Discontinuous Sturm Liouville Research Articles

A generalized discontinuous Sturm–Liouville problem with boundary conditions rationally dependent on the eigenparameter

The present paper deals with a class of generalized discontinuous Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter. Using operator theoretic formulation under the new inner product based on boundary conditions rationally dependent on the eigenparameter and general transmission conditions, the self-adjointness of this operator is proved. We also obtain that the eigenvalues of the problems depend not only continuously but also smoothly on the parameters of the problem: the coefficients, the boundary conditions, the general transmission conditions, as well as the endpoints. Moreover, we find the differential expressions for each parameters, respectively.

Inverse nodal problem for discontinuous Sturm–Liouville operator by new Prüfer Substitutions

In the present paper, a boundary value problem consisting of a Sturm–Liouville equation with boundary conditions dependent on the eigenparameter and discontinuous conditions inside the interval is investigated. We present new Prufer substitutions and obtain the asymptotic form of eigenvalues, nodal points and nodal lengths. Then, we prove the uniqueness theorem for the solution of the inverse nodal problem, present a constructive procedure for the potential function by using nodal lengths. Finally, we study Lipschitz stability for the inverse problem.

Uniqueness Theorems for Inverse Problems of Discontinuous Sturm–Liouville Operator

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to

Solvability of an inverse problem for discontinuous Sturm–Liouville operators

In this paper, we consider the Sturm–Liouville equation with the jump conditions inside the interval (0,π). The inverse problem is studied, which consists in recovering operator coefficients from two spectra, corresponding to different boundary conditions. We prove the uniqueness theorem and provide necessary and sufficient conditions for solvability of the inverse problem. We also obtain the oscillation theorem for the eigenfunctions of the considered discontinuous boundary value problem.

An Inverse Problem for Discontinuous Sturm–Liouville Equations with Non-separated Boundary Conditions

In this article, a discontinuous Sturm–Liouville boundary value problem with non-separated boundary conditions is considered. It is proven that the potential function is uniquely established by either the Weyl function or by two spectra.

Riesz Basisness of Root Functions of a Sturm–Liouville Operator with Conjugate Conditions

In this paper we are interested in Riesz basisness of root functions of the non-selfadjoint a discontinuous Sturm–Liouville operator with periodic boundary condition which are not strong regular and with conjugate conditions. Here we assume that the potentials are complex valued and continuously differentiable functions. One of conjugate conditions have different finite one-sided limits at point zero. In order to prove Riesz basisness of root functions, we firstly obtain asymptotic expressions of fundamental solutions. Putting these solutions into characteristic determinant, we get asymptotic formulas of eigenvalues by means of Rouche theorem. Asymptotic formulas of eigenfunctions acquired by obtained relation and fundamental solutions. By the aid of asymptotic formulas of eigenfunctions and Bessel properties of eigenfunctions we prove the basisness of the root functions of the boundary value problem. We also prove the Riesz basisness of root functions of the same operator with antiperiodic boundary conditions and with same conjugate conditions.

On the Riesz basis property of the root functions of a discontinuous boundary problem

In this paper, we consider the nonself‐adjoint discontinuous Sturm Liouville operator with periodic (antiperiodic) boundary condition and compatibility conditions. Asymptotic formulas of eigenvalues and eigenfunctions of the operator are obtained. Using these accurate asymptotic formulas for eigenvalues and eigenfunctions, we prove the basisness of the root functions of the boundary value problem.

Discontinuous fractional Sturm–Liouville problems with transmission conditions

The main purpose of this study is the investigation of discontinuous Sturm–Liouville problem with fractional derivatives. We give an operator theoretic framework of the problem under consideration. Namely, we define an operator A in the Hilbert space L2[−1,1], the eigenvalues and corresponding eigenfunctions of which coincide with the eigenvalues and corresponding eigenfunctions of the boundary value problem, respectively. Then, we establish the characteristic function and prove that the eigenvalues of the considered problem coincide with the roots of this characteristic function.

On the Boundary Value Problem for Discontinuous Sturm–Liouville Operator

In this paper, we study the Sturm–Liouville problem which has not only the discontinuous coefficient, but also the discontinuity condition at an interior point of a finite interval. The new integral representation for the solution of the discontinuous Sturm–Liouville equation is constructed and the properties of the its kernel function are given. The asymptotic formulas of the eigenvalues and eigenfunctions of this problem are examined. The completeness theorem and expansion theorem of eigenfunctions are proved.

Inverse method identification of thermophysical properties based on solotone effect analysis for discontinuous Sturm–Liouville systems

ABSTRACTWe consider one-dimensional Sturm–Liouville systems where conditions for the existence of oscillatory terms within the distribution of the eigenvalues have been established. This so-called solotone effect arises when the underlying partial differential equations contain discontinuities within their coefficients. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. In this paper, we consider heat conduction in a layered composite and provide what is probably the first example of the presence of a solotone effect within a heat transfer context. Furthermore, we devise a novel method whereby relative values of the specific heat capacity and thermal conductivity of a composite rod may be accurately estimated from oscillations within the eigenvalue spectra. That is, we show how the solotone effect may be used to solve a particular heat transfer inverse problem. The method we present could be used to determine thermophysical properties of bonds or welds, for example when developing new adhesives or solders.

Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type

We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems.

The Basis Property of the System of Weak Eigenfunctions of a Discontinuous Sturm–Liouville Problem

The purpose of this study is to investigate some important spectral properties of one discontinuous Sturm–Liouville problem. By applying our own approaches the considered problem is transformed into an eigenvalue problem for suitable integral equation in terms of which it is defined as a concept of weak eigenfunctions. Then we introduce for consideration some compact operators in such a way that this integral equation can be reduced to the appropriate operator-pencil equation and prove that this operator-pencil is self-adjoint and positive definite for sufficiently large negative values of the eigenparameter. Finally, it is established that the spectrum is discrete and the system of corresponding weak eigenfunctions forms an orthonormal basis of the appropriate Hilbert space.

Spectral Problem for the Sturm–Liouville Operator with Retarded Argument Containing a Spectral Parameter in the Boundary Condition

We consider a discontinuous Sturm–Liouville problem with retarded argument containing a spectral parameter in the boundary condition. First, we investigate the simplicity of eigenvalues and then prove the existence theorem. As a result, we obtain the asymptotic formulas for eigenvalues and eigenfunctions.

Completeness of eigenfunctions of discontinuous Sturm–Liouville problems

In this work we investigate the resolvent operator and completeness of eigenfunctions of a Sturm–Liouville problem with discontinuities at two points. The problem contains an eigenparameter in one of the boundary conditions. For operator-theoretic formulation of the considered problem we define an equivalent inner product in the Hilbert space \( L_{2} [ - 1,1] \oplus C \) and suitable self-adjoint linear operator in it.

Sturm-Liouville problem with moving discontinuity points

In this paper, we present a new discontinuous Sturm Liouville problem with symmetrically located discontinuities which are defined depending on a neighborhood of a midpoint of the interval. Also the problem contains an eigenparameter in one of the boundary conditions and has coupled transmission conditions at the discontinuity points. We investigate the properties of the eigenvalues, obtain asymptotic formulas for the eigenvalues and the corresponding eigenfunctions and construct Green's function of this problem.

On the inverse nodal problems for discontinuous Sturm–Liouville operators

In this paper, we discuss the inverse nodal problems for discontinuous Sturm–Liouville operators with Robin boundary conditions. We show that the potential q up to its mean value on the interval [0,1] and coefficients h, H, ba can be uniquely determined by the twin-dense nodal subset on the subinterval [a0,b0], a0<12<b0. In addition, we still establish several uniqueness theorems by the left twin-dense nodal subset on the interval [a0,12] and additional information, respectively.

Spectral Properties of Discontinuous Sturm–Liouville Problems with a Finite Number of Transmission Conditions

In this paper we investigate discontinuous two-point boundary value problems with eigenparameter in the boundary conditions and with transmission conditions at the finitely many points of discontinuity. A self-adjoint linear operator A is defined in a suitable Hilbert space H such that the eigenvalues of the considered problem coincide with those of A. We obtain asymptotic formulas for the eigenvalues and eigenfunctions. Also we show that the eigenelements of A are complete in H.

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions

In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.

Inverse nodal problems of discontinuous Sturm–Liouville operator

We study inverse nodal problems for the second-order differential operator with discontinuity inside a finite interval. In particular, we solve the uniqueness, reconstruction and stability problems using the nodal set of its eigenfunctions. Furthermore, we show that the space of discontinuous Sturm–Liouville operators characterized by B=(q,h,H,a1,a2)∈L1(0,1)×R4 such that ∫01q(x)dx=0 is homeomorphic to the partition set of the space of quasinodal sequences, which are all admissible sequences X={Xnj}n⩾2,j=1,n−1¯ which form sequences that converge to q, h, H, and 2a2a1+a1−1 individually.

Trace and inverse problem of a discontinuous Sturm–Liouville operator with retarded argument

In this study a discontinuous boundary value problem with retarded argument and with transmission conditions at the point of discontinuity is investigated. Regularized sums from the eigenvalues, oscillations of eigenfunctions and the solutions of inverse nodal problem for the problem are obtained. In the special case that the transmission coefficient δ=1 and retarded argument Δ≡0 in the results obtained below coincide with corresponding results in the classical Sturm–Liouville operator.