Self-adjoint Boundary Value Problems Research Articles

On Self-Adjoint Boundary Value Problems for a Functional Equation of Even Order

Investigating a functional differential equation of even order, we explore a class of self-adjoint boundary value problems with two-point conditions. We establish the basis property of the system of eigenfunctions and compare the eigenvalues.

Kramer-type sampling theorems associated with higher-order differential equations

For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.

An inverse boundary value problem for transverse vibrations of a bar

In this article, we study an inverse problem (IP) for a fourth-order hyperbolic equation with nonlocal boundary conditions. This IP is reduced to the not self-adjoint boundary value problem (BVP) with corresponding boundary condition. Then, we use the separation of variables method, to reduce the not self-adjoint BVP to an integral equation. The existence and uniqueness of the integral equation are established by the contraction mappings principle and it is concluded that this solution is unique for a not-adjoint BVP. The existence and uniqueness of a nonlocal BVP with integral condition is proved. In addition, the fourth-order hyperbolic PDE is discretized using a collocation technique based on the quintic B-spline (QnB-spline) functions and reformed by the Tikhonov regularization function. The noise and analytical data are considered. The numerical outcome for a standard numerical example is discussed. Furthermore, the stability of the discretized system is also analyzed. The rate of convergence (ROC) of the method is also obtained.

Spectral problem for the Laplacian and a selfadjoint nonlinear elliptic boundary value problem

In this paper, we present some connections between the spectral problem,
 −Δu(x) = λ1u(x) in Ω,u(x) = 0 on ∂Ω
 and selfadjoint boundary value problem,
 Δu(x) − λ1u(x) + g(x, u(x)) = h(x) in Ω,u(x) = 0 on ∂Ω,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.
 In this paper, we present some connections between the spectral problem,
 and selfadjoint boundary value problem,
 where λ1 is the smallest eigenvalue of −∆, Ω ⊆ Rn is a bounded domain, h ∈ L2(Ω) and the nonlinear function g is a Caratheodory function satisfying a growth condition. We initially investigate the existence of solutions for the spectral problem by considering the selfadjoint boundary value problem. The selfadjoint boundary value problem is then considered for both existence and estimation results. We use degree argument in order to show that the selfadjoint boundary value problem has a solution instead of the Landesman-Lazer condition or the monotonocity assumption on the second argument of the function g.

Kronecker Product of matrices and their applications to self-adjoint two-point boundary value problems associated with first order matrix differential systems

In this paper, we shall be concerned with Kronecker product or Tensor product of matrices and develop their properties in a systematic way. The properties of the Kronecker product of matrices is used as a tool to establish existence and uniqueness of solutions to two-point boundary value problems associated with system of first order differential systems. A new approach is described to solve the Kronecker product linear systems and establish best least square solutions to the problem. Several interesting examples are given to highlight the importance of Kronecker product of matrices. We present adjoint boundary value problems and deduce a set of necessary and sufficient conditions for the Kronecker product boundary value problem to be self-adjoint.

A type of full multigrid method for non-selfadjoint Steklov eigenvalue problems in inverse scattering

In this paper, a type of full multigrid method is proposed to solve non-selfadjoint Steklov eigenvalue problems. Multigrid iterations for corresponding selfadjoint and positive definite boundary value problems generate proper iterate solutions that are subsequently added to the coarsest finite element space in order to improve approximate eigenpairs on the current mesh. Based on this full multigrid, we propose a new type of adaptive finite element method for non-selfadjoint Steklov eigenvalue problems. We prove that the computational work of these new schemes are almost optimal, the same as solving the corresponding positive definite selfadjoint boundary value problems. In this case, these type of iteration schemes certainly improve the overfull efficiency of solving the non-selfadjoint Steklov eigenvalue problem. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

Compound operators on vector spaces with applications to linear differential equations

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given.

On Oscillation Properties of Self-Adjoint Boundary Value Problems of Fourth Order

The connection between the number of internal zeros of nontrivial solutions to fourth-order self-adjoint boundary value problems and the inertia index of these problems is studied. We specify the types of problems for which such a connection can be established. In addition, we specify the types of problems for which a connection between the inertia index and the number of internal zeros of the derivatives of nontrivial solutions can be established. Examples demonstrating the effectiveness of the proposed new approach to an oscillatory problem are considered.

Numerical solution of singularly perturbed self-adjoint boundary value problem using Galerkin method

This paper presents numerical solution of second order singularly perturbed self-adjoint boundary value problems using weighted residual method of Galerkin type. First, for the given problem, the residue was computed using appropriate approximated basis function which satisfies all the boundary conditions. Then, using the chosen weighting function, integrating the weighted residue over the domain and the given differential equation is transformed to linear systems of algebraic equations. Further, these algebraic equations were solved using Galerkin method. To validate the applicability of the proposed method, two model examples have been considered and solved for different values of perturbation parameter and with different order of basis function. Additionally, convergence of error bounds has been established for the method. As it can be observed from the numerical results, the present method approximates the exact solution very well. Moreover, the present method gives better accuracy when the order of basis function is increased and it also improves the result of the methods existing in the literature.
 Keywords: Singularly perturbed problems, Self-adjoint problem, Galerkin method, Boundary value problems.

The nonlocal boundary value problem with perturbations of mixed boundary conditions for an elliptic equation with constant coefficients. II

In this paper we continue to investigate the properties of the problem with nonlocal conditions, which are multipoint perturbations of mixed boundary conditions, started in the first part. In particular, we construct a generalized transform operator, which maps the solutions of the self-adjoint boundary-value problem with mixed boundary conditions to the solutions of the investigated multipoint problem. The system of root functions $V(L)$ of operator $L$ for multipoint problem is constructed. The conditions under which the system $V(L)$ is complete and minimal, and the conditions under which it is the Riesz basis are determined. In the case of an elliptic equation the conditions of existence and uniqueness of the solution for the problem are established.

Eigenvalues with respect to a weight for general boundary value problems on networks

In this work we analyze self-adjoint boundary value problems on networks for Schrödinger operators, in which a part of the boundary with a Neumann condition is always considered. We first characterize when the energy is positive semi-definite on the space of functions satisfying the null boundary conditions. To do this, the fundamental tools are the Doob transform and the discrete version of the trace function. Then, we raise eigenvalue problems with respect to a weight for general boundary value problems and we prove the discrete version of the Mercer Theorem. Finally, we apply the obtained results to a Dirichlet-Robin boundary value problem on a star-like network.

One-dimensional conformable fractional Dirac system

In this article, we consider a conformable fractional Dirac system. We prove an existence and uniqueness theorem for this system and formulate a self-adjoint boundary-value problem. We also construct the associated Green matrix of the conformable fractional Dirac system, and we give the eigenfunction expansions. Finally, we give some examples.

Eigencurves for linear elliptic equations

This paper provides results forvariational eigencurvesassociated with self-adjoint linear elliptic boundary value problems. The elliptic problems are treated as a general two-parameter eigenproblem for a triple (a,b,m) of continuous symmetric bilinear forms on a real separable Hilbert spaceV.Geometric characterizationsof eigencurves associated with (a,b,m) are obtained and are based on their variational characterizations described here. Continuity, differentiability, as well as asymptotic, results for these eigencurves are proved. Finally, two-parameter Robin–Steklov eigenproblems are treated to illustrate the theory.

Green’s Function of Ordinary Differential Operators and an Integral Representation of Sums of Certain Power Series

The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.

Computational Approach for Fourth-Order Self-Adjoint Singularly Perturbed Boundary Value Problems via Non-polynomial Quintic Spline

This paper proposes a non-polynomial quintic spline method for approximate solutions of fourth-order self-adjoint singular perturbation boundary value problems, which comprises the polynomial part of degree three and two non-polynomial terms, i.e., sine and cosine. It is explicitly shown that the free parameter \(\tau\) of the non-polynomial part can be used to raise the order of accuracy of the scheme. The method has been proved for the second and fourth-order convergence. The proposed scheme is tested on two examples. The experimental results are compared with the existing method.

Matslise 2.0

The Matslise 2.0 software package is a thorough revision of the successful Matlab package Matslise of 2005. The package can be used to compute the eigenvalues and eigenfunctions of regular and some important classes of singular self-adjoint Sturm-Liouville boundary value problems. The code uses new or improved algorithms, offers some new features, and has an updated graphical user interface.

On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator

The complete manifold of ground-state eigenfunctions for the purely magnetic two-dimensional Pauli operator is considered as a byproduct of a new reduction (found by the authors several years ago) for the algebro-geometric inverse spectral data (that is, Riemann surfaces and divisors). This reduction is associated with a -soliton hierarchy containing a 2D analogue of the famous `Burgers system'. This paper also surveys previous papers since 1980, including the first topological ideas in the space of quasi-momenta, and presents new results on self-adjoint boundary-value problems for the Pauli operator. The `non-spectral' Bloch–Floquet functions of zero 2D level give discrete points of additional spectrum analogous to the `boundary states' of finite-gap 1D potentials in the gaps. Bibliography: 35 titles.

Matrix representations of fourth-order boundary value problems with coupled or mixed boundary conditions

An equivalence between a class of regular self-adjoint fourth-order boundary value problems with coupled or mixed boundary conditions and a certain class of matrix problems is investigated. Such an equivalence was previously known only in the second-order case and fourth-order case with separated boundary conditions.

Adjoint and self-adjoint boundary value problems on a geometric graph

We consider boundary value problems of arbitrary order for linear differential equations on a geometric graph. Solutions of boundary value problems are coordinated at the interior vertices of the graph and satisfy given conditions at the boundary vertices. For considered boundary value problems, we construct adjoint boundary value problems and obtain a self-adjointness criterion. We describe the structure of the solution set of homogeneous self-adjoint boundary value problems with alternating coefficients of a differential equation and obtain nondegeneracy conditions for these boundary value problems.

Indexing of eigenvalues of boundary value problems for Hamiltonian systems of ordinary differential equations

For a self-adjoint spectral boundary value problem for a linear Hamiltonian system of ordinary differential equations, an indexing of the eigenvalues is found that is invariant under smooth variations in the parameters of the problem.