Fourth-order Operator Research Articles

The influence of the spectrum of the symmetrized fourth order operator on the geometry of a Riemannian manifold

On etudie l'influence du spectre d'un operateur d'ordre 4 special, appele l'operateur de Laplace symetrise d'ordre 4, sur la geometrie d'une variete de Riemann compacte

The accuracy of the solution of a difference boundary value problem for a fourth-order elliptic operator with mixed boundary conditions

An estimate of the accuracy of a conservative difference scheme in an energy net metric is established, when the solution of the initial problem belongs to Sobolev spaces. The accuracy of difference schemes for fourth-order elliptic differential operators when free-boundary type boundary conditions are imposed were investigated in [1–4] assuming that the solution and the coefficients belong to the spaces C (4)(Ω) and C (4)(Ω) respectively. A new approach, proposed in [5], when investigating the accuracy of difference schemes uses special averaging operators and enables one to obtain an estimate of the accuracy of the net method when there are requirements regarding the smoothness of the coefficients and the solution of the problem in terms of Sobolev spaces. Estimates of the accuracy of these schemes for certain problems with fourth-order elliptic operators were obtained in [6–8]. In this paper we investigate the accuracy of a difference scheme from [4] assuming that the solution and the coefficients of the initial problem belong to the spaces W 2 (k)(Ω) , kϵ [3, 4] and W 2 (m)(Ω) , mϵ (1, 2) respectively.

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

Effective methods for solving eigenvalue problems with fourth-order elliptic operators

A nine-noded quadratic control-volume-based finite element for heat conduction

A control-volume -based finite element formulation is presented for the nine-noded, quadratic, quadrilateral finite element for application to conduction heat transfer. The formulation is presented with the nine-noded element being used rather than the eight-noded element for reasons of convenience of control volume definition. The quadratic nature of the element in itself offers greater geometric flexibility and the potential for significantly higher accuracy than does the previous linear control volume element. It was observed that for steady-state conduction, the central control volume 'numerical molecule' is the optimal, fourth order operator on a uniform grid; however, due to the large connectivity for the corner control volume, such a 'molecule' interpretation is not readily facilitated. Performance of the element was evaluated through application to two test problems. It was observed that on the predominantly flux prescribed problem the performance is comparable to that of the Galerkin formulation. However, for the Dirichlet problem, a reduction of error by two orders of magnitude over the Galerkin formulation was observed.

The Dirichlet index under minimal conditions

SynopsisWe extend several known properties of the Dirichlet index to the case of minimal conditions and prove that the index is invariant under positive t bounded perturbations of the associated quadratic form. The index is also shown to be minimal for fourth order operators with certain growth conditions on some of their coefficients.

An operator compact implicit method of exponential type

A fourth-order accurate tridiagonal operator compact implicit finite-difference method for diffusion-convection equations is developed. The coefficients of the difference operator contain exponentials of the coefficients of the differential operator. In this way the method avoids spurious oscillations when the cell Reynolds number is large. Numerical results and comparisons to other methods are presented.

The Operator Compact Implicit Method for Reservoir Simulation

Abstract This paper describes the application of the fourth-order operator compact implicit (OCI) method to multiphase flow through porous media. A conservative form of the method is developed and used to solve a one-dimensional, two-phase, immiscible waterflood problem. The results of numerical experiments demonstrate the ability to track sharp fronts with relatively few mesh points. Comparisons with conventional finite-difference methods and variational methods are given.

The finite difference eigenvalue problem for fourth-order elliptic operator

The accuracy of the mesh method is proved for the case when the eigenfunction and its normal derivative are specified on the boundary of a rectangle. Error estimates for the mesh method are obtained in the context of an increase in the derivatives of the solution in the neighbourhood of the corner points. An error estimate is obtained for the eigenfunctions in a uniform metric.

Disconjugacy and the limit-point condition for fourth-order operators

SynopsisLetLdenote the ordinary differential operator given byLf= (pf″)″ + (qf′)′ +rf, withp″,q′ andrcontinuous functions on [0,∞), and withp>0,q≦ 0, andr≧ 0. It is proved that if the equationLg= 0 possesses a non-oscillatory solution, then any non-trivial solutionftoLf= 0 such thatf(0) =f′(0) = 0 is eventually bounded away from zero.This theorem is used to prove that, for a general class of functionsqandrcontaining the polynomials as a very special case, the equationLg= 0 has at most two linearly independent square integrable solutions, whenpis identically one,q≦ 0 andr≧ 0.Finally, the main theorem is applied to show that certain sixth-order operators are limit-3.

Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)

We consider the ``missing label'' problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3)⊃O(3)⊃O(2). We prove that only two independent O (3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C(2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C(3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

On the completeness of eigenfunctions of some higher order operators

The completeness of diverse eigenfunction systems is of interest, e.g., in connection with their use for expansion purposes. Whereas second-order operators have been treated extensively, it is more difficult to find valid proofs even for comparatively simple fourth- and higher-order operators. A criterion is developed that allows known properties of second-order differential operators to be used as a basis for conclusions regarding the completeness of the eigenfunctions of related fourth-order operators. The completeness of the “flatclamped-plate modes” is proved as an example, and it is demonstrated that the detailed form of the “conditions of finiteness at the singular point” can be crucial for the definition of operators corresponding to differential expressions with singular points.

On the inverse problem for differential operators of fourth order with rational reflection coefficients

On the inverse problem for differential operators of fourth order with rational reflection coefficients

A remark on the derivation of the indirect interaction between nuclear or ionic spins

AbstractIn order to analyse the effect of many‐particle excitations on the indirect interaction between nuclear or ionic spins the fourth order contribution to the ground state energy is calculated. This contribution appears to be divergent. This is due to the fact that the contributions of the disconnected diagrams and the irregular part of the fourth order operator do not cancel each other, which is a consequence of the commutator properties of angular momentum operators. It is therefore concluded that the derivation of the R.K.K.Y. interaction on the basis of Rayleigh‐Schrödinger perturbation theory is of doubtful validity.