Higher-order Differential Operators Research Articles

A local tangential lifting differential method for triangular meshes

In this note we present a local tangential lifting (LTL) algorithm to compute differential quantities for triangular meshes obtained from regular surfaces. First, we introduce a new notation of the local tangential polygon and lift functions and vector fields on a triangular mesh to the local tangential polygon. Then we use the centroid weights proposed by Chen and Wu [4] to define the discrete gradient of a function on a triangular mesh. We also use our new method to define the discrete Laplacian operator acting on functions on triangular meshes. Higher order differential operators can also be computed successively. Our approach is conceptually simple and easy to compute. Indeed, our LTL method also provides a unified algorithm to estimate the shape operator and curvatures of a triangular mesh and derivatives of functions and vector fields. We also compare three different methods : our method, the least square method and Akima’s method to compute the gradients of functions.

The Algebraic Multiplicity of Eigenvalues and the Evans Function Revisited

This paper is related to the spectral stability of traveling wave solutions of partial differential equations. In the first part of the paper we use the Gohberg-Rouche Theorem to prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstract operator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of the corresponding Birman-Schwinger type operator pencil. In the second part of the paper we apply this result to discuss three particular classes of problems: the Schrödinger operator, the operator obtained by linearizing a degenerate system of reaction diffusion equations about a pulse, and a general high order differential operator. We study relations between the algebraic multiplicity of an isolated eigenvalue for the respective operators, and the order of the eigenvalue as the zero of the Evans function for the corresponding first order system.

A signed measure on rough paths associated to a PDE of high order: results and conjectures

Following old ideas of V. Yu. Krylov we consider the possibility that high order differential operators of dissipative type and constant coefficients might be associated, at least formally, with signed measures on path space in the same way that Wiener measure is associated with the Laplacian. There are fundamental difficulties with this idea because the measure would always have locally infinite mass. However, this paper provides evidence that if one considers equivalence classes of paths corresponding to distinct parameterisations of the same path, the measures might really exist on this quotient space. Precisely, we consider the measures on piecewise linear paths with given time partition defined using the semigroup associated to the differential operator and prove that these measures converge in distribution when the test functions on path space are the iterated integrals of the paths. Given a random piecewise-linear path, we evaluate its expected signature in terms of an explicit tensor series in the tensor algebra. Our approach uses an integration by parts argument under very mild conditions on the polynomial corresponding to the PDE of high order.

Lp–Lqestimates for dispersive equations and related applications

This paper is mainly concerned with the L p – L q estimates of solutions for a class of general dispersive equations under the condition ( H b ) with an index b ∈ ( 0 , 1 ] . To the end, the new pointwise decay estimates of oscillatory integrals related to the fundamental solutions are proved. If b = 1 , then the pointwise estimates are particularly consistent with the sharp results of the nondegenerate cases. Moreover, as an application of the L p – L q estimates, we also show that higher-order differential operator i P ( D ) + V ( x , D ) generates a fractionally integrated group on some L p ( R n ) , from which certain L p -estimates for the solutions of generalized Schrödinger equations are obtained.

On approximation of a certain differential operator

This paper consider the approximation problem of the image of an analytic function under the action of a higher-order differential operator with polynomial coefficients. The obtained formula generalizes the corresponding formulas of numerical differentiation.

Finite-gap systems, tri-supersymmetry and self-isospectrality

We show that an n-gap periodic quantum system with parity-even smooth potential admits 2n − 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral tri-supersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lamé equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite-period limit.

Isogeometric analysis of the Cahn–Hilliard phase-field model

The Cahn–Hilliard equation involves fourth-order spatial derivatives. Finite element solutions are not common because primal variational formulations of fourth-order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally C 1 -continuous. There are a very limited number of two-dimensional finite elements possessing C 1 -continuity applicable to complex geometries, but none in three-dimensions. We propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two- and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C 1 and higher-order continuity. A NURBS-based variational formulation for the Cahn–Hilliard equation was tested on two- and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

Numerical method for nonlinear two-phase displacement problem and its application

For the three-dimensional nonlinear two-phase displacement problem, the modified upwind finite difference fractional steps schemes were put forward. Some techniques, such as calculus of variations, induction hypothesis, decomposition of high order difference operators, the theory of prior estimates and techniques were used. Optimal order estimates were derived for the error in the approximation solution. These methods have been successfully used to predict the consequences of seawater intrusion and protection projects.

Reconstruction of higher-order differential operators on compact graphs

Reconstruction of higher-order differential operators on compact graphs

Necessary and sufficient conditions for the Riesz basis property of the eigen- and associated functions of high-order differential operators on an interval

Necessary and sufficient conditions for the Riesz basis property of the eigen- and associated functions of high-order differential operators on an interval

Regularized traces of higher-order singular differential operators

We consider singular differential operators of order 2m, m ∈ ℕ, with discrete spectrum in L2[0, + ∞). For self-adjoint extensions given by the boundary conditions y(0) = y″(0) = ⋯ = y(2m−2)(0) = 0 or y′(0) = y‴(0) = ⋯ = y(2m−1)(0) = 0, we obtain regularized traces. We present the explicit form of the spectral function, which can be used for calculating regularized traces.

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac–Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac–Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.

Generalized higher-order nonlinear energy operators

We extend and generalize the Teager-Kaiser [in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (1993), Vol. 3, p. 149] and the higher-order differential energy operators [IEEE Signal Process. Lett.2, 152 (1995)] to a large class of operators called higher-order energy operators. We show that for AM-FM signal demodulation, the introduced partial derivative orders have to satisfy certain conditions. These operators are parameterized for local processing of AM-FM signals. The operators are illustrated using synthetic signals and a real signal from light scanning interferometry.

Index theorems for quantum graphs

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher order differential operators as an intermediary. In this paper, the case of quantum graphs is addressed. A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a ‘Laplacian’) with suitable conditions at vertices. For the case of scale-invariant vertex conditions (i.e. conditions that do not mix the values of functions and of their derivatives), the constant term of the heat-kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph. This observation is placed into the index-theory context by factoring the Laplacian into two first-order operators, H = A*A, and relating the constant term to the index of A. An independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions. It is also found that the algebraic multiplicity of 0 as a root of the secular determinant of H is the sum of the nullities of A and A*.

Discrete Orthogonal Decomposition and Variational Fluid Flow Estimation

We exploit the mimetic finite difference method introduced by Hyman and Shashkov to present a framework for estimating vector fields and related scalar fields (divergence, curl) of physical interest from image sequences. Our approach provides a basis for consistent definitions of higher-order differential operators, for the analysis and a novel stability result concerning second-order div-curl regularizers, for novel variational schemes to the estimation of solenoidal (divergence-free) image flows, and to convergent numerical methods in terms of subspace corrections.

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

AbstractThe mathematical model of the three‐dimensional semiconductor devices of heat conduction is described by a system of four quasi‐linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection‐dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L2 norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Associative homotopy Lie algebras and Wronskians

We analyze representations of Schlessinger-Stasheff associative homotopy Lie algebras by higher-order differential operators. W-transformations of chiral embeddings of a complex curve related with the Toda equations into Kahler manifolds are shown to be endowed with the homotopy Lie-algebra structures. Extensions of the Wronskian determinants preserving Schlessinger-Stasheff algebras are constructed for the case of n ≥ 1 independent variables.

Parasupersymmetry and [formula omitted]-fold supersymmetry in quantum many-body systems. I: General formalism and second order

We propose an elegant formulation of parafermionic algebra and parasupersymmetry of arbitrary order in quantum many-body systems without recourse to any specific matrix representation of parafermionic operators and any kind of deformed algebra. Within our formulation, we show generically that every parasupersymmetric quantum system of order p consists of N -fold supersymmetric pairs with N ⩽ p and thus has weak quasi-solvability and isospectral property. We also propose a new type of non-linear supersymmetries, called quasi-parasupersymmetry, which is less restrictive than parasupersymmetry and is different from N -fold supersymmetry even in one-body systems though the conserved charges are represented by higher-order linear differential operators. To illustrate how our formulation works, we construct second-order parafermionic algebra and three simple examples of parasupersymmetric quantum systems of order 2, one is essentially equivalent to the one-body Rubakov–Spiridonov type and the others are two-body systems in which two supersymmetries are folded. In particular, we show that the first model admits a generalized 2-fold superalgebra.

Existence and Completeness of the Wave Operators for Higher Order Differential Operators

The aim of the present paper is to study the existence and the completeness of the wave operators $$\cal{W}_{\pm}(H,H_{0})$$ for elliptic operators of higher order (Schrodinger operator) with short-range potential of the form: $$ H = \sum_{|\alpha|,|\beta| \leq m}(-1)^{|\alpha|} D^{\alpha}(b_{\alpha \beta}(x)D^{\beta})+V_{0}+V$$ and other related results by using the trace class method.

The Upwind Finite Difference Fractional Steps Method for Nonlinear Coupled System of Dynamics of Fluids in Porous Media

For nonlinear coupled system of multilayer dynamics of fluids in porous media, the second order and first order upwind finite difference fractional steps schemes applicable to parallel arithmetic are put forward, and two-dimensional and three-dimensional schemes are used to form a complete set. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high order difference operators and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the second order approximate solution. This method has already been applied to the numerical simulation of migration-accumulation of oil resources.