Second-order Operator Research Articles

Global regularity of second order twisted differential operators

In this paper we characterize global regularity in the sense of Shubin of twisted partial differential operators of second order in dimension 2. These operators form a class containing the twisted Laplacian, and in bi-unique correspondence with second order ordinary differential operators with polynomial coefficients and symbol of degree 2. This correspondence is established by a transformation of Wigner type. In this way the global regularity of twisted partial differential operators turns out to be equivalent to global regularity and injectivity of the corresponding ordinary differential operators, which can be completely characterized in terms of the asymptotic behavior of the Weyl symbol. In conclusion we observe that we have obtained a new class of globally regular partial differential operators which is disjoint from the class of hypo-elliptic operators in the sense of Shubin.

Upper Bounds for the Approximation of Some Classes of Bivariate Functions by Triangular Fourier-Hermite Sums in the Space L2,ρ(ℝ2)

We evaluate the suprema of approximations of bivariate functions by triangular partial sums of the double Fourier-Hermite series on the class of functions Lr2(D) in the space L2,ρ(ℝ2), where D is the second-order Hermite operator. Sharp Jackson-Stechkin type inequalities on the sets L2,ρ(ℝ2) are obtained, in which the best approximation is estimated from above both in terms of moduli of continuity of order m and in terms of K-functionals. N-widths of some classes of functions in L2,ρ(ℝ2) are evaluated.

Understanding the Performance of an Unstructured-Mesh Global Shallow Water Model on Kinetic Energy Spectra and Nonlinear Vorticity Dynamics

A strategy for evaluating a global shallow water model based on aspects of kinetic energy spectra and nonlinear vorticity dynamics is proposed in this study. The kinetic energy spectra and nonlinear vorticity dynamics of a recently developed global shallow water model on an unstructured mesh are evaluated in comparison with the benchmark solutions from a global high-resolution spectral model. The results show that the kinetic energy spectra, the rotational and divergent components, the stationary and transient components, and the nonlinear spectral fluxes of the developed shallow water model agree well with those generated by the reference model. In addition, the influence of different flux operators for transporting the potential vorticity (PV) is assessed specifically. It is indicated that the second-order flux operator leads to a spurious increase in the kinetic energy at the tail of the spectrum, whereas the upwind third-order flux operator does not support this behavior owing to implicit numerical diffusion. Moreover, the nonlinear vorticity dynamics is studied by using colliding modons. It is found that the grid-point model maintains the symmetrical pattern of vortices, and generates similar kinetic energy spectra and nonlinear spectral fluxes to the reference model. The evaluation provides a reference for assessing the shallow water model in terms of nonlinear dynamics, and the developed global shallow water model presents a good example.

Comparison of operator splitting schemes for the numerical solution of the Allen-Cahn equation

In this paper, we have analyzed the first- and second-order operator splitting schemes for the numerical solution of the Allen-Cahn equation. Different possibilities are considered for the derivation of the schemes. The numerical errors are computed in order to verify the effectiveness of each scheme. Validation of the schemes is provided by numerically solving some well-known examples.

Second-order numerical methods for the tempered fractional diffusion equations

In this paper, a class of second-order tempered difference operators for the left and right Riemann–Liouville tempered fractional derivatives is constructed. And a class of second-order numerical methods is presented for solving the space tempered fractional diffusion equations, where the space tempered fractional derivatives are evaluated by the proposed tempered difference operators, and in the time direction is discreted by the Crank–Nicolson method. Numerical schemes are proved to be unconditionally stable and convergent with order O(h^{2}+tau ^{2}). Numerical experiments demonstrate the effectiveness of the numerical schemes.

On finite difference schemes for partial integro-differential equations of Lévy type

In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of possibly degenerate integro-differential equations by treating the nonlocal operator as a second-order operator on the whole unit ball, eliminating the need for truncation of the Lévy measure which is present in the existing literature. This yields an approximation scheme with significantly reduced computational cost, especially for Lévy measures corresponding to processes with jumps of infinite variation.

Disentangling the IR spectra of 2,3,3,3-tetrafluoropropene using an ab initio description of vibrational polyads by means of canonical Van Vleck perturbation theory

The vibrational spectra of 2,3,3,3-tetrafluoropropene (2333TFP) are studied in the infrared experimentally and theoretically by the canonical second-order Van Vleck operator perturbation theory (CVPT2) and full quartic potential energy surface (PES). 2333TFP belongs to hydrofluoroolefins (HFOs) considered among the most promising alternatives to the hydrofluorocarbons presently in use in refrigeration. The medium resolution infrared spectra of gaseous 2333TFP were recorded in the range 9500–30 cm−1. The integrated IR intensities for all bands falling between 6300–400 cm−1 were accurately determined. Theoretical descriptions of up to four-quanta anharmonic vibrational states of 2333TFP are performed using the state-of-the-art numerical-analytic implementation of CVPT2. High quality harmonic frequencies, obtained at different coupled-cluster levels of theory up to the CCSD(T*)-F12c/VTZ-F12 model were combined with the full quartic MP2/cc-pVTZ anharmonic force field to form the hybrid PES. All Fermi and Darling-Dennison resonances were detected using a universal criterion and treated with the variational CVPT2/VCI stage by using two-, three-, and four-quanta harmonic oscillator basis sets. General recommendations for choosing such basis sets in future studies are formulated. In addition to all fundamentals and their resonance satellites, the vibrational analysis led to the assignment of virtually all observed first/second overtone and binary/ternary combination bands. Computed dipole moment first derivatives in the principal axis system yielded reliable predictions of band shapes for fundamental transitions in good agreement with observed counterparts. Computed anharmonic IR intensities showed an excellent agreement with observed data in the measured ranges. The fundamental spectroscopic effect of excited resonances is observed experimentally and explained theoretically as a universal mechanism of formation of vibrational polyads. The efficiency of the CVPT2/VCI approach employed is clearly demonstrated and can be considered as a benchmark for modeling vibrational states and interpretation of vibrational spectra of semi-rigid molecules.

A Tractable Logic for Molecular Biology

We introduce a logic for knowledge representation and reasoning on protein-protein interactions. Modulo a theory, formulas describe protein structures and dynamic changes. They can be composed in order to add or remove static and dynamic observations. A second-order circumscription operator then enables nonmonotonic reasoning on the changes implied by a formula. We introduce deduction rules that produce formulas which are, up to equivalence, in a first-order fragment with decidable satisfiability and validity. Importantly, the rules can produce circumscribed formulas.

Approximate Controllability of Second-order Neutral Stochastic Non-autonomous Integrodifferential Inclusionsby Resolvent Operators

In this paper, we formulate a set of sufficient conditions for the approximate controllability for a class of second-order neutral stochastic nonautonomous integrodifferential inclusions in Hilbert space. We establish the results with the help of resolvent operators and Bohnenblust-Karlin’s fixed point theorem is to prove the main result. An application is given to illustrate the main result.

Triangulation-based isogeometric analysis of the Cahn–Hilliard phase-field model

This paper presents triangulation-based Isogeometric Analysis of the Cahn–Hilliard phase-field model. The Cahn–Hilliard phase-field model is governed by a time-dependent fourth-order partial differential equation. The corresponding primal variational form involves second-order operators, making it difficult to be directly analyzed with traditional C0 finite element analysis. In this paper, we construct C1 Bernstein–Bézier simplicial elements through macro-element techniques, including various triangle-split based macro-elements in both 2D and 3D space. We extend triangulation-based isogeometric analysis to solving the primal variational form of the Cahn–Hilliard equation. We validate our method by convergence analysis, showing the nodal and degree-of-freedom advantages over C0 Finite Element Analysis. We then demonstrate detailed system evolution from randomly perturbed initial conditions in periodic two-dimensional squares and three-dimensional cubes. We incorporate an adaptive time-stepping scheme in these numerical experiments. Our numerical study demonstrates that triangulation-based isogeometric analysis offers optimal convergence and time step stability, is applicable to complex geometry and allows local refinement.

Backward uniqueness for general parabolic operators in the whole space

We prove the backward uniqueness for general parabolic operators of second order in the whole space under assumptions that the leading coefficients of the operator are Lipschitz and their gradients satisfy certain decay conditions. The point is that the decay rate is related to the exponential growth rate of the solution, which is quite different from the case of the half-space (Wu and Zhang in Commun Contemp Math 18(1):1550011, 2016). This result extends in some ways a classical result of Lions and Malgrange (Math Scand 8:277–286, 1960 ) and a recent result of Wu and Zhang (2016).

The Dirichlet-to-Neumann operator on $C(\partial \Omega)$

Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous coefficients and a positive potential $V$. We show that the semigroup generated by $-D_V$ leaves $C(\Gamma)$ invariant and that the restriction of this semigroup to $C(\Gamma)$ is a $C_0$-semigroup. We investigate positivity and spectral properties of this semigroup. We also present results where $V$ is allowed to be negative. Of independent interest is a new criterium for semigroups to have a continuous kernel.

Solutions of the Al-Salam–Chihara and allied moment problems

We study the moment problem associated with the Al-Salam–Chihara polynomials in some detail providing raising (creation) and lowering (annihilation) operators, Rodrigues formula, and a second-order operator equation involving the Askey–Wilson operator. A new infinite family of weight functions is also given. Sufficient conditions for functions to be weight functions for the [Formula: see text]-Hermite, [Formula: see text]-Laguerre and Stieltjes–Wigert polynomials are established and used to give new infinite families of absolutely continuous orthogonality measures for each of these polynomials.

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

The Heun–Askey–Wilson algebra is introduced through generators $$\{{\textsf {X}},{\textsf {W}}\}$$ and relations. These relations can be understood as an extension of the usual Askey–Wilson ones. A central element is given, and a canonical form of the Heun–Askey–Wilson algebra is presented. A homomorphism from the Heun–Askey–Wilson algebra to the Askey–Wilson one is identified. On the vector space of the polynomials in the variable $$x=z+z^{-1}$$ , the Heun operator of Askey–Wilson type realizing $${\textsf {W}}$$ can be characterized as the most general second-order q-difference operator in the variable z that maps polynomials of degree n in $$x=z+z^{-1}$$ into polynomials of degree $$n+1$$ .

Principal functions of impulsive difference operators on semi axis

In this paper, we investigate the continuous spectrum and resolvent operator of a second-order difference operator with an impulsive condition. Then, under certain conditions, we prove finiteness of eigenvalues, spectral singularities. At last, we present principal functions of corresponding impulsive operator.

A Sobolev space theory for stochastic partial differential equations with time-fractional derivatives

In this article, we present an $L_{p}$-theory ($p\geq 2$) for the semi-linear stochastic partial differential equations (SPDEs) of type \begin{equation*}\partial^{\alpha }_{t}u=L(\omega ,t,x)u+f(u)+\partial^{\beta }_{t}\sum_{k=1}^{\infty }\int^{t}_{0}(\Lambda^{k}(\omega,t,x)u+g^{k}(u))\,dw^{k}_{t},\end{equation*} where $\alpha \in (0,2)$, $\beta <\alpha +\frac{1}{2}$ and $\partial^{\alpha }_{t}$ and $\partial^{\beta }_{t}$ denote the Caputo derivatives of order $\alpha $ and $\beta $, respectively. The processes $w^{k}_{t}$, $k\in \mathbb{N}=\{1,2,\ldots \}$, are independent one-dimensional Wiener processes, $L$ is either divergence or nondivergence-type second-order operator, and $\Lambda^{k}$ are linear operators of order up to two. This class of SPDEs can be used to describe random effects on transport of particles in medium with thermal memory or particles subject to sticking and trapping. We prove uniqueness and existence results of strong solutions in appropriate Sobolev spaces, and obtain maximal $L_{p}$-regularity of the solutions. By converting SPDEs driven by $d$-dimensional space–time white noise into the equations of above type, we also obtain an $L_{p}$-theory for SPDEs driven by space–time white noise if the space dimension $d<4-2(2\beta -1)\alpha^{-1}$. In particular, if $\beta <1/2+\alpha /4$ then we can handle space–time white noise driven SPDEs with space dimension $d=1,2,3$.

A PDE approach to fractional diffusion: a space-fractional wave equation

We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $$s \in (0,1)$$ , of symmetric, coercive, linear, elliptic, second-order operators in bounded domains $$\varOmega $$ . We realize fractional diffusion as the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on the semi-infinite cylinder $$\mathcal {C}= \varOmega \times (0,\infty )$$ . We thus rewrite our evolution problem as a quasi-stationary elliptic problem with a dynamic boundary condition and derive space, time, and space–time regularity estimates for its solution. The latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation. We propose and analyze two fully discrete schemes. The discretization in time is based on finite difference discretization techniques: the trapezoidal and leapfrog schemes. The discretization in space relies on the tensorization of a first-degree FEM in $$\varOmega $$ with a suitable hp-FEM in the extended variable. For both schemes we derive stability and error estimates. We consider a first-degree FEM in $$\varOmega $$ with mesh refinement near corners and the aforementioned hp-FEM in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open, bounded, polytopal but not necessarily convex domains $$\varOmega \subset {\mathbb {R}}^2$$ . We discuss implementation details and report several numerical examples.

A priori estimates in Sobolev spaces for a class of hyperbolic operators in presence of transition

We establish several a priori estimates of local or global nature in Sobolev spaces with general exponent [Formula: see text] for a class of second-order hyperbolic operators with double characteristics in presence of a transition in a domain of the Euclidian space [Formula: see text].

Factorization of Darboux—Laplace Transformations for Discrete Hyperbolic Operators

We classify elementary Darboux—Laplace transformations for semidiscrete and discrete second-order hyperbolic operators. We prove that there are two types of elementary Darboux—Laplace transformations in the (semi) discrete case as in the continuous case: Darboux transformations constructed from a particular element in the kernel of the initial hyperbolic operator and classical Laplace transformations that are defined by the operator itself and are independent of the choice of an element in the kernel. We prove that on the level of equivalence classes in the discrete case, any Darboux—Laplace transformation is a composition of elementary transformations.

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

We establish the rectifiability of measures satisfying a linear PDE constraint. The obtained rectifiability dimensions are optimal for many usual PDE operators, including all first-order systems and all second-order scalar operators. In particular, our general theorem provides a new proof of the rectifiability results for functions of bounded variations (BV) and functions of bounded deformation (BD). For divergence-free tensors we obtain refinements and new proofs of several known results on the rectifiability of varifolds and defect measures.