Differential Operators Of Arbitrary Order Research Articles

Higher Airy Structures, 𝒲 Algebras and Topological Recursion

We define higher quantum Airy structures as generalizations of the Kontsevich–Soibelman quantum Airy structures by allowing differential operators of arbitrary order (instead of only quadratic). We construct many classes of examples of higher quantum Airy structures as modules of W ( g ) \mathcal {W}(\mathfrak {g}) algebras at self-dual level, with g = g l N + 1 \mathfrak {g}= \mathfrak {gl}_{N+1} , s o 2 N \mathfrak {so}_{2 N } or e N \mathfrak {e}_N . We discuss their enumerative geometric meaning in the context of (open and closed) intersection theory of the moduli space of curves and its variants. Some of these W \mathcal {W} constraints have already appeared in the literature, but we find many new ones. For g l N + 1 \mathfrak {gl}_{N+1} our result hinges on the description of previously unnoticed Lie subalgebras of the algebra of modes. As a consequence, we obtain a simple characterization of the spectral curves (with arbitrary ramification) for which the Bouchard–Eynard topological recursion gives symmetric ω g , n \omega _{g,n} s and is thus well defined. For all such cases, we show that the topological recursion is equivalent to W ( g l ) \mathcal {W}(\mathfrak {gl}) constraints realized as higher quantum Airy structures, and obtain a Givental-like decomposition for the corresponding partition functions.

Continuity of families of Calderón projections

We consider a continuous family of linear elliptic differential operators of arbitrary order over a smooth compact manifold with boundary. Assuming constant dimension of the spaces of inner solutions, we prove that the orthogonalized Calderón projections of the underlying family of elliptic operators form a continuous family of projections. Hence, its images (the Cauchy data spaces) form a continuous family of closed subspaces in the relevant Sobolev spaces. We use only elementary tools and classical results: basic manipulations of operator graphs and other closed subspaces in Banach spaces, elliptic regularity, Green's formula and trace theorems for Sobolev spaces, well-posed boundary conditions, duality of spaces and operators in Hilbert space, and the interpolation theorem for operators in Sobolev spaces.

Theoretical Analysis and Simulation of a Fractional-Order Compartmental Model with Time Delay for the Propagation of Leprosy

This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the fractional-order epidemic model. The stability of the continuous model at different points of equilibria is investigated. The basic reproduction number, R0, is obtained for the leprosy model. It is observed that the leprosy system is locally asymptotically stable at both steady states when R0<1. On the other hand, the fractional-order system is globally asymptotically stable when R0>1. To find the approximate solutions for the continuous epidemic model, a non-standard numerical scheme is constructed. The main features of the non-standard scheme (such as positivity and boundedness of the numerical method) are also confirmed by applying some benchmark results. Simulations and a feasible test example are presented to discern the properties of the numerical method. Our computational results confirm both the analytical and the numerical properties of the finite-difference scheme.

On Symmetry and Uniqueness of Ground States for Linear and Nonlinear Elliptic PDEs

We study ground state solutions for linear and nonlinear elliptic PDEs in $\mathbb{R}^n$ with (pseudo-)differential operators of arbitrary order. We prove a general symmetry result in the nonlinear case as well as a uniqueness result for ground states in the linear case. In particular, we can deal with problems (e.g., higher order PDEs) that cannot be tackled by usual methods such as maximum principles, moving planes, or Polya--Szegö inequalities. Instead, we use arguments based on the Fourier transform and we apply a rigidity result for the Hardy--Littlewood majorant problem in $\mathbb{R}^n$ recently obtained by the last two authors of the present paper.

Integral representations in the complex plane and iterated boundary value problems

Fundamental solutions to differential operators lead to integral operators providing integral representation formulas for solutions to related differential equations. Proper modifications of the fundamental solutions result in integral operators which are related to certain boundary value problems. For complex partial differential operators of arbitrary order in the plane, fundamental solutions are achievable by properly integrating the Cauchy kernel. Particular such complex model differential operators are the polyanalytic and the polyharmonic operators. A hierarchy of integral operators is available for these model operators leading to polyanalytic Cauchy–Schwarz and to polyharmonic Green, Neumann, Robin, and hybrid Green integral operators. The theory is supplemented here by constructing a new polyanalytic Pompeiu (Pompeiu–Vekua) integral operator of any order adjusted to (iterated) Neumann boundary conditions.

Essential spectra of singular Hamiltonian differential operators of arbitrary order under a class of perturbations

AbstractThe main object of this paper is to study the essential spectrum of a Hamiltonian system of arbitrary order with one singular endpoint under a class of perturbations. We first present a characterization of the essential spectrum in terms of singular sequences and then give the concept of perturbations small at singular endpoints of Hamiltonian systems. Based on the above characterization, the invariance of essential spectra of Hamiltonian systems under these perturbations is shown. It is noted that these perturbations are given by using the associated pre‐minimal operator , which provides great convenience in the study of essential spectra of Hamiltonian systems since each element of the domain of has compact support. As applications, some sufficient conditions for the invariance of essential spectra of some systems are obtained in terms of coefficients of systems and perturbations terms. Further, essential spectra of Hamiltonian systems with different weight functions are discussed. Here, Hamiltonian systems may be non‐symmetric.

Higher-order imperfect interface modeling via complex variables based asymptotic analysis

We present a new methodology to derive imperfect interface models for the problems with interphase layers. The test case is potential problems, e.g., thermal conductivity, antiplane elasticity, etc. The methodology combines classical asymptotic analysis with concepts from the theory of complex-valued functions. Its major advantage over existing asymptotic approaches is the straightforward derivation of jump conditions that involve surface differential operators of arbitrary order, resulting in a hierarchy of models that maintain arbitrary-order accuracy with respect to the layer thickness and its curvature. Unlike low-order models, the derived higher-order models can accurately represent layers that are significantly softer or stiffer than the adjacent bulk materials, exhibit varying curvature, or are of finite thickness with respect to the characteristic length scale of the adjacent bulk regions. The interface models obtained via our methodology are compared with existing models of different orders, their limiting behavior is validated with respect to known interface regimes, and the improved accuracy of higher-order variants is illustrated for a benchmark example. While here we limited ourselves to scalar problems in two dimensions, the extension to vector problems in two-dimensions is straightforward. We also discuss the pathway to extend our methodology to scalar and vector problems in three-dimensions.

Continuity Points Via Riesz Potentials for ℂ-Elliptic Operators

Abstract We establish a Riesz potential criterion for Lebesgue continuity points of functions of bounded $\mathbb{A}$-variation, where $\mathbb{A}$ is a $\mathbb{C}$-elliptic differential operator of arbitrary order. This result generalizes a potential criterion that is known for full gradients to the case where full gradient estimates are not available by virtue of Ornstein’s non-inequality.

On symmetry of traveling solitary waves for dispersion generalized NLS

We consider dispersion generalized nonlinear Schrödinger equations (NLS) of the form where denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers . The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein–Gordon and half-wave equations.

On a Caputo-type fractional derivative

Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

Spectral asymptotics for problems with integral constraints

The eigenvalue problem for differential operators of arbitrary order with integral constraints is considered. The asymptotics of the eigenvalues is obtained. The results are applied to finding the asymptotics of the probability of small deviations for some detrended processes of nth order.

Two-scale relations for B-L-splines with uniform knots

Analogs of scaling relations are constructed for basis exponential splines with uniform knots corresponding to a linear differential operator of arbitrary order with constant coefficients and real pairwise distinct roots of the characteristic polynomial; the construction does not employ techniques from harmonic analysis.

The CMV Bispectral Problem

A classical result due to Bochner classifies the orthogonal polynomials on the real line which are common eigenfunctions of a second order linear differential operator. We settle a natural version of the Bochner problem on the unit circle which answers a similar question concerning orthogonal Laurent polynomials and can be formulated as a bispectral problem involving CMV matrices. We solve this CMV bispectral problem in great generality proving that, except the Lebesgue measure, no other one on the unit circle yields a sequence of orthogonal Laurent polynomials which are eigenfunctions of a linear differential operator of arbitrary order. Actually, we prove that this is the case even if such an eigenfunction condition is imposed up to finitely many orthogonal Laurent polynomials.

Inverse spectral problems for differential operators on spatial networks

A short survey is given of results on inverse spectral problems for ordinary differential operators on spatial networks (geometrical graphs). The focus is on the most important non-linear inverse problems of recovering coefficients of differential equations from spectral characteristics when the structure of the graph is known a priori. The first half of the survey presents results related to inverse Sturm–Liouville problems on arbitrary compact graphs. Results on inverse problems for differential operators of arbitrary order on compact graphs are then presented. In the conclusion the main results on inverse problems on non-compact graphs are given. Bibliography: 55 titles.

Hadamard type operators on spaces of real analytic functions in several variables

We consider multipliers on the space of real analytic functions of several variables A ( Ω ) , Ω ⊂ R d open, i.e., linear continuous operators for which all monomials are eigenvectors. If zero belongs to Ω these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Euler differential operators of arbitrary order are multipliers. We represent all multipliers via a kind of multiplicative convolution with analytic functionals and characterize the corresponding sequences of eigenvalues as moments of suitable analytic functionals. Moreover, we represent multipliers via suitable holomorphic functions with Laurent coefficients equal to the eigenvalues of the operator. We identify in some standard cases what topology should be put on the suitable space of analytic functionals in order that the above mentioned isomorphism becomes a topological one when the space of multipliers inherits the topology of uniform convergence on bounded sets from the space of all endomorphisms on A ( Ω ) . We also characterize in the same cases when the discovered topology coincides with the classical topology of bounded convergence on the space of analytic functionals. We provide several examples of multipliers and show surjectivity results for multipliers on A ( Ω ) if Ω ⊂ R + d .

A family of hyper-Bessel functions and convergent series in them

Abstract The Delerue hyper-Bessel functions that appeared as a multi-index generalizations of the Bessel function of the first type, are closely related to the hyper-Bessel differential operators of arbitrary order, introduced by Dimovski. In this work we consider an enumerable family of hyper-Bessel functions and study the convergence of series in such a kind of functions. The obtained results are analogues to the ones in the classical theory of the widely used power series, like Cauchy-Hadamard, Abel and Fatou theorem.

Polynomial supersymmetry for matrix Hamiltonians

We study intertwining relations for matrix one-dimensional, in general, non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any matrix intertwining operator Q_N^- of minimal order N there is a matrix operator Q_{N'}^+ of different, in general, order N' that intertwines the same Hamiltonians as Q_N^- in the opposite direction and such that the products Q_{N'}^+Q_N^- and Q_N^-Q_{N'}^+ are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and of reducibility of a matrix intertwining operator are considered and the criteria of minimizability and of reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.

Analog of the Riesz theorem and the basis property in L p of a system of root functions of a differential operator: I

An ordinary differential operator of arbitrary order is considered. We find necessary conditions for the Riesz property of systems of normalized root functions, prove an analog of the Riesz theorem, and use it to obtain sufficient conditions for the basis property of a system of root functions of this operator in Lp.

Riemann-Liouville and Caputo type multiple Erdélyi-Kober operators

AbstractIn this paper some generalized operators of Fractional Calculus (FC) are investigated that are useful in modeling various phenomena and systems in the natural and human sciences, including physics, engineering, chemistry, control theory, etc., by means of fractional order (FO) differential equations. We start, as a background, with an overview of the Riemann-Liouville and Caputo derivatives and the Erdélyi-Kober operators. Then the multiple Erdélyi-Kober fractional integrals and derivatives of R-L type of multi-order (δ 1,…,δ m) are introduced as their generalizations. Further, we define and investigate in detail the Caputotype multiple Erdélyi-Kober derivatives. Several examples and both known and new applications of the FC operators introduced in this paper are discussed. In particular, the hyper-Bessel differential operators of arbitrary order m > 1 are shown as their cases of integer multi-order. The role of the so-called special functions of FC is emphasized both as kernel-functions and solutions of related FO differential equations.

Refinement of Two-Factor Factorizations of a Linear Partial Differential Operator of Arbitrary Order and Dimension

Given a right factor and a left factor of a Linear Partial Differential Operator (LPDO), under which conditions we can refine these two-factor factorizations into one three-factor factorization? This problem is solved for LPDOs of arbitrary order and number of variables. A more general result for the incomplete factorizations of LPDOs is proved as well.