9 research outputs found
Fourth order Birkhoff regular problems with eigenvalue parameter dependent boundary conditions
A regular fourth order differential equation which depends quadratically on the eigenvalue parameter is considered with classes of separable boundary conditions independent of or depending on linearly. Conditions are given for the problems to be Birkhoff regular
Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions
The eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with boundary conditions
y(¸; 0) = 0;
y00(¸; 0) = 0;
y(¸; a) = 0;
y00(¸; a) + i®¸y0(¸; a) = 0;
where g 2 C1[0; a] is a real valued function and ® > 0, has an operator pencil L(¸) =
¸2 ¡ i®¸K ¡ A realization with self-adjoint operators A, M and K. It was shown that the
spectrum for the above boundary eigenvalue problem is located in the upper-half plane and
on the imaginary axis. This is due to the fact that A, M and K are self-adjoint. We consider
the eigenvalue problem y(4)(¸; x) ¡ (gy0)0(¸; x) = ¸2y(¸; x) with more general ¸-dependent
separated boundary conditions Bj(¸)y = 0 for j = 1; ¢ ¢ ¢ ; 4 where Bj(¸)y = y[pj ](aj) or
Bj(¸)y = y[pj ](aj) + i²j®¸y[qj ](aj), aj = 0 for j = 1; 2 and aj = a for j = 3; 4, ® > 0, ²j = ¡1
or ²j = 1. We assume that at least one of the B1(¸)y = 0, B2(¸)y = 0, B3(¸)y = 0, B4(¸)y = 0
is of the form y[p](0)+i²®¸y[q](0) = 0 or y[p](a)+i²®¸y[q](a) = 0 and we investigate classes of
boundary conditions for which the corresponding operator A is self-adjoint
Spectral theory of self-adjoint higher order differential operators with eigenvalue parameter dependent boundary conditions
We consider on the interval [0; a], rstly fourth-order di erential operators with eigenvalue
parameter dependent boundary conditions and secondly a sixth-order di erential operator
with eigenvalue parameter dependent boundary conditions. We associate to each of these
problems a quadratic operator pencil with self-adjoint operators. We investigate the spectral
proprieties of these problems, the location of the eigenvalues and we explicitly derive the rst
four terms of the eigenvalue asymptotics
A new inertial condition on the subgradient extragradient method for solving pseudomonotone equilibrium problem
In this paper we study the pseudomonotone equilibrium problem. We consider a
new inertial condition for the subgradient extragradient method with
self-adaptive step size for approximating a solution of the equilibrium problem
in a real Hilbert space. Our proposed method contains inertial factor with new
conditions that only depend on the iteration coefficient. We obtain a weak
convergence result of the proposed method under weaker conditions on the
inertial factor than many existing conditions in the literature. Finally, we
present some numerical experiments for our proposed method in comparison with
existing methods in the literature. Our result improves, extends and
generalizes several existing results in the literature
Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem
An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal.
Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods.
The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems.
The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one.
As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes
SELF-ADJOINT FOURTH ORDER DIFFERENTIAL OPERATORS WITH EIGENVALUE PARAMETER OPERATORS WITH EIGENVALUE PARAMETER
Click on the link to view the abstract.Keywords: Fourth order differential equation, eigenvalue dependent boundary conditions, quadratic operator pencil, self-adjoint operatorQuaestiones Mathematicae 34(2011), 393–40
SIXTH ORDER DIFFERENTIAL OPERATORS WITH EIGENVALUE DEPENDENT BOUNDARY CONDITIONS
We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper halfplane, except for finitely many eigenvalues on the negative imaginary axis
Analytic vs. numerical solutions to a Sturm-Liouville transmission eigenproblem
An elliptic one-dimensional second order boundary value problem involving discontinuous coefficients, with or without transmission conditions, is considered. For the former case by a direct sum spaces method we show that the eigenvalues are real, geometrically simple and the eigenfunctions are orthogonal.
Then the eigenpairs are computed numerically by a local linear finite element method (FEM) and by some global spectral collocation methods.
The spectral collocation is based on Chebyshev polynomials (ChC) for problems on bounded intervals respectively on Fourier system (FsC) for periodic problems.
The numerical stability in computing eigenvalues is investigated by estimating their (relative) drift with respect to the order of approximation. The accuracy in computing the eigenvectors is addressed by estimating their departure from orthogonality as well as by the asymptotic order of convergence. The discontinuity of coefficients in the problems at hand reduces the exponential order of convergence, usual for any well designed spectral algorithm, to an algebraic one.
As expected, the accuracy of ChC outcomes overpasses by far that of FEM outcomes