Matrix representations of fourth order boundary value problems with finite spectrum

We show that a class of regular self-adjoint fourth order boundary value problems is equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues. Such an equivalence was previously known only in the second order case.     

带谱参数边界条件的四阶边值问题的矩阵表示

研究了一类具有有限谱的带有谱参数边界条件的四阶微分方程边值问题及其矩阵表示,证明了对任意正整数m,所考虑的问题至多有2m+6个特征值,进一步给出这类带有谱参数边条件的四阶边值问题与一类矩阵特征值问题之间在具有相同特征值的意义下是等价的.     

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.     

Matrix representations of Sturm–Liouville problems with eigenparameter-dependent boundary conditions

We show that a class of regular self-adjoint Sturm–Liouville problems with eigenparameter-dependent boundary conditions are equivalent to a certain class of matrix problems. Equivalent here means that they have exactly the same eigenvalues.     

求解一类不连续的Sturm-Liouville问题

研究一类边界条件中有谱参数的不连续的Sturm-Liouville问题.首先在Hilbert空间中定义了一个自共轭的线性算子A,使得该类Sturm-Liouville问题的特征值与算子A的特征值相一致.进一步证明了算子A是自共轭的,且这类Sturm-Liouville问题特征值是解析单的.最后展示了一个具体问题的特征值以及特征函数的逼近解.     

Inverse spectral problem of fourth‐order boundary value problems with finite spectrum

In the present paper, we consider a class of inverse spectral problem of fourth‐order boundary value problems. Under the so‐called “Atkinson type” conditions, the problem has finite spectrum and corresponding matrix representations. By using the method of inverse matrix eigenvalue problems of two‐banded matrix, the leading coefficient and potential functions are reconstructed from the given three sets of interlacing real numbers satisfying certain conditions.     

双锥不动点定理及其在非线性边值问题中的应用

本文研究了具可变号非线性项的非线性边值问题的正解存在性,推广了 Krasnoselskii 不动点定理,得到了新的锥上不动点定理,并应用这些定理给出这类边值问题正解的存在性．     

Indefinite Boundary Eigenvalue Problems in a Pontrjagin Space Setting

We study eigenvalue problems Fy = λ Gy consisting of Hamiltonian systems of ordinary differential equations on a compact interval with symmetric λ-linear boundary conditions. The problems we are interested in are non-definite: neither left-nor right-definite. Instead of this, we give some weak condition on one coefficient of the Hamiltonian system which ensures that a hermitian form associated with the operator F has at most finitely many negative squares. This enables us to study the problem by the help of a compact self-adjoint operator in a Pontrjagin space and we obtain as a main result uniformly convergent eigenfunction expansions. In the final section, applications to formally self-adjoint differential equations of higher order are given.     

Limitations of Richardson’s extrapolation for a high order fitted mesh method for self-adjoint singularly perturbed problems

The aim of this paper is to investigate whether we can accelerate the order of convergence of existing high order methods to solve some singularly perturbed two-point BVPs. To this end, we consider a fitted mesh finite difference method of Patidar (Appl. Math. Comput., 188:720–733, 2007) applied on a mesh of Shishkin type for the solution of self-adjoint problem which is ε-uniformly convergent of order four. We attempted to increase the order of convergence by Richardson’s extrapolation and discovered that this well-known convergence acceleration technique has some limitations. We observe that even though this extrapolation technique improves the accuracy slightly, it does not increase the rate of convergence which is originally four for the underlying method for the problem above. Theoretical investigations are demonstrated by some numerical experiments.     

General Transmission Problems in the Theory of Elastic Oscillations of Anisotropic Bodies (Basic Interface Problems)

Here we discuss three-dimensional so-called basic and mixed boundary value problems (BVP) for steady state oscillations of piecewise homogeneous anisotropic bodies imbedded into an infinite elastic continuum. Uniqueness is shown with the help of generalized Sommerfeld–Kupradze radiation conditions, while existence follows for arbitrary values of the oscillation parameter by the reduction of the original interface transmission BVPs to equivalent uniquely solvable boundary integral or pseudodifferential equations on the interfaces. For the basic BVPs, we show classical regularity and, in addition for the mixed BVPs that the solutions are Hölder continuous with exponent α ∈ (0, 1/2) in the neighbourhood of the curves of discontinuity of the boundary and transmission conditions.     

辛矩阵和四阶微分算子自共轭边界条件的基本型

给出了辛矩阵的定义,讨论了它的性质,并通过使用辛矩阵的方法研究四阶自共轭的边界条件,得到了四阶自共轭边界条件的基本型,从而使得其它各种自共轭的边界条件都可以通过基本型的辛变换得到.     

Existence of solutions to the nonlinear,singular second order Bohr boundary value problems

This paper investigates the existence of solutions for nonlinear systems of second order, singular boundary value problems (BVPs) with Bohr boundary conditions. A key application that arises from this theory is the famous Thomas–Fermi equations for the model of the atom when it is in a neutral state. The methodology in this paper uses an alternative and equivalent BVP, which is in the class of resonant singular BVPs, and thus this paper obtains novel results by implementing an innovative differential inequality, Lyapunov functions and topological techniques. This approach furnishes new results in the area of singular BVPs for a priori bounds and existence of solutions, where the BVP has unrestricted growth conditions and subject to the Bohr boundary conditions. In addition, the results can be relaxed and hold for the non-singular case too.     

Second Root Vectors for Multiparameter Eigenvalue Problems of Fredholm Type

A class of multiparameter eigenvalue problems involving (generally) non self-adjoint and unbounded operators is studied. A basis for the second root subspace, at eigenvalues of Fredholm type, is computed in terms of the underlying multiparameter system. A self-adjoint version of this result is given under a weak definiteness condition, and Sturm-Liouville and finite-dimensional examples are considered.

     

Lie symmetries of nonlinear boundary value problems

Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.     

Dependence of Eigenvalues of Certain Closely Discrete Sturm-Liouville Problems

This paper is concerned with dependence of eigenvalues of certain closely discrete Sturm-Liouville problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.     

Numerical Solution of Boundary Value Problems for Selfadjoint Differential Equations of 2nth Order

The paper is devoted to solving boundary value problems for self-adjoint linear differential equations of 2nth order in the case that the corresponding differential operator is self-adjoint and positive semidefinite. The method proposed consists in transforming the original problem to solving several initial value problems for certain systems of first order ODEs. Even if this approach may be used for quite general linear boundary value problems, the new algorithms described here exploit the special properties of the boundary value problems treated in the paper. As a consequence, we obtain algorithms that are much more effective than similar ones used in the general case. Moreover, it is shown that the algorithms studied here are numerically stable.     

A Decomposition Theorem for Higher-Order Sturm-Liouville Problems on the Line and Numerical Approximation of the Spectrum

In this work, we present a decomposition of the scattering matrix for higher-order Sturm-Liouville problems in terms of scattering matrices associated to disjointly supported potentials. Consequently, we propose a numerical method to approximate the eigenvalue problem. It is shown that the theory and numerics apply to the non self-adjoint case.     

The Generalized Order Tensor Complementarity Problems

The main propose of this paper is devoted to studying the solvability of the generalized order tensor complementarity problem. We define two problems: the generalized order tensor complementarity problem and the vertical tensor complementarity problem and show that the former is equivalent to the latter. Using the degree theory, we present a comprehensive analysis of existence, uniqueness and stability of the solution set of a given generalized order tensor complementarity problem.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Order Reduced Schemes for the Fourth Order Eigenvalue Problems on Multi-Connected Planar Domains

In this paper, we study the order reduced finite element method for the fourth order eigenvalue problems on multi-connected planar domains. Particularly, we take the biharmonic and the Helmholtz transmission eigenvalue problems as model problems, present for each an equivalent order reduced formulation and a corresponding stable discretization scheme, and present rigorous theoretical analysis. The schemes are readily fit for multilevel correction algorithms with optimal computational costs. Numerical experiments are given for verifications.