Hikmet Koyunbakan

A new inverse problem for the diffusion operator

Abstract In this work, we have estimated nodal points and nodal lengths for the diffusion operator. Furthermore, by using these new spectral parameters, we have shown that the potential function of the diffusion operator can be established uniquely. An analogous inverse problem was solved for the Sturm–Liouville problem in recent years.

A uniqueness theorem for inverse nodal problem

In this article, it is found that the asymptotic formulas for nodal points and nodal length for the differential operators having singularity type at the points 0 and π, it is shown that the potential function can be determined from the positions of the nodes for the eigenfunctions.

Inverse nodal problem for p-Laplacian energy-dependent Sturm-Liouville equation

In this study, the inverse nodal problem is solved for p-Laplacian Schrödinger equation with energy-dependent potential function with the Dirichlet conditions. Asymptotic estimates of eigenvalues, nodal points and nodal lengths are given by using Prüfer substitution. Especially, an explicit formula for a potential function is given by using nodal lengths. Results are more general than the classical p-Laplacian Sturm-Liouville problem. For the proofs, methods previously developed by Law et al. and Wang et al., in 2009 and 2011, respectively, are used. In there, they solved an inverse nodal problem for the classical p-Laplacian Sturm-Liouville equation with eigenparameter boundary conditions.MSC:34A55, 34L20.

Reconstruction of potential function and its derivatives for Sturm–Liouville problem with eigenvalues in boundary condition

The purpose of this article is solving inverse nodal problem for Sturm–Liouville equation with a boundary condition depending on spectral parameter. Taking into account Law and Chens method, we construct the potential function q and its derivatives by using nodal data. We give several lemmas in order to complete proof of the main theorem. Especially, we obtain an explicit formula for potential function and its derivatives from the nodal data by a pointwise limit.

Reconstruction of the Potential Function and its Derivatives for the Diffusion Operator

We solve the inverse nodal problem for the diffusion operator. In particular, we obtain a reconstruction of the potential function and its derivatives using only nodal data. Results are a generalization of Law’s and Yang’s works.

Reconstruction formula for the potential function of Sturm–Liouville problem with eigenparameter boundary condition

It is known that the uniqueness of potential function of the Sturm–Liouville problem can be shown from the nodal points. In this article, we solve the inverse nodal problem of the reconstruction of the potential function q from the nodal data by a pointwise limit. We show that this convergence is in the L1. It is mentioned that this method is based on the works of Law and Yang, but we have applied the method for the Sturm–Liouville problem depending on eigenparameter boundary conditions.

Reconstruction of Potential Function for Diffusion Operator

Inverse spectral problem for diffusion operator consists in reconstruction of this operator by its spectrums and norming constants. In this paper, we are concerned with inverse problem for diffusion operator using a new kind of spectral data that is known as nodal points. We give a reconstruction of q as a limit of a sequence of functions whose nth term is dependent only eigenvalue and its associated nodal data. The technique we use to obtain the results is an adaptation of the method discussed in the reference Chen et al. (Proc. Ann. Math. Ser. 2002; 130:2319–2324).

Solution of a discontinuous inverse nodal problem on a finite interval

In this paper, the inverse problem of recovering the potential function, on a general finite interval, of a singular Sturm-Liouville problem with a new spectral parameter, called the nodal point, is studied. In addition, we give an asymptotic formula for nodal points and the density of the nodal set.

Half inverse problem for singular differential operator

A spectral analysis for the Sturm–Liouville equation defined on (0,1] and singularity of type at zero is investigated (l is an integer). As known, the potential function q(x) in the singular Sturm–Liouville problem can be uniquely determined from two spectrum. In this study, we show that if q(x) is prescribed on (1/2,1], then only one spectrum is sufficient to determine q(x) on the interval (0,1/2).

Inverse problem for a singular differential operator

In this paper, we give the solution of the inverse Sturm-Liouville problem on two partially coinciding spectra. In particular, in this case we obtain Hochstadts theorem concerning the structure of the difference q(x)-q@?(x) for the singular Sturm Liouville problem defined on the finite interval (0,@p) having the singularity type 14sin^2x at the points 0 and @p.