Fourth-order Operator Research Articles

FDTD3C—A FORTRAN program to model multi-component seismic waves for vertically heterogeneous attenuative media

Full-waveform seismic response of horizontally layered media can be calculated by semi-analytical methods. However, for gradient velocity and randomly heterogeneous structures the semi-analytical methods face difficulties. In such cases, numerical methods such as the finite-difference (FD) method have to be used. We develop an efficient numerical scheme to calculate plane-wave response of vertically heterogeneous attenuative media by applying Radon transform to the three-dimensional wave equation. The scheme employs fourth-order FD operator in space and second-order FD operator in time to solve the wave equation. In order to facilitate applicability of the scheme we introduce the FORTRAN code FDTD3C which implements the algorithm and provides multi-component response of the media to oblique incident P-, SV-, and SH-waves incoming from arbitrary azimuth. The calculated components are three particle velocity components in three Cartesian directions, and divergence and rotation of the wavefield. The code is extremely efficient and is capable of incorporating highly fluctuating subsurface velocity and attenuation models. This program is intended for all FD users who are concerned with full-waveform seismic modelling and inversion. Wide range of applicability of the code is demonstrated with a set of numerical examples.

A Discontinuous Galerkin Method for the Fourth-Order Curl Problem

In this paper, we present a discontinuous Galerkin (DG) method based on the Nedelec finite element space for solving a fourth-order curl equation arising from a magnetohy- drodynamics model on a 3-dimensional bounded Lipschitz polyhedron. We show that the method has an optimal error estimate for a model problem involving a fourth-order curl operator. Furthermore, some numerical results in 2 dimensions are presented to verify the theoretical results.

A higher‐order finite element approach to the Kuramoto‐Sivashinsky equation

AbstractThe Kuramoto‐Sivashinsky equation has emerged as a fundamental evolution equation to describe highly nonlinear physical processes in unstable systems. In general, it constitutes a nonlinear initial‐valued problem involving fourth‐order spatial derivatives. Finite element solutions for the Kuramoto‐Sivashinsky equation are not common because the primal variational formulation of fourth‐order operators requires finite element basis functions which are piecewise smooth and globally at least C1‐continuous. In this paper a novel B‐spline based finite element approach to the solution of the one‐dimensional Kuramoto‐Sivashinsky equation is presented. Extensive numerical studies of different scenarios in the Kuramoto‐Sivashinsky equation will illustrate the quality of our approach.

The theory of regularized traces of operators as applied to approximate computation of eigenvalues and eigenfunctions of fluid dynamics problems

Some results concerning the computation of eigenvalues and eigenfunctions of fluid dynamics problems by applying methods of regularized traces of differential operators are presented. The presentation is focused primarily on the non-self-adjoint fourth-order Orr-Sommerfeld operator, which arises in the hydrodynamic stability theory of viscous flows.

On nonlocal symmetries for the Krichever–Novikov equation

We construct new infinite hierarchies of nonlocal symmetries and cosymmetries for the Krichever–Novikov equation using the inverse of the fourth-order recursion operator of the latter.

Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in $ {\mathbb{R}^n} $ . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.

On the completeness of eigen and root vector systems for fourth-order operator matrices and their applications

In this paper, we consider the eigenvalue problem of a class of fourth-order operator matrices appearing in mechanics, including the geometric multiplicity, algebraic index, and algebraic multiplicity of the eigenvalue, the symplectic orthogonality, and completeness of eigen and root vector systems. The obtained results are applied to the plate bending problem.

Spectral estimates for a periodic fourth-order operator

Spectral estimates for a periodic fourth-order operator

Spectral theory of difference operators with almost constant coefficients

The spectrum of higher even order difference operators with almost constant coefficients is determined. With appropriate smoothness and decay conditions on the coefficients, we show that singular continuous spectrum is absent and that the absolutely continuous spectrum agrees with that of the constant coefficient limiting operator. For such operators, the absolutely continuous spectrum is determined uniquely by the range of the characteristic polynomial. This result extends a similar result for even order differential operators. The methods of proof are closely related likewise. Finally, some results on fourth order operators with unbounded coefficients are shown.

New two-dimensional quantum models with shape invariance

Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional supersymmetric quantum mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

Asymptotic analysis for fourth order Paneitz equations with critical growth

We investigate fourth order Paneitz equations of critical growth in the case of n-dimensional closed conformally at manifolds, n 5. Such equations arise from conformal geometry and are modelized on the Einstein case of the geometric equation describing the eects of conformal changes of metrics on the Q-curvature. We obtain sharp asymptotics for arbitrary bounded energy sequences of solutions of our equations from which we derive stability and compactness properties. In doing so we establish the criticality of the geometric equation with respect to the trace of its second order terms. In 1983, Paneitz (25) introduced a conformally covariant fourth order operator extending the conformal Laplacian. Branson and rsted (5), and Branson (2, 3), introduced the associated notion of Q-curvature when n = 4 and in higher di- mensions when dealing with the conformally covariant extensions of the Paneitz operator by Graham-Jenne-Mason-Sparling. The scalar and the Q-curvatures are respectively, up to the conformally invariant Weyl's tensor in dimension four, the integrands in dimensions two and four for the Gauss-Bonnet formula for the Euler characteristic. The articles by Branson and Gover (4), Chang (6, 7), Chang and Yang (8), and Gursky (15) contain several references and many interesting material on the geometric and physics aspects associated to this notion of Q-curvature. In what follows we let (M;g) be a smooth compact Riemannian manifold of dimension n 5 and consider the fourth order variational Paneitz equations of critical Sobolev growth which are written as 2 gu +b gu +cu = u

Preconditioning a class of fourth order problems by operator splitting

We develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space. We consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two (discrete) linear second order elliptic operators, and exploit this property in designing the preconditioners. The underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available. We propose symmetric and non-symmetric preconditioners, along with theory and numerical experiments. They both document crucial properties of the preconditioners as well as their practical performance. It is important to note that we neither need H s -regularity, s > 1, of the continuous problem nor quasi-uniform grids.

Critical Hardy–Lieb–Thirring Inequalities for Fourth-Order Operators in Low Dimensions

This paper considers Hardy-Lieb-Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality tr ...

Trace anomaly, massless scalars, and the gravitational coupling of QCD

The anomalous effective action describing the coupling of gravity to a non-abelian gauge theory can be determined by a variational solution of the anomaly equation, as shown by Riegert long ago. It is given by a nonlocal expression, with the nonlocal interaction determined by the Green's function of a conformally covariant operator of fourth order. In recent works it has been shown that this interaction is mediated by a simple pole in an expansion around a Minkowski background, coupled in the infrared in the massless fermion limit. This result relies on the local formulation of the original action in terms of two auxiliary fields, one physical scalar and one ghost, which take the role of massless composite degrees of freedom. In the gravity case, the two scalars have provided ground in favour of some recent proposals of an infrared approach to the solution of the dark energy problem, entirely based on the behaviour of the vacuum energy at the QCD phase transition. As a test of this general result, we perform a complete one-loop computation of the effective action describing the coupling of a non-abelian gauge theory to gravity. We confirm the appearance of an anomaly pole which contributes to the trace part of the $TJJ$ correlator and of extra poles in its trace-free part, in the quark and gluon sectors, describing the coupling of the energy momentum tensor ($T$) to two non abelian gauge currents ($J$).

Magnetostructural relations from a combined ab initio and ligand field analysis for the nonintuitive zero-field splitting in Mn(III) complexes

The zero-field splitting (ZFS) of a model monometallic Mn(III) complex is theoretically studied as function of a systematic symmetry lowering. First, we treat the octahedral case for which the standard S.D.S model Hamiltonian cannot be applied due to a zero-field splitting in the absence of anisotropy induced by the spin-orbit coupling between the two spatial components of the (5)E(g) state at second-order of perturbation. Next, the symmetry is lowered to D(4h) and D(2h) and the anisotropic spin Hamiltonian is extracted using effective Hamiltonian theory. A simple relation is derived between the ratio E//D/ and the applied rhombic and axial distortions. Moreover, it is shown that close to O(h) symmetry, the orbital mixing due to spin-orbit coupling can be accurately described with Stevens fourth-order operators. The calculated tendencies are interpreted within a refined Racah plus ligand field model and it is shown that the ZFS parameters in Mn(III) complexes follow special rules that are nonintuitive compared to other d(n) configurations. Finally, some angular distortions are applied to study their effect on the anisotropy.

PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces.

Magnetic anisotropy in binuclear complexes in the weak-exchange limit: From the multispin to the giant-spin Hamiltonian

This paper studies the physical basis of the giant-spin Hamiltonian, which is usually used to describe the anisotropy of single-molecule magnets. A rigorous extraction of the model has been performed in the weak-exchange limit of a binuclear centrosymmetric Ni(II) complex, using correlated ab initio calculations and effective Hamiltonian theory. It is shown that the giant-spin Hamiltonian is not appropriate to describe polynuclear complexes as soon as spin mixing becomes non-negligible. A relevant model is proposed involving fourth-order operators, different from the traditionally used Stevens operators. The new giant-spin Hamiltonian correctly reproduces the effects of the spin mixing in the weak-exchange limit. A procedure to switch on and off the spin mixing in the extraction has been implemented in order to separate this effect from other anisotropic effects and to numerically evaluate both contributions to the tunnel splitting. Furthermore, the new giant-spin Hamiltonian has been derived analytically from the multispin Hamiltonian at the second order of perturbation and the theoretical link between the two models is studied to gain understanding concerning the microscopic origin of the fourth-order interaction in terms of axial, rhombic, or mixed (axial-rhombic) character. Finally, an adequate method is proposed to extract the proper magnetic axes frame for polynuclear anisotropic systems.

A Stampacchia-type inequality for a fourth-order elliptic operator on Kähler manifolds and applications

In this paper we will prove an integral inequality of Stampacchia-type for a fourth-order elliptic operator on complete and connected Kähler manifolds. Our inequality implies a Hodge–Kodaira orthogonal decomposition for the Sobolev-type space Wp,q ( X ). In particular we will able to prove, under suitable topological conditions on the manifold X , the existence of an isomorphism between the Aeppli groups Λ_p,q_( X ) and the groups Hp,q ( X ) of all global harmonic forms of bidegree ( p , q ).