An inverse problem for a differential pencil using nodal points as data

An inverse nodal problem is studied for a differential pencil with non-separated boundary conditions. We prove that a dense subset of nodal points uniquely determines the boundary data and potential functions. We also provide a constructive procedure for the solution of the inverse nodal problem.     

Inverse Nodal Problem for Dirac System with Spectral Parameter in Boundary Conditions

An inverse nodal problem lies in constructing operators from the given zeros of their eigenfunctions. In this work, we deal with an inverse nodal problem of reconstructing the Dirac system with the spectral parameter in the boundary conditions. We prove that a set of nodal points of one of the components of the eigenfunctions uniquely determines all the parameters of the boundary conditions and the coefficients of the Dirac equations. We also provide a constructive procedure for solving this inverse nodal problem.     

Reconstruction of the Dirac Operator From Nodal Data

Inverse nodal problems consist in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, we deal with the inverse nodal problem of reconstructing the Dirac operator on a finite interval. We prove that a dense subset of nodal points uniquely determine the parameters of the boundary conditions, the mass of a particle and the potential function of the Dirac system. We also provide a constructive procedure for the solution of the inverse nodal problem.     

Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data

Abstract

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.     

具有特征参数多项式边界条件的Sturm-Liouville 方程的逆结点问题

逆结点问题是通过特征函数的零点重构算子. 本文主要讨论具有特征参数多项式边界条件的 Sturm-Liouville 方程的逆结点问题. 20世纪50年代以后,人们发现在许多工程领域, Sturm-Liouville 问题的谱参数不仅出现在方程中, 而且也出现在边界条件中,因此带参数边界条件的逆结点问题对数学物理方面的研究有重要意义. 本文讨论区间 $[0,1]$ 上边界条件为参数多项式的 Sturm-Liouville 方程的逆结点问题, 并证明在 $[0,b]$ \big($ b\in \big(\frac{1}{2},1\big]$\big) 上结点的稠密子集可唯一确定 $[0,1]$ 上的势函数和边界条件中多项式的未知系数.     

Inverse nodal problem for the Sturm-Liouville operator with a weight

In this work,we consider the inverse nodal problem for the Sturm-Liouville problem with a weight and the jump condition at the middle point.It is shown that the dense nodes of the eigenfunctions can uniquely determine the potential on the whole interval and some parameters.     

Partial inverse nodal problems for differential pencils on a star-shaped graph

In this paper, the authors study partial inverse nodal problems for differential pencils on a star-shaped graph. We firstly show that the potential on each edge can be uniquely determined by twin-dense nodal subsets on some interior intervals under certain conditions. Without any nodal information on some potential on the fixed edge, we may add some spectral information to guarantee these uniqueness theorems. We still consider the case of arbitrary intervals having the internal vertex. In particular, we pose and solve a new partial inverse nodal problem for differential pencils on the star-shaped graph from the Weyl m-function and the theory concerning densities of zeros of entire functions.     

A new inverse problem for the diffusion operator

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.     

Determination of differential pencils from dense nodal subset in an interior subinterval

The uniqueness problem of the inverse nodal problem for the differential pencils defined on interval [0, 1] with the Dirichlet boundary conditions is considered. We prove that a bilaterally dense subset of the nodal set in interior subinterval (a ₁, a ₂)(? [0, 1]) can determine the pencil uniquely. However, in the case of 1/2 ? [a ₁, a ₂] we need additional spectral information to treat this problem, which is associated with the derivatives of eigenfunctions at some known nodal points.     

Inverse nodal problem for p–laplacian dirac system

In this study, we solve an inverse nodal problem for p‐Laplacian Dirac system with boundary conditions depending on spectral parameter. Asymptotic formulas of eigenvalues, nodal points and nodal lengths are obtained by using modified Prüfer substitution. The key step is to apply modified Prüfer substitution to derive a detailed asymptotic estimate for eigenvalues. Furthermore, we have shown that the functions r(x) and q(x) in Dirac system can be established uniquely by using nodal parameters with the method used by Wang et al. Obtained results are more general than the classical Dirac system. Copyright © 2016 John Wiley & Sons, Ltd.     

Incomplete Inverse Spectral and Nodal Problems for Differential Pencils

We prove uniqueness theorems for so-called half inverse spectral problem (and also for some its modification) for second order differential pencils on a finite interval with Robin boundary conditions. Using the obtained result we show that for unique determination of the pencil it is sufficient to specify the nodal points only on a part of the interval slightly exceeding its half.     

Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data

Partial inverse nodal problems for Sturm–Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin‐dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm–Liouville operator on the compact equilateral star graph.     

A new adaptive approach of the Metropolis-Hastings algorithm applied to structural damage identification using time domain data

In the present work, the formulation and solution of the inverse problem of structural damage identification is presented based on the Bayesian inference, a powerful approach that has been widely used for the formulation of inverse problems in a statistical framework. The structural damage is continuously described by a cohesion field, which is spatially discretized by the finite element method, and the solution of the inverse problem of damage identification, from the Bayesian point of view, is the posterior probability densities of the nodal cohesion parameters. In this approach, prior information about the parameters of interest and the quantification of the uncertainties related to the magnitudes measured can be used to estimate the sought parameters. Markov Chain Monte Carlo (MCMC) method, implemented via the Metropolis-Hastings (MH) algorithm, is commonly used to sample such densities. However, the conventional MH algorithm may present some difficulties, for instance, in high dimensional problems or when the parameters of interest are highly correlated or the posterior probability density is very peaked. In order to overcome these difficulties, a new adaptive MH algorithm (P-AMH) is proposed in the present work. Numerical results related to an inverse problem of damage identification in a simply supported Euler-Bernoulli beam are presented. Synthetic experimental time domain data, obtained with different damage scenarios, and noise levels, were addressed with the aim at assessing the proposed damage identification approach. An adaptive MH algorithm (H-AMH) and the conventional MH algorithm, already consolidated in the literature, were also considered for comparison purposes. The numerical results show that both adaptive algorithms outperformed the conventional MH. Besides, the P-AMH provided Markov chains with faster convergence and better mixing than the ones provided by the H-AMH.     

Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

旋转壳边界条件广义表达式

旋转壳的边界条件,传统的表达方式是在中面位移μ,μ,ω,ψ或相应的四个力共八个量当中给定四个.而以节圆广义位移作为基本未知数,一个节圆上未知数的数目超过四个[1][2][3][4].在这种情况下关于边界条件的处理问题尚无令人满意的解决办法.本文利用虚功原理,导出一组壳边广义量与非广义量关系公式.研究了七种类型常见边界,给出用广义力与广义位移表示的边界条件公式.每一种边界条件公式的数目可以和一个节圆上所采用的未知数数目相一致.有了这些公式,即可直接将边界条件代入广义位移法运动方程以求解广义位移.这样做,避免了文献[2]关于未知数的变换与逆变换过程,不仅道理上简明而且也简化了计算.有了边界条件广义表达式,使得旋转壳广义位移法在理论上也更为完善.     

Solutions to open problems of Yang concerning inverse nodal problems

Yang [X. F. Yang, A new inverse nodal problem, Journal of Differential Equations 169 (2001), 633–653] considered a new inverse nodal problem for the Sturm-Liouville operator L(q, α, β) in L ²[0, 1]: an s-dense subset of the nodal set in (0, b) (for any fixed b ∈ ( \(\frac{1}{2}\) , 1]) determines the potential q and boundary data α, β. (1) Since the s-dense condition is stronger than the dense condition, X. F. Yang proposed an open problem “It is open if the boundary parameter α can be determined by a dense subset of the nodal set in (0, b) but not necessarily by an s-dense subset of the nodal set in (0, b).” Cheng et al. have solved this problem and shown that a dense subset of the nodal set in (0, b) completely determines the potential q and boundary data α, β. (2) Another interesting open question: “It remains open if the result holds true for b ∈ (0, \(\frac{1}{2}\) ]” is also proposed by X. F. Yang. In this paper we provide a counterexample to claim that the result does not hold true for b ∈ (0, \(\frac{1}{2}\) ), and a uniqueness theorem for b = \(\frac{1}{2}\) .     

Uniqueness For An Inverse Boundary Value Problem For Dirac Operators

The uniqueness of both the inverse boundary value problem and inverse scattering problem for Dirac equation with a magnetic potential and an electrical potential are proved. Also, a relation between the Dirichlet to Dirichlet map for the inverse boundary value problem and the scattering amplitude for the inverse scattering problem is given     

Inverse coefficient problem for a quasilinear hyperbolic equation with final overdetermination

The inverse problem of recovering a solution-dependent coefficient multiplying the lowest derivative in a hyperbolic equation is investigated. As overdetermination is required in the inverse problem, an additional condition is imposed on the solution to the equation with a fixed value of the timelike variable. Global uniqueness and local existence theorems are proved for the solution to the inverse problem. An iterative method is proposed for solving the inverse problem.     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

Inverse problems for the boundary value problem with the interior nodal subsets

In this paper, inverse nodal problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter were studied. The authors showed that some uniqueness theorems on the potential function hold by the Weyl function, respectively.