Self-adjoint Boundary Value Problems Research Articles

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.

Self-adjoint boundary conditions and interlacing of eigenvalues for the Sturm-Liouville equation on graphs

Applying the approach of Kostrykin and Schrader, [14], an explicit characterisation of self-adjoint boundary conditions at the nodes or vertices of a graph for the Sturm-Liouville equation is given. This is then proven to be equivalent to the conditions for self-adjointness of the corresponding system boundary value problem with separated boundary conditions. In addition, using an example, it is shown that the complete graph configuration is incorporated in the system boundary condition at the terminal end point. Making use of the separated system formulation, via matrix Prufer angle techniques, an interlacing property of the eigenvalues for a self-adjoint Sturm-Liouville boundary value problem on a graph is ascertained. Mathematics subject classification (2010): 34B45, 34L05, 34B24, 34L15.

Matrix representations of fourth order boundary value problems with periodic boundary conditions

We show that a class of regular self-adjoint fourth order boundary value problems (BVPs) is equivalent to a certain class of matrix problems. Conversely, for any given matrix problem in this class, there exist fourth order self-adjoint BVPs which are equivalent to the given matrix problem. Equivalent here means that they have exactly the same eigenvalues. Such an equivalence is considered with the special coupled boundary conditions, i.e. periodic boundary conditions.

An approach for computing eigenvalues of discrete symplectic boundary-value problems

We propose a new method for calculating eigenvalues of discrete symplectic boundaryvalue problems. This method is based on the discrete oscillation theory and on a modification of the Abramov double sweep method for discrete self-adjoint boundary-value problems.

Morse–Smale index theorems for elliptic boundary deformation problems

Morse-type index theorems for self-adjoint elliptic second order boundary value problems arise as the second variation of an energy functional corresponding to some variational problem. The celebrated Morse index theorem establishes a precise relation between the Morse index of a geodesic (as critical point of the geodesic action functional) and the number of conjugate points along the curve. Generalization of this theorem to linear elliptic boundary value problems appeared since seventies. (See, for instance, Smale (1965) [12], Uhlenbeck (1973) [15] and Simons (1968) [11] among others.) The aim of this paper is to prove a Morse–Smale index theorem for a second order self-adjoint elliptic boundary value problem in divergence form on a star-shaped domain of the N-dimensional Euclidean space with Dirichlet and Neumann boundary conditions. This result will be achieved by generalizing a recent new idea introduced by authors in Deng and Jones (2011) [5], based on the idea of shrinking the boundary.

A modification of the method of continuous prolongation for finding the bifurcation solutions of steady-state self-adjoint boundary value problems

An algorithm for showing solution of systems of non-linear algebraic equations describing the steady-state behaviour of objects in the mechanics of a deformable solid is considered. The existence of limit points and simple bifurcation points on the trajectory of the solution of the system is admitted and, at these points, the Jacobian matrix of the system, assumed to be real, symmetric and continuous, degenerates. The basis of the algorithm is a transformation of the space of the arguments of the solution of systems of non-linear algebraic equations using a rotation matrix formed from the eigenvectors of the Jacobian matrix.

Artificial boundary conditions for elliptic systems on polyhedral truncation surfaces

For a sufficiently broad class of formally selfadjoint boundary value problems in the domains with conical outlets to infinity, including exterior boundary value problems, we suggest an algorithm for constructing some artificial boundary conditions on polyhedral truncation surfaces that guarantees higher precision of approximation as the infinite domain is replaced with a large but bounded domain. The errors are estimated. Anisotropic 3-dimensional elasticity problems are discussed as an example.

Finite energy solutions of self-adjoint elliptic mixed boundary value problems

This paper describes existence, uniqueness and special eigenfunction representations of H1-solutions of second order, self-adjoint, elliptic equations with both interior and boundary source terms. The equations are posed on bounded regions with Dirichlet conditions on part of the boundary and Neumann conditions on the complement. The system is decomposed into separate problems defined on orthogonal subspaces of H1(Ω). One problem involves the equation with the interior source term and the Neumann data. The other problem just involves the homogeneous equation with Dirichlet data. Spectral representations of the solution operators for each of these problems are found. The solutions are described using bases that are, respectively, eigenfunctions of the differential operator with mixed null boundary conditions, and certain mixed Steklov eigenfunctions. These series converge strongly in H1(Ω). Necessary and sufficient conditions for the Dirichlet part of the boundary data to have a finite energy extension are described. The solutions for a problem that models a cylindrical capacitor is found explicitly. Copyright © 2009 John Wiley & Sons, Ltd.

The best constant of Sobolev inequality corresponding to a bending problem of a beam on an interval

Green function of 2-point simple-type self-adjoint boundary value problem for 4-th order linear ordinary differential equation, which represents bending of a beam with the boundary condition as clamped, Dirichlet, Neumann and free. The construction of Green function needs the symmetric orthogonalization method in some cases. Green function is the reproducing kernel for suitable set of Hilbert space and inner product. As an application, the best constants of the corresponding Sobolev inequalities are expressed as the maximum of the diagonal values of Green function.

A robust second-order numerical method for global solution and global normalized flux of singularly perturbed self-adjoint boundary-value problems

In this paper, we consider the finite difference hybrid scheme constructed by Natesan et al. for obtaining uniformly convergent global solution and uniformly convergent normalized flux for self-adjoint singularly perturbed boundary value problems. The global solution is obtained from the numerical solution at the mesh points of this scheme, having almost second-order uniform convergence at the nodal points when it is constructed on a piecewise uniform Shishkin mesh. Using a classical cubic spline, we define the solution and the normalized flux on the entire domain. We prove that the uniform order of convergence of the global solution is the same as that of the hybrid scheme at the mesh points. In addition, the global normalized flux is also almost second-order uniformly convergent in the whole domain. We provide theoretical error bounds and some numerical examples showing the efficiency of the proposed technique for obtaining the global solution and the normalized flux.

Initial-value technique for self-adjoint singular perturbation boundary value problems

We have developed an initial-value technique for self-adjoint singularly perturbed two-point boundary value problems. The original problem is reduced to its normal form, and the reduced problem is converted into first-order initial-value problems. These initial-value problems are solved by the cubic spline method. Numerical illustrations are given at the end to demonstrate the efficiency of our method. Graphs are also depicted in support of the results.

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.

Efficient preconditioning for the discontinuous Galerkin finite element method by low-order elements

We derive and analyze a block diagonal preconditioner for the linear problems arising from a discontinuous Galerkin finite element discretization. The method can be applied to second-order self-adjoint elliptic boundary value problems and exploits the natural decomposition of the discrete function space into a global low-order subsystem and components of higher polynomial degree. Similar to results for the p-version of the conforming FEM, we prove for the interior penalty discontinuous Galerkin discretization that the condition number of the preconditioned system is uniformly bounded with respect to the mesh size of the triangulation. Numerical experiments demonstrate the performance of the method.

Existence and uniqueness of periodic solution for a class of semilinear evolution equations

This paper deals with the existence and uniqueness for the periodic boundary value problem of the semilinear evolution equation in a Hilbert space H { u ′ ( t ) + A u ( t ) = f ( t , u ( t ) ) , 0 < t < ω , u ( 0 ) = u ( ω ) , where A : D ( A ) ⊂ H → H is a positive definite self-adjoint operator, ω > 0 and f : [ 0 , ω ] × H → H satisfy Caratheodory condition. We present some spectral conditions for the nonlinearity f ( t , u ) to guarantee the existence and uniqueness. These spectral conditions are the generalization for nonresonance condition of the self-adjoint elliptic boundary value problems.

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.

A parallel approach for self-adjoint singular perturbation problems using Numerov's scheme

In this paper, we considered singularly perturbed self-adjoint boundary-value problems and proposed a computational technique based on Numerov's scheme, which is also suitable for parallel computing. The whole domain is divided into three non-overlapping subdomains, and corresponding subproblems are obtained by using zeroth-order approximations of the solution at the boundaries of these subproblems. The subproblems corresponding to boundary-layer regions are solved using Numerov's method after the introduction of suitable stretching variables and the solution of the reduced problem is taken as an approximate solution in the outer region. A numerical example is provided to show the efficiency and accuracy of the technique.

Geometric mesh FDM for self-adjoint singular perturbation boundary value problems

A numerical method based on finite difference method with variable mesh is given for second order singularly perturbed self-adjoint two point boundary value problems. The original problem is reduced to its normal form and the reduced problem is solved by FDM taking variable mesh(geometric mesh). The maximum absolute errors max i | y ( x i ) - y i | , for different values of parameter ϵ, number of points N, and the mesh ratio r, for three examples have been given in tables to support the efficiency of the method.

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.

Bounds on the first nonzero eigenvalue for self-adjoint boundary value problems on networks

We aim here at obtaining bounds on the first non-null eigenvalue for self-adjoint boundary value problems on a weighted network by means of equilibrium measures, that includes the study of Dirichlet, Neumann and Mixed problems. We also show the sharpness of these bounds throughout the analysis of some known examples. In particular, we emphasize the case of distance-regular graphs, and we show that the bounds obtained are better than the known until now.