Fourth-order Operator Research Articles

Spectral analysis of a fourth-order nonselfadjoint operator with nonsmooth coefficients

Using the method of similar operators, we study the spectral properties of a fourth-order differential operator with two types of classical boundary conditions. We obtain asymptotics for the spectrum and estimates for the spectral decompositions of this operator. We construct the semigroup generated by the opposite fourth-order differential operator.

Trace formulas for fourth order operators on unit interval, II

We consider self-adjoint fourth order operators on the unit interval with the Dirichlet type boundary conditions. For such operators we determine few trace formulas, similar to the case of Gelfand--Levitan formulas for second order operators.

Efficient fourth-order split operator for solving the triatomic reactive Schrödinger equation in the time-dependent wavepacket approach.

An efficient fourth-order split operator (named 4A6a in the main text), which was presented in the work by Blanes and Moan and was a partitioned Runge-Kutta method ( J. Comput. Appl. Math. 2002 , 142 , 313 ), is recommended for general usage in a reactive scattering calculation by the time-dependent quantum wavepacket method. This 4A6a propagator is constructed in a TVT form, that is, splitting in kinetic-potential-kinetic form, which is an optimal one among a series of higher-order split operators in examining with several typical triatomic reactive scattering processes, H + H2, H + H2(+), H + NH, H + O2, and F + HD reactions. A detailed comparison between the performances of higher-order split operators in the VTV form, that is, splitting in a potential-kinetic-potential, which was reported by Sun et al. ( Phys. Chem. Chem. Phys. 2012 , 14 , 1827 ), and in the TVT form reported in the current work suggests that the recommended 4A6a operator in the TVT form always has good numerical efficiency. This fact may suggest that this fourth propagator in the TVT form can be safely chosen without any further examination, at least among all of the higher-order split operators tested in this work, to apply in an efficient time-dependent wavepacket numerical calculation for describing a triatomic reactive scattering process.

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity.

We consider the nonlinear Dirac equation in 1 + 1 dimension with scalar-scalar self interaction g(2)/κ+1(̅ΨΨ)(κ+1) and with mass m. Using the exact analytic form for rest frame solitary waves of the form Ψ(x,t)=ψ(x)e(-iωt) for arbitrary κ, we discuss the validity of various approaches to understanding stability that were successful for the nonlinear Schrödinger equation. In particular we study the validity of a version of Derrick's theorem and the criterion of Bogolubsky as well as the Vakhitov-Kolokolov criterion, and find that these criteria yield inconsistent results. Therefore, we study the stability by numerical simulations using a recently developed fourth-order operator splitting integration method. For different ranges of κ we map out the stability regimes in ω. We find that all stable nonlinear Dirac solitary waves have a one-hump profile, but not all one-hump waves are stable, while all waves with two humps are unstable. We also find that the time t(c), it takes for the instability to set in, is an exponentially increasing function of ω and t(c) decreases monotonically with increasing κ.

Two-sided bounds of eigenvalues of second-and fourth-order elliptic operators

This article presents an idea in the finite element methods (FEMs) for obtaining two-sided bounds of exact eigenvalues. This approach is based on the combination of nonconforming methods giving lower bounds of the eigenvalues and a postprocessing technique using conforming finite elements. Our results hold for the second and fourth-order problems defined on two-dimensional domains.

Sharp eigenvalue asymptotics for fourth order operators on the circle

We determine high energy asymptotics of eigenvalues of fourth order operator on the circle.

A fourth-order approximate projection method for the incompressible Navier–Stokes equations on locally-refined periodic domains

In this follow-up of our previous work [30], the author proposes a high-order semi-implicit method for numerically solving the incompressible Navier–Stokes equations on locally-refined periodic domains. Fourth-order finite-volume stencils are employed for spatially discretizing various operators in the context of structured adaptive mesh refinement (AMR). Time integration adopts a fourth-order, semi-implicit, additive Runge–Kutta method to treat the non-stiff convection term explicitly and the stiff diffusion term implicitly. The divergence-free condition is fulfilled by an approximate projection operator. Altogether, these components yield a simple algorithm for simulating incompressible viscous flows on periodic domains with fourth-order accuracies both in time and in space. Results of numerical tests show that the proposed method is superior to previous second-order methods in terms of accuracy and efficiency. A major contribution of this work is the analysis of a fourth-order approximate projection operator.

Self-adjoint commuting ordinary differential operators

In this paper we study self-adjoint commuting ordinary differential operators of rank two. We find sufficient conditions when an operator of fourth order commuting with an operator of order 4g+2 is self-adjoint. We introduce an equation on potentials V(x),W(x) of the self-adjoint operator \(L=(\partial_{x}^{2}+V)^{2}+W\) and some additional data. With the help of this equation we find the first example of commuting differential operators of rank two corresponding to a spectral curve of higher genus. These operators have polynomial coefficients and define commutative subalgebras of the first Weyl algebra.

Nonlocal action for the super-Weyl anomalies: a new representation

Using the recently discovered N=1 supersymmetric extension of the conformal fourth-order scalar operator (introduced originally by Fradkin and Tseytlin and also known as the "Paneitz operator" or "Riegert operator"), we derive a new representation for the nonlocal action generating the super-Weyl anomalies.

Functional Model of Dissipative Fourth Order Differential Operators

Abstract In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

Spectral theory of higher order difference operators

The spectral theory of higher order difference operators is studied by means of asymptotic summation, thereby extending many results of differential operators to the discrete setting. The spectra of degenerate fourth-order operators are also investigated. The results are then compared with those of corresponding differential operators. Even though there are many similarities between both classes of operators, the spectral results may be quite distinct.

The spatial fourth-order energy-conserved S-FDTD scheme for Maxwell’s equations

In this paper we develop a new spatial fourth-order energy-conserved splitting finite-difference time-domain method for Maxwell’s equations. Based on the staggered grids, the splitting technique is applied to lead to a three-stage energy-conserved splitting scheme. At each stage, using the spatial fourth-order difference operators on the strict interior nodes by a linear combination of two central differences, one with a spatial step and the other with three spatial steps, we first propose the spatial high-order near boundary differences on the near boundary nodes which ensure the scheme to preserve energy conservations and to have fourth-order accuracy in space step. The proposed scheme has the important properties: energy-conserved, unconditionally stable, non-dissipative, high-order accurate, and computationally efficient. We first prove that the scheme satisfies energy conversations and is in unconditional stability. We then prove the optimal error estimates of fourth-order in spatial step and second-order in time step for the electric and magnetic fields and obtain the convergence and error estimate of divergence-free as well. Numerical dispersion analysis and numerical experiments are presented to confirm our theoretical results.

Eigenvalue problem of a class of fourth-order Hamiltonian operators

The eigenvalue problem of a class of fourth-order Hamiltonian operators is studied. We first obtain the geometric multiplicity, the algebraic index and the algebraic multiplicity of each eigenvalue of the Hamiltonian operators. Then, some necessary and sufficient conditions for the completeness of the eigen or root vector system of the Hamiltonian operators are given, which is characterized by that of the vector system consisting of the first components of all eigenvectors. Moreover, the results are applied to the plate bending problem.

Extension of split perfectly matched absorbing layer for 2D wave propagation in porous transversely isotropic media

The perfectly matched layer (PML) has proven to be efficient in absorbing outgoing waves in elastic and poroelastic media. It has not, however, been applied for porous anisotropic media. We develop the velocity–stress formulation for propagation of seismic waves for fluid-saturated porous anisotropic media with Biot’s equations. Then we extend the split perfectly matched absorbing layer (SPML) to these media and describe the staggered-grid finite-difference scheme. Using fourth-order spatial operators and a second-order temporal operator under 2D Cartesian coordinates, we numerically solve the equations for the solid and fluid particle velocity components, and for the solid stress components and fluid pressure. The energy decay curve we show demonstrates that the algorithm can run stably. Results from the horizontally layered model show that the SPML model absorbs the outgoing wave well, which illustrates the algorithm is efficient for modelling in porous transversely isotropic media.

More hyperelastic models for rubber-like materials: consistent tangent operators and comparative study

AbstractRubber-like materials can deform largely and nonlinearly upon loading, and they return to the initial configuration when the load is removed. Such rubber elasticity is achieved due to very flexible long-chain molecules and a three-dimensional network structure that is formed via cross-linking or entanglements between molecules. Over the years, to model the mechanical behavior of such randomly oriented microstructures, several phenomenological and micromechanically motivated network models for nearly incompressible hyperelastic polymeric materials have been proposed in the literature. To implement these models for polymeric material (undoubtedly with widespread engineering applications) in the finite element framework for solving a boundary value problem, one would require two important ingredients, i.e., the stress tensor and the consistent fourth-order tangent operator, where the latter is the result of linearization of the former. In our previous work, 14 such material models are reviewed by deriving the accurate stress tensors and tangent operators from a group of phenomenological and micromechanical models at large deformations. The current contribution will supplement some further important models that were not included in the previous work. For comparison of all selected models in reproducing the well-known Treloar data, the analytical expressions for the three homogeneous defomation modes, i.e., uniaxial tension, equibiaxial tension, and pure shear, have been derived and the performances of the models are analyzed.

Inequalities for lower-order eigenvalues of a fourth-order elliptic operator

In this paper, we investigate the Dirichlet weighted eigenvalues problem of a fourth-order elliptic operator with variable coefficients on a bounded domain with smooth boundary in ℝ n . We establish some inequalities for lower-order eigenvalues of this problem. In particular, our results contain an inequality for eigenvalues of the biharmonic operator derived by Cheng, Huang, and Wei.

A Fourth-Order Block-Grid Method for Solving Laplace's Equation on a Staircase Polygon with Boundary Functions in

The integral representations of the solution around the vertices of the interior reentered angles (on the “singular” parts) are approximated by the composite midpoint rule when the boundary functions are from These approximations are connected with the 9-point approximation of Laplace's equation on each rectangular grid on the “nonsingular” part of the polygon by the fourth-order gluing operator. It is proved that the uniform error is of order where and is the mesh step. For the -order derivatives () of the difference between the approximate and the exact solutions, in each “ singular” part order is obtained; here is the distance from the current point to the vertex in question and is the value of the interior angle of the th vertex. Numerical results are given in the last section to support the theoretical results.

Trace formula for fourth order operators on the circle

We determine the trace formula for the fourth order operator on the circle. This formula is similar to the famous trace formula for the Hill operator obtained by Dubrovin, Its-Matveev and McKean-van Moerbeke.

Quantization and the Hessian of Mabuchi energy

Let L be an ample bundle over a compact complex manifold X. Fix a Hermitian metric in L whose curvature defines a K\"ahler metric on X. The Hessian of Mabuchi energy is a fourth-order elliptic operator D on functions which arises in the study of scalar curvature. We quantise D by the Hessian E(k) of balancing energy, a function appearing in the study of balanced embeddings. E(k) is defined on the space of Hermitian endomorphisms of H^0(X, L^k), endowed with the L^2-innerproduct. We first prove that the leading order term in the asymptotic expansion of E(k) is D. We next show that if Aut(X,L) is discrete modulo scalars, then the eigenvalues and eigenspaces of E(k) converge to those of D. We also prove convergence of the Hessians in the case of a sequence of balanced embeddings tending to a constant scalar curvature K\"ahler metric. As consequences of our results we prove that a certain estimate of Phong-Sturm is sharp and give a negative answer to a question of Donaldson. We also discuss some possible applications to the study of Calabi flow.

The heat kernel and Green function of a fourth-order hypoelliptic operator on the Heisenberg group

We construct the heat kernel and Green function of a fourth-order hypoelliptic operator on the Heisenberg group.