Parameter-dependent Boundary Research Articles

Spectral properties for discontinuous Dirac system with eigenparameter‐dependent boundary condition

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems.

Eta-invariant of elliptic parameter-dependent boundary-value problems

In this paper, we study the eta-invariant of elliptic parameter-dependent boundary value problems and its main properties. Using Melrose’s approach, we de ne the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundary value problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary value problems and is based on the reduction of parameter-dependent boundary value problems to boundary value problems with no parameter.

Eigenvalues of one-dimensional Hamiltonian operators with an eigenparameter in the boundary condition

In this paper, one-dimensional Hamiltonian operators with spectral parameter-dependent boundary conditions are investigated. First, the eigenvalues of the problem under consideration are transformed into the eigenvalues of an operator in an appropriate Hilbert space. Then, some properties of the eigenvalues are given. Moreover, the continuity and differentiability of the eigenvalues of the problem are obtained, and the differential expressions of the eigenvalues concerning each parameter are also given. Finally, Green’s function is also involved.

On the solvability of a parameter-dependent cantilever-type BVP

We discuss the solvability of a parameter dependent cantilever-type boundary value problem. We provide an existence and localization result for the positive solutions via a Birkhoff–Kellogg type theorem. We also obtain, under additional growth conditions, upper and lower bounds for the involved parameters. An example is presented in order to illustrate the theoretical results.

Singular Green operators in the edge algebra formalism

We give a new description of the class of singular Green operators introduced by Grubb in the study of parameter-dependent boundary value problems, employing the concept of operator-valued symbols introduced by Schulze in the framework of pseudodifferential operators on manifolds with edges.

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

On one class eigenvalue problem with eigenvalue parameter in the boundary condition at one end-point

In the present paper we investigate the spectrum of operator corresponding to eigenvalue problem with parameter dependent boundary condition. Trace formula for that operator is also established.

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

Let D be an open connected subset of the complex plane C with sufficiently smooth boundary ∂D. Perturbing the Cauchy problem for the Cauchy–Riemann system ∂u = f in D with boundary data on a closed subset S ⊂ ∂D, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter e ∈ (0, 1] in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on ∂D\S, each of them has a unique solution in some appropriate Hilbert space H +(D) densely embedded in the Lebesgue space L 2(∂D) and the Sobolev–Slobodetskiĭ space H 1/2−δ(D) for every δ > 0. The corresponding family of the solutions {u e} converges to a solution to the Cauchy problem in H +(D) (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in H +(D) is equivalent to boundedness of the family {u e} in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.

Isogeometric analysis for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

Sixth-order boundary value problems of a one-parameter gradient-elastic Kirchhoff plate model are formulated in a weak form within an H3 Sobolev space setting with the corresponding equilibrium equations and general boundary conditions. The corresponding conforming Galerkin method is proposed with error estimates for discretizations satisfying C2 continuity requirements. Continuity, coercivity and consistency of the corresponding bilinear form are utilized for proving the theoretical results. Numerical computations with conforming isogeometric discretizations of Cp−1-continuous NURBS basis functions of order p≥3 confirm the theoretical results and illustrate the features of the problem for both statics and free vibrations. In particular, the effects of the additional boundary conditions and parameter-dependent boundary layers corresponding to the gradient elasticity theory are addressed by the numerical examples.

Functional Equations Related to Sine Type Functions

Functional equations in three classes of entire functions related to sine type functions are considered. These functional equations occur in inverse Sturm–Liouville problems with eigenvalue parameter dependent boundary conditions. In the first class \(\mathcal {A}\), which consists of certain pairs of entire functions (P, Q) with leading terms \(-\lambda \sin \lambda a\) and \(\cos \lambda a\), respectively, it is shown that, for each \(\eta \in \mathbb {R}\), the functional equation \(Q(\lambda )Y(\lambda )-P(\lambda )Z(\lambda )=\eta \) has a unique solution (Y, Z) in \(\mathcal {A}\). As a consequence, the functional equation \(Q(\lambda )S_1(\lambda )-P(\lambda )S_0(\lambda )=1 \) has a unique solution \((S_1,S_0)\) in a class of functions with leading terms \(\cos \lambda a\) and \(\lambda ^{-1}\sin \lambda a\), respectively. A third class of families \(\Phi _\alpha \), \(\alpha >0\), \(\alpha \not =1\), consists of entire functions with leading terms \(-\lambda \sin \lambda a+i\lambda \alpha \cos \lambda a\). It is shown that for each \(\phi \in \Phi _\alpha \) and \(\Delta \in \mathbb {R}\), the functional equation \(\phi (\lambda )X(-\lambda )-\phi (-\lambda )X(\lambda )=2i\alpha \lambda \Delta \) has a unique solution \(X\in \Phi _\alpha \).

Spectral parameter power series for Sturm-Liouville equations with a potential polynomially dependent on the spectral parameter and Zakharov-Shabat systems

A spectral parameter power series (SPPS) representation for solutions of Sturm-Liouville equations of the form pu′′+qu=u∑k=1Nλkrk is obtained. Such equations are known as Sturm-Liouville equations with potentials polynomially dependent on the spectral parameter or Sturm-Liouville pencils (SLP). It allows one to write a general solution of the equation as a power series in terms of the spectral parameter λ. The coefficients of the series are given in terms of recursive integrals involving a particular solution of the equation pu0′′+qu0=0. The convenient form of the solution of SLP provides an efficient numerical method for solving corresponding initial value, boundary value, and spectral problems. A special case of the SLP arises in relation with the Zakharov-Shabat system. We derive a SPPS representation for its general solution and consider other applications as the one-dimensional Dirac system and the equation describing a damped string. Several numerical examples illustrate the efficiency and the accuracy of the numerical method based on the SPPS representations which besides its natural advantages like the simplicity in implementation and accuracy is applicable to the problems admitting complex coefficients, spectral parameter dependent boundary conditions, and complex spectrum.

Eigenvalues of Vectorial Sturm-Liouville Problems with Parameter Dependent Boundary Conditions

We generalize theregularized sampling methodintroduced in 2005 by the author to compute the eigenvalues of scalar Sturm-Liouville problems (SLPs) to the case of vectorial SLP with parameter dependent boundary conditions. A few problems are worked out to illustrate the effectiveness of the method and show by the same token that we have indeed a general method capable of handling with ease very broad classes of SLPs, whether scalar or vectorial.

Numerical comparison of sinc-collocation and Chebychev-collocation methods for determining the eigenvalues of Sturm–Liouville problems with parameter-dependent boundary conditions

In this paper, we will compare the performance of sinc-collocation method and the Chebychev-collocation method applied to determine the eigenvalues of Sturm–Liouville problems with parameter-dependent boundary conditions. The numerical results indicate that the chebychev method possesses a significant computational advantage over the sinc-collocation method.

Evaluation of the Optimal Utility of Some Investment Projects with Irreversible Environmental Effects

In this work, the authors propose efficient numerical methods to solve mathematical models for different optimal investment problems with irreversible environmental effects. A relevant point is that both the benefits of the environment and the alternative project are uncertain. The cases with instantaneous and progressive transformation of the environment are addressed. In the first case, an augmented Lagrangian active set (ALAS) algorithm combined with finite element methods are proposed as a more efficient technique for the numerical solution to the obstacle problem associated with a degenerated elliptic PDE. In the second case, the mathematical model can be split into two subsequent steps: first we solve numerically a set of parameter dependent boundary value problems (the parameter being the level of progressive transformation), and secondly an evolutive nonstandard obstacle problem is discretized, thus leading to an obstacle problem at each time step. Also, an ALAS algorithm is proposed at each time step. Numerical solutions are validated through qualitative properties theoretically proven in the literature for different examples.

Generalized Sturm-Liouville Problems and Chebychev Collocation Method

In this paper, we present an algorithm for approximating the eigenvalues of Sturm-Liouville problems with parameter-dependent boundary conditions. The algorithm is based on the Chebychev method. A few examples shall be presented to illustrate the proposed method and a comparison made with the regularized sampling method. It is shown that the Chebychev method yields better results.

On the Solution of a Boundary Value Problem related to the Heat Transmission

In this study, we consider a heat transmission problem which has derivative with respect to the time in boundary condition. Applying the seperation of variables method, we get a Sturm-Liouville equation with discontinuous coefficient and a spectral parameter dependent boundary condition. For this spectral problem, the operator theoretic formula is given, the resolvent operator constructed and the expansion formula with respect to the eigenfunctions obtained. Using the expansion formula, the solution of the heat problem expressed.

Asymptotic properties of eigenvalues and eigenfunctions of a Sturm-Liouville problem with discontinuous weight function

In this paper, we extend some spectral properties of regular Sturm-Liouville problems to those which consist of a Sturm-Liouville equation with discontinuous weight at two interior points together with spectral parameter-dependent boundary conditions. By modifying some techniques of [C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 293-308; O. Sh. Mukhtarov and M. Kadakal, Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 46 (2005) 681-694], we give an operator-theoretic formulation for the considered problem and obtain asymptotic formulas for the eigenvalues and eigenfunctions. 2010 Mathematics Subject Classification: 34L20; 35R10

Inverse eigenvalue problems for a discontinuous Sturm-Liouville operator with two discontinuities

In this paper, we consider a discontinuous Sturm-Liouville operator with parameter-dependent boundary conditions and two interior discontinuities. We obtain eigenvalues and eigenfunctions together with their asymptotic approximate formulas. Then, we give some uniqueness theorems by using Weyl function and spectral data, which are called eigenvalues and normalizing constants for solution of inverse problem.

Determination of differential pencils with spectral parameter dependent boundary conditions from interior spectral data

Second-order differential pencils L(p,q,h0,h1,H0,H1) on a finite interval with spectral parameter dependent boundary conditions are considered. We prove the following: (i) a set of values of eigenfunctions at the mid-point of the interval [0,π] and one full spectrum suffice to determine differential pencils L(p,q,h0,h1,H0,H1); and (ii) some information on eigenfunctions at some an internal point and parts of two spectra suffice to determine differential pencils L(p,q,h0,h1,H0,H1). Copyright © 2013 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.