Remarks on a New Inverse Nodal Problem (original) (raw)
Related papers
Solving inverse nodal problem with spectral parameter in boundary conditions
Inverse Problems in Science and Engineering, 2019
In this paper, uniqueness theorem is studied for the diffusion operator on a finite interval with separated boundary conditions. The oscillation of the eigenfunctions corresponding to large modulus eigenvalues is established and an asymptotic of the nodal points is obtained. By using these new spectral parameters, uniqueness theorem is proved.
The inverse nodal problem for a differential operator with an eigenvalue in the boundary condition
Applied Mathematics Letters, 2008
We consider the Sturm-Liouville problem with an eigenvalue dependent boundary condition. In this work, by using method of Yang [X.F. Yang, A solution of the inverse nodal problem, Inverse Problems 13 (1997) 203-213.], we reconstruct the potential of the Sturm-Liouville problem with an eigenvalue in the boundary condition from nodal points (zeros of eigenfunctions). Also, we give a uniqueness theorem.
Inverse nodal problems for Dirac operators and their numerical approximations
Electronic Journal of Differential Equations, 2023
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.
The inverse nodal problem and the Ambarzumyan problem for the p-Laplacian
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2009
We study the issues of the reconstruction and stability of the inverse nodal problem for the one-dimensionalp-Laplacian eigenvalue problem. A key step is the application of a modified Prüfer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths. Two associated Ambarzumyan problems are also solved.
On one inverse spectral problem relatively domain
Different practical problems, espesially, problems of hydrodynamics, elasticity theory, geophysics and aerodynamics can be reduced to finding of an optimal shape. The investigation of these problems is based on the study of depending of the functiuonals on the domain, their first variation and gradient. In the paper the inverse problem relatively domain is considered for two-dimensional Schrodinger operator and operator Lu = ∆ 2 u and the definition of s−functiuons is introduced. The method is proposed for the determination of the domain by given set of functions.
2018
In this article, we extend solution of inverse nodal problem for one-dimensional p-Laplacian equation to the case when the boundary condition is polynomially eigenparameter. To find the spectral data as eigenvalues and nodal parameters, a Prüfer substitution is used. Then, we give a reconstruction formula of the potential function by using nodal lengths. This method is similar to used in [24], and our results are more general.
Inverse Problems for Partial Differential Equations
Oberwolfach Reports, 2012
This workshop brought together mathematicians engaged in different aspects of inverse problems for partial differential equations. Classical topics such as the inverse problems of impedance tomography and scattering theory as well as new developments such as interior transmission eigenvalues were discussed.
An Introduction to Classical Inverse Eigenvalue Problems
CISM International Centre for Mechanical Sciences, 2011
Aim of these notes is to present an elementary introduction to classical inverse eigenvalue problems in one dimension. Attention is mainly focused on Sturm-Liouville differential operators given in canonical form on a finite interval. A uniqueness result for a fourth order Euler-Bernoulli operator is also discussed.
Related topics
Cited by
Inverse Problems in Science and Engineering
Inverse nodal problems for the Sturm–Liouville equation in a finite interval with boundary conditions depending polynomially on the spectral parameter are studied. We prove a uniqueness theorem: nodal points uniquely determine the polynomials of the boundary conditions and the potential function of the Sturm–Liouville equation. For these inverse nodal problems we provide constructive procedures.
Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data
Numerical Functional Analysis and Optimization
The inverse nodal problem for Dirac type integro-differential operator with the spectral parameter in the boundary conditions is studied. We prove that dense subset of the nodal points determines the coefficients of differential part of operator and gives partial information for integral part of it.
A Conformable Inverse Problem with Constant Delay
Journal of Advances in Applied & Computational Mathematics
This paper aims to express the solution of an inverse Sturm-Liouville problem with constant delay using a conformable derivative operator under mixed boundary conditions. For the problem, we stated and proved the specification of the spectrum. The asymptotics of the eigenvalues of the problem was obtained and the solutions were extended to the Regge-type boundary value problem. As such, a new result, as an extension of the classical Sturm-Liouville problem to the fractional phenomenon, has been achieved. The uniqueness theorem for the solution of the inverse problem is proved in different cases within the interval (0,π). The results in the classical case of this problem can be obtained at α=1. 2000 Mathematics Subject Classification. 34L20,34B24,34L30.
Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition
Inverse Problems in Science and Engineering, 2016
In this study, we consider Sturm-Liouville problem with a boundary condition depending on spectral parameter. We apply the nodal points as input data and calculate the approximate solution of the inverse nodal problem using Chebyshev polynomials of the first kind. Finally, we illustrate the numerical results by providing several examples.
Solving inverse nodal problem with spectral parameter in boundary conditions
Inverse Problems in Science and Engineering, 2019
In this paper, uniqueness theorem is studied for the diffusion operator on a finite interval with separated boundary conditions. The oscillation of the eigenfunctions corresponding to large modulus eigenvalues is established and an asymptotic of the nodal points is obtained. By using these new spectral parameters, uniqueness theorem is proved.
Inverse nodal problems for Sturm–Liouville operators on star-type graphs
Journal of Inverse and Ill-posed Problems, 2008
Inverse nodal problems are studied for second-order differential operators on star-type graphs with standard matching conditions in the internal vertex. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.
Related papers
Survey of the inverse spectral problem
2004
This is a survey of the inverse spectral problem on (mainly compact) Riemannian manifolds, with or without boundary. The emphasis is on wave invariants: on how wave invariants have been calculated and how they have been applied to concrete inverse spectral problems.
Numerical solution of inverse nodal problem with an eigenvalue in the boundary condition
Inverse Problems in Science and Engineering, 2016
In this study, we consider Sturm-Liouville problem with a boundary condition depending on spectral parameter. We apply the nodal points as input data and calculate the approximate solution of the inverse nodal problem using Chebyshev polynomials of the first kind. Finally, we illustrate the numerical results by providing several examples.
We prove that given any compact subset of the complex plane containing zero, there exists a Hankel operator having this set as its spectrum.
2012
This booklet relates the major developments of the evolution of inverse problems. Three sections are contained within. The first offers definitions and the fundamental characteristics of an inverse problem, with a brief history of the birth of inverse problem theory in geophysics given at the beginning. Then, the most well known Internet sites and scientific reviews dedicated to inverse problems are presented, followed by a description of research undertaken along these lines at IFSTTAR (formerly LCPC) since the 1990s. The final section concerns the different approaches available to solve an inverse problem. These approaches are divided into three categories: functional analysis, regularization techniques for ill-posed problems, and stochastic or Bayesian inversion.
Inverse Nodal Problems for Second Order Differential Operators with a Regular Singularity
2006
The inverse nodal problem for the Sturm-Liouville operator is the problem of finding the potential function q and the boundary conditions using the nodal points. The purpose of this paper is to present a method for solving the inverse nodal problem for a singular differential operator on a finite interval. We find asymptotic formulas for nodal points and the nodal lengths for differential operators having singularity type l(l + 1) x 2 + 2 x at the point 0. We also determine the potential function from the position of nodes.
Inverse nodal problems for Dirac-type integro-differential operators
2016
The inverse nodal problem for Dirac differential operator perturbated by a Volterra integral operator is studied. We prove that dense subset of the nodal points determines the coefficients of differential and integral part of the operator. We also provide a uniqueness theorem and an algorithm to reconstruct the coefficients of the problem by using the nodal points.
Inverse problems: Theory and application to science and engineering
2014
It is a real pleasure to announce the publication of this special issue. Inverse problems arise naturally in many branches of science and engineering where the values of some model parameters must be obtained from the observed data. In recent years, theory and applications of inverse problems have undergone tremendous growth. Inverse problems can be formulated in many mathematical areas and analyzed by different theoretical and computational techniques. This special issue contains 13 papers, and it aims to highlight recent research, development, and applications of inverse problems in science and engineering.
Inverse nodal problems for Sturm–Liouville operators on star-type graphs
Journal of Inverse and Ill-posed Problems, 2008
Inverse nodal problems are studied for second-order differential operators on star-type graphs with standard matching conditions in the internal vertex. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.