \(q\) where \(c<\dfrac{\pi}2\) and give some suggestions for further research.
scientific article; zbMATH DE number 5956402
scientific article