Higher-order Differential Operators Research Articles

SPECTRAL DATA ASYMPTOTICS FOR THE HIGHER-ORDER DIFFERENTIAL OPERATORS WITH DISTRIBUTION COEFFICIENTS

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the case when, for the two boundary value problems, some coefficients of the differential expressions and of the boundary conditions coincide. We estimate the difference of their spectral data in this case. Although the asymptotic behaviour of spectral data is well-studied for differential operators with regular (integrable) coefficients, to the best of the author’s knowledge, there were no results in this direction for the higher-order differential operators with distribution coefficients (generalized functions) in a general form. The technique of this paper relies on the recently obtained regularization and the Birkhoff-type solutions for differential operators with distribution coefficients. Our results have applications to the theory of inverse spectral problems as well as a separate significance.

Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators

Continuous/Discontinuous Galerkin Difference Discretizations of High-Order Differential Operators

Uniqueness Criteria for Solving a Time Nonlocal Problem for a High-Order Differential Operator Equation l(·)—A with a Wave Operator with Displacement

This article presents a criterion for the uniqueness of the solution of a problem nonlocal in time for a differential-operator equation with a symmetric operator part on space variables. The symmetry of the operator part of the operator-differential equation guarantees the existence of good basic properties of its system of root elements. The spectral properties of the symmetric operator part make it possible not only to prove the necessity of the criterion formulated by us, but also to substantiate their sufficiency. In contrast to previously known works, in this work the semiboundedness of the symmetric part of the differential-operator equation can be violated. In this article, the differential-operator equation is represented as the difference of two commuting operators. The uniqueness of the solution is guaranteed when the spectra of the commuting operators do not intersect. It is important that only one of the operators should be symmetrical.

Bispectral Jacobi type polynomials

We study the bispectrality of Jacobi type polynomials, which are eigenfunctions of higher-order differential operators and can be defined by taking suitable linear combinations of a fixed number of consecutive Jacobi polynomials. Jacobi type polynomials include, as particular cases, the Krall-Jacobi polynomials. As the main results we prove that the Jacobi type polynomials always satisfy higher-order recurrence relations (i.e., they are bispectral). We also prove that the Krall-Jacobi families are the only Jacobi type polynomials which are orthogonal with respect to a measure on the real line.

Notes on Higher-Spin Diffeomorphisms

Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and interactions. Several yes-go and no-go results on higher-spin diffeomorphisms are collected from the mathematical literature in order to propose a generalisation of the algebra of differential operators on which higher-spin diffeomorphisms are well-defined. This work is dedicated to the memory of Christiane Schomblond, who taught several generations of Belgian physicists the formative rigor and delicate beauty of theoretical physics.

Inverse Spectral Problems for Arbitrary-Order Differential Operators with Distribution Coefficients

In this paper, we propose an approach to inverse spectral problems for the n-th order (n≥2) ordinary differential operators with distribution coefficients. The inverse problems which consist in the reconstruction of the differential expression coefficients by the Weyl matrix and by several spectra are studied. We prove the uniqueness of solution for these inverse problems, by developing the method of spectral mappings. The results of this paper generalize the previously known results for the second-order differential operators with singular potentials and for the higher-order differential operators with regular coefficients. In the future, the approach of this paper can be used for constructive solution and for investigation of solvability of the considered inverse problems.

An infinite family of higher-order difference operators that commute with Ruijsenaars operators of type A

We introduce a new infinite family of higher-order difference operators that commute with the elliptic Ruijsenaars difference operators of type A. These operators are related to Ruijsenaars’ operators through a formula of Wronski type.

DNT preconditioner for one-sided space fractional diffusion equations

In this paper, we consider diagonal-times-nonsymmetric-Toeplitz (DNT) preconditioner for Toeplitz-like linear systems arising from one-sided space fractional diffusion equations (OSFDE) with variable coefficients. We first correct the result of Lemma 3.3 in Lin et al. (BIT Numer Math 58, 729–748, 2018) and prove the corrected result. We then extend the DNT preconditioner to Toeplitz-like linear systems arising from OSFDEs, where high order difference operators are applied to discretize the fractional derivative. Theoretically, we prove that the condition number of the preconditioned matrix is uniformly bounded by a constant under mild assumptions and verify that several discretization schemes from the literature satisfy the required assumptions. Numerical results are reported to show the efficiency of the DNT preconditioner.

Isogeometric analysis for a phase-field constrained optimization problem of morphological evolution of vesicles in electrical fields

In this paper, we solve a constrained Willmore problem coupled with an electrical field using IsoGeometric Analysis (IGA) to simulate the morphological evolution of vesicles subjected to static electrical fields. The model consist of two phases, the lipid bilayer and the electrolyte. The two-phases problem is modeled using the phase-field method, a subclass of the diffusive interface models. The bending, flexoelectric and dielectric energies of the model are reformulated using the phase-field parameter. A modified Augmented-Lagrangian approach was used to satisfy the constraints while maintaining numerical stability and a relatively large time step. This approach guarantees the satisfaction of the constraints at each time step over the entire temporal domain. The results show the superiority of the isogeometric analysis in solving high-order differential operators without the need for additional intermediate equations to account for classical mesh-based methods limited continuity. On the physical side, the morphological evolution of the vesicles can be simulated accurately using IGA, even when considering the flexoelectric response of the biomembrane, which adds another layer of numerical complexity to the system. The effect of the flexoelectricity, the conductivity ratio and other aspects of the problem are studied through several 3D numerical examples.

Power shaping control of DC–DC converters with constant power loads

This paper deals with the stabilization of DC–DC converters feeding constant power loads (CPLs). It is shown that power/energy dynamics are easily computed via the image of the input/output voltages/currents of the converter. Such image is represented by a high-order differential operator, obtained from first principles, acting on the variables of a stabilizing interconnected controller. This approach is based upon the premise that the destabilizing effect of CPLs, is caused due to a power imbalance in a dissipation equality, which prevents passivity. The proposed stabilizing mechanism shapes the power dynamics via an interconnected controller, inducing a feasible power flow and thus passivity. To do so, we developed a stability test, as well as control design tools in terms of linear matrix inequalities (LMIs), constructed from coefficient matrices of the system dynamics. As a practical advantage, the proposed power shaping approach permits to link stability conditions with the nominal power rate specification of power converters. Finally, we show that the proposed framework can also accommodate other traditional frequency domain, eigenvalue and immittance criteria. The main results are validated via a theoretical analysis and experimental results.

Spectral Analysis of Higher-Order Differential Operators with Discontinuity Conditions at Interior Points

Higher-order differential operators on a finite interval with discontinuity conditions inside the interval are studied. Properties of spectral characteristics are obtained, and completeness and expansion theorems are proved for that class of operators.

On the spectral properties of a class of high-order differential operators with operator coefficients

On the spectral properties of a class of high-order differential operators with operator coefficients

Higher-order nonlocal theory of Updated Lagrangian Particle Hydrodynamics (ULPH) and simulations of multiphase flows

In this work, we present a higher-order nonlocal continuum theory of a recently developed updated Lagrangian particle hydrodynamics (ULPH) (see: Tu and Li, 2017 and Yan et al., 2019) and its applications to multiphase flows. The original nonlocal differential operators used in ULPH have only the first-order accuracy, and they may not be able to simulate some complicated and delicate three-dimensional flow problems. In order to improve the computational accuracy and stability, we adopt the nonlocal synchronized differential operator from the reproducing kernel particle method (RKPM), which is regarded as the higher-order nonlocal differential operator, and we apply them to build higher-order ULPH formulations. The main advantage of the higher-order nonlocal differential operator over the original one is its high accuracy in unstructured particle distribution, and it is fully controlled by the approximation polynomial basis.Numerical verifications have been carried out to validate the accuracy of the proposed approach by using various polynomial bases. Several challenging and delicate three-dimensional multiphase flow benchmark problems are solved to demonstrate the capability of the proposed method. The numerical results of the higher-order ULPH show good agreement with theoretical and numerical solutions in the literature, demonstrating the promising potential of higher-order nonlocal ULPH formulation in modeling and simulating multiphase flows with substantial topological variations of interfaces and high density ratios.

High order difference schemes using the local anisotropic basis function method

Mesh-free methods have significant potential for simulations in complex geometries, as the time consuming process of mesh-generation is avoided. Smoothed Particle Hydrodynamics (SPH) is the most widely used mesh-free method, but suffers from a lack of consistency. High order, consistent, and local (using compact computational stencils) mesh-free methods are particularly desirable.Here we present a novel framework for generating local high order difference operators for arbitrary node distributions, referred to as the Local Anisotropic Basis Function Method (LABFM). Weights are constructed from linear sums of anisotropic basis functions (ABFs), chosen to ensure exact reproduction of polynomial fields up to a given order. The ABFs are based on a fundamental Radial Basis Function (RBF), and the choice of fundamental RBF has small effect on accuracy, but influences stability. LABFM is able to generate high order difference operators with compact computational stencils (4th order with N≈25 nodes, 8th order with N≈60 nodes in two dimensions). At domain boundaries (with incomplete support) LABFM automatically provides one-sided differences of the same order as the internal scheme, up to 4th order. We use the method to solve elliptic, parabolic and mixed hyperbolic-parabolic partial differential equations (PDEs), showing up to 8th order convergence. The inclusion of hyperviscosity is straightforward, and can effectively provide stability when solving hyperbolic problems.LABFM is a promising new mesh-free method for the numerical solution of PDEs in complex geometries. The method is highly scalable, and for Eulerian schemes, the computational efficiency is competitive with RBF-FD for a given accuracy. A particularly attractive feature is that in the low order limit, LABFM collapses to Smoothed Particle Hydrodynamics (SPH), and there is potential for Arbitrary Lagrangian-Eulerian schemes with natural adaptivity of resolution and accuracy.

Sufficient conditions for regular solvability of an arbitrary order operator-differential equation with initial-boundary conditions

On this paper, for an arbitrary order operator-differential equation with the weight e^{frac{-alpha t}{2}}, alpha in (-infty ,+ infty ), in the space W^{n+m}_{2}(R_{+};H), we attain sufficient conditions for the well-posedness of a regular solvable of the boundary value problem. These conditions are provided only by the operator coefficients of the investigated equation where the leading part of the equation has multiple characteristics. We prove the connection between the lower bound of the spectrum of the higher-order differential operator in the main part and the exponential weight and also obtain estimations of the norms of operator intermediate derivatives. We apply the results of this paper to a mixed problem for higher-order partial differential equations (HOPDs).

Uniform, on the Real Line, Equiconvergence of Spectral Expansions for the Higher-Order Differential Operators

Result on the uniform, over the entire real line, equiconvergence of spectral expansions related to the self-adjoint extension of a general differential operation of any even order with coefficients from the one-dimensional Kato class, with the Fourier integral expansion is presented. The statement is based on the obtained uniform estimates for the spectral function of this operator.

BCJ numerators from differential operator of multidimensional residue

In previous works, we devised a differential operator for evaluating typical integrals appearing in the Cachazo–He–Yuan (CHY) forms and in this paper we further streamline this method. We observe that at tree level, the number of free parameters controlling the differential operator depends solely on the number of external lines, after solving the constraints arising from the scattering equations. This allows us to construct a reduction matrix that relates the parameters of a higher-order differential operator to those of a lower-order one. The reduction matrix is theory-independent and can be obtained by solving a set of explicitly given linear conditions. The repeated application of such reduction matrices eventually transforms a given tree-level CHY-like integral to a prepared form. We also provide analytic expressions for the parameters associated with any such prepared form at tree level. We finally give a compact expression for the multidimensional residue for any CHY-like integral in terms of the reduction matrices. We adopt a dual basis projector which leads to the CHY-like representation for the non-local Bern–Carrasco–Johansson (BCJ) numerators at tree level in Yang–Mills theory. These BCJ numerators are efficiently computed by the improved method involving the reduction matrix.

Heat kernel for higher-order differential operators and generalized exponential functions

We consider the heat kernel for higher-derivative and nonlocal operators in $d$-dimensional Euclidean space-time and its asymptotic behavior. As a building block for operators of such type, we consider the heat kernel of the minimal operator - generic power of the Laplacian - and show that it is given by the expression essentially different from the conventional exponential Wentzel-Kramers-Brillouin (WKB) ansatz. Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox-Wright $\varPsi$-functions and Fox $H$-functions. The structure of its essential singularity in the proper time parameter is different from that of the usual exponential ansatz, which invalidated previous attempts to directly generalize the Schwinger-DeWitt heat kernel technique to higher-derivative operators. In particular, contrary to the conventional exponential decay of the heat kernel in space, we show the oscillatory behavior of GEF for higher-derivative operators. We give several integral representations for the generalized exponential function, find its asymptotics and semiclassical expansion, which turns out to be essentially different for local operators and nonlocal operators of noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.

Trace Formula for a High-Order Differential Operator on an Interval under a Perturbation of the Lower-Order Term by a Finite Charge

In this paper, a regularized trace formula is obtained for a higher-order differential operator on an interval under a perturbation of the lower-order term by a finite charge. Arbitrary regular boundary conditions and an arbitrary order n ⩾ 3 of the operator are considered. A new effect is discovered: for an even order of the operator, an additional term appears depending on the jump of the charge distribution function at the midpoint of the interval.

Bispectrality of Charlier type polynomials

ABSTRACTGiven a finite set of positive integers G and polynomials , , with degree of equal to g, we associate to them a sequence of Charlier type polynomials defined from the Charlier polynomials by using certain Casoratian determinants (whose entries are the polynomials ). Charlier type polynomials are eigenfunctions of higher order difference operators. When , the polynomials are also orthogonal, and then satisfy a three term recurrente relation. In this paper, we prove that whatever the polynomials 's are, the Charlier type polynomials always satisfy higher order recurrence relations. We also introduce and characterize the algebra of difference operators associated to the recurrence relations satisfied by the sequence . Our characterization is constructive and surprisingly simple. As a consequence, we prove that is, essentially, the unique choice such that the polynomials are orthogonal with respect to a measure.