Inverse Nodal Problem Research Articles

Inverse problems for discrete Sturm–Liouville operator having Bessel-type potential

In this paper, inverse problems are studied for Sturm–Liouville difference operators having p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes.

English

english

Inverse nodal problem with fractional order conformable type derivative

Inverse nodal problem with fractional order conformable type derivative

An inverse nodal problem for a fourth-order self-adjoint binomial operator

In this work we deal with inverse nodal problems of reconstructing a fourth-order self-adjoint binomial operator d4dx4+q with Dirichlet boundary conditions. We prove that a dense subset of nodal points uniquely determines the potential function q.

Inverse nodal problems for Dirac operators and their numerical approximations

In this article, we consider an inverse nodal problem of Dirac operators and obtain approximate solution and its convergence based on the second kind Chebyshev wavelet and Bernstein methods. We establish a uniqueness theorem of this problem from parts of nodal points instead of a dense nodal set. Numerical examples are carried out to illustrate our method. 
 For more information see https://ejde.math.txstate.edu/Volumes/2023/81/abstr.html

Inverse nodal problem for singular Sturm–Liouville operator on a star graph

Abstract In this study, singular Sturm–Liouville operators on a star graph with edges are investigated. First, the behavior of sufficiently large eigenvalues is learned. Then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points. Lastly, a constructive solution to the inverse problems of this class is obtained.

Inverse nodal problems for perturbed spherical Schrödinger operators

Inverse nodal problems for perturbed spherical Schrödinger operators

Inverse Nodal Problem for a Conformable Fractional Diffusion Operator With Parameter-Dependent Nonlocal Boundary Condition

In this paper, we consider the inverse nodal problem for the conformable fractional diffusion operator with parameter-dependent Bitsadze–Samarskii type nonlocal boundary condition. We obtain the asymptotics for the eigenvalues, the eigenfunctions, and the zeros of the eigenfunctions (called nodal points or nodes) of the considered operator, and provide a constructive procedure for solving the inverse nodal problem, i.e., we reconstruct the potential functions p(x) and q(x) by using a dense subset of the nodal points.

Inverse nodal problem for diffusion operator on a star graph with nonhomogeneous edges

Abstract In this study, a diffusion operator is investigated on a star graph with nonhomogeneous edges. First, the behaviors of sufficiently large eigenvalues are learned, and then the solution of the inverse problem is given to determine the potential functions and parameters of the boundary condition on the star graph with the help of a dense set of nodal points and to obtain a constructive solution to the inverse problems of this class.

Inverse nodal problem for the quadratic pencil of the Sturm$-$Liouville equations with parameter-dependent nonlocal boundary condition

In this paper, we consider the inverse nodal problem for a quadratic pencil of the Sturm$-$Liouville equations with parameter-dependent Bitsadze$-$Samarskii type nonlocal boundary condition and we give an algorithm for the reconstruction of the potential functions by obtaining the asymptotics of the nodal points.

Inverse nodal problems for Sturm-Liouville equation with nonlocal boundary conditions

In this paper, a Sturm–Liouville problem with some nonlocal boundary conditions of the Bitsadze-Samarskii type is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of the potential function and some other coefficients in the boundary conditions.

Numerical Solutions of Inverse Nodal Problems for a Boundary Value Problem

In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of the potential function for the boundary value problem by the first kind Chebyshev wavelet method. Two numerical examples show that the first kind Chebyshev wavelet method for solving the inverse nodal problems for the boundary value problem is valid.

The Partial Inverse Spectral and Nodal Problems for Sturm–Liouville Operators on a Star-Shaped Graph

We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subset and prove uniqueness theorems from paired-dense nodal subsets in interior subintervals having a central vertex. In particular, we obtain some uniqueness theorems by replacing the information of nodal data on some fixed edge with part of the eigenvalues under some conditions.

Solving an inverse nodal problem with Herglotz–Nevanlinna functions in boundary conditions using the second‐kind Chebyshev wavelets method

This study considers Sturm–Liouville (S–L) equation with a special function which includes spectral parameter at boundary conditions and expresses asymptotic forms of inverse problem solution, nodal points, and lengths. Furthermore, it proves some uniqueness theorems. Doing this, approximate solution of inverse S–L problem is obtained by second‐kind Chebyshev polynomials whereby the second‐kind Chebyshev wavelets (SCW) method is used to solve these types of problems. Finally, effectiveness of the method is demonstrated in a few examples.

Inverse Nodal Problem for p-Laplacian String Equation with Prüfer Substitution

We consider an inverse nodal problem for p-Laplacian string equation under some boundary conditions. Asymptotic formulas for eigenvalues and nodal parameters are constructed by modified Pr¨ufer substitution. The most important process is to apply modified Pr¨ufer substitution to get an exhaustive asymptotic estimate for eigenvalues. Moreover, a reconstruction formula for density function of p-aplacian string equation is obtained by nodal parameters. Generated outcomes are the generalization of the known string problem.

The numerical solution of inverse nodal problem for integro-differential operator by Legendre wavelet method

In this study, integro-differential equation under separated boundary conditions is considered. In fact, we study the inverse nodal problem, which includes differential part of Sturm–Liouville equation and Volterra type integral to obtain approximative solutions by Legendre polynomials. And, we use Legendre wavelets method to solve these types of problems. In addition, we present the error bound of for continuous derivatives with its approximative solution. Once and for all, the strength of the method can be seen in a few examples.

Inverse Nodal Problems for Dirac-Type Integro-Differential Operators with Linear Functions in the Boundary Condition

In this article, Dirac-type integro-differential operator with linear functions in the boundary condition is considered. We obtain asymptotic expressions for the solution of the differential system and derive the large eigenvalues and nodal points. We also give a constructive procedure for solving an inverse nodal problem. We prove that a dense subset of the nodes determines the coefficients of the differential part of the operator and gives partial information for the integral part of it.

Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions

In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of some coefficients of the operator.

Inverse problem for Hill equation with jump conditions

In the present study, the inverse nodal problem is discussed for the discontinuous periodic (or anti-periodic) Sturm–Liouville problem. Such types of problems are different from regular problems because of discontinuity in the boundary conditions. Firstly, results of the Sturm–Liouville problem including jump condition are present. Then, by deferring the zeros of eigenfunctions, the inverse problem is solved as we desired. The method is based on considering a translation so that the periodic (or anti-periodic) problem is reduced to a Dirichlet problem as in Cheng and Law (Inverse Prob 22: 891–901, 2006). But, our problem including also discontinuous conditions. In addition to all of these, although there are so many results on this subject, we combine both periodic conditions and discontinuity.

Inverse nodal problems for Dirac type integro differential system with a nonlocal boundary condition

In this paper, the Dirac type integro differential system\ with a nonlocal integral boundary condition is considered. First, we derive the asymptotic expressions for the solutions and large eigenvalues. Second, we provide asymptotic expressions for the nodal points and prove that a dense subset of nodal points uniquely determines the boundary condition parameter and the potential function of the considered differential system. We also provide an effective procedure for solving the inverse nodal problem.