Second-order Linear Differential Operator Research Articles

An optimally convergent Fictitious Domain method for interface problems

We introduce a novel Fictitious Domain (FD) unfitted method for interface problems associated with a second-order elliptic linear differential operator, that achieves optimal convergence without the need for adaptive mesh refinements nor enrichments of the Finite Element spaces. The key aspect of the proposed method is that it extends the solution into the fictitious domain in a way that ensures high global regularity. Continuity of the solution across the interface is enforced through a boundary Lagrange multiplier. The subdomains coupling, however, is not achieved by means of the duality pairing with the Lagrange multiplier, but through an L2 product with the H1 Riesz representative of the latter, thus avoiding gradient jumps across the interface. Thanks to the enhanced regularity, the proposed method attains an increase, with respect to standard FD methods, of up to one order of convergence in energy norm. The Finite Element formulation of the method is presented, followed by its analysis. Numerical tests on a model problem demonstrate its effectiveness and its superior accuracy compared to standard unfitted methods.

Best Constant in Ulam Stability for the Third Order Linear Differential Operator with Constant Coefficients

The authors of the present paper previously proved the Ulam stability for the n-th-order linear differential operator with constant coefficients. They obtained its best Ulam constant for the case of distinct roots of the characteristic equation. However, a complete answer to the problem of the best Ulam constant was later obtained only for the second-order linear differential operator. This paper deals with the Ulam stability of the third-order linear differential operator with constant coefficients acting in a Banach space. The paper’s main purpose is to obtain the best Ulam constant of this operator, thus completing the previous research in the field.

Financial Interpretation of Feller’s Factorization

The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.

Classical orthogonal polynomials via a second-order linear differential operators

Let Tc := D(x - c)((x - c)D + 2II) be a second-order linear differential operator, where c is an arbitrary complex number, $$D: = \frac{d}{{dx}}$$ and II represents the identity on the linear space of polynomials with complex coefficients. The aim of this paper is to describe all of the Tc-classical orthogonal polynomials. Two canonical situations appear: the Laguerre $$\{L_n^{(2)}\}_{n\geq0}$$ and the Jacobi $$\{P_n^{(\alpha-2,2)}\}_{n\geq0}$$

Bounds for the extremal parameter of nonlinear eigenvalue problems and applications

We consider the nonlinear eigenvalue problem L u = λ f ( u ), posed in a smooth bounded domain Ω ⊆ R N with Dirichlet boundary condition, where L is a uniformly elliptic second-order linear differential operator, λ > 0 and f : [ 0 , a f ) → R + ( 0 < a f ⩽ ∞) is a smooth, increasing and convex nonlinearity such that f ( 0 ) > 0 and which blows up at a f . First we present some upper and lower bounds for the extremal parameter λ ∗ and the extremal solution u ∗ . Then we apply the results to the operator L A = − Δ + A c ( x ) with A > 0 and c ( x ) is a divergence-free flow in Ω. We show that, if ψ A , Ω is the maximum of the solution ψ A ( x ) of the equation L A u = 1 in Ω with Dirichlet boundary condition, then for any incompressible flow c ( x ) we have, ψ A , Ω ⟶ 0 as A ⟶ ∞ if and only if c ( x ) has no non-zero first integrals in H 0 1 ( Ω ). Also, taking c ( x ) = − x ρ ( | x | ) where ρ is a smooth real function on [ 0 , 1 ] then c ( x ) is never divergence-free in unit ball B ⊂ R N , but our results completely determine the behaviour of the extremal parameter λ A ∗ as A ⟶ ∞.

Optimization of Mayer Problem with Sturm–Liouville-Type Differential Inclusions

The present paper studies a new class of problems of optimal control theory with Sturm–Liouville-type differential inclusions involving second-order linear self-adjoint differential operators. Our main goal is to derive the optimality conditions of Mayer problem for differential inclusions with initial point constraints. By using the discretization method guaranteeing transition to continuous problem, the discrete and discrete-approximation inclusions are investigated. Necessary and sufficient conditions, containing both the Euler–Lagrange and Hamiltonian-type inclusions and “transversality” conditions are derived. The idea for obtaining optimality conditions of Mayer problem is based on applying locally adjoint mappings. This approach provides several important equivalence results concerning locally adjoint mappings to Sturm–Liouville-type set-valued mappings. The result strengthens and generalizes to the problem with a second-order non-self-adjoint differential operator; a suitable choice of coefficients then transforms this operator to the desired Sturm–Liouville-type problem. In particular, if a positive-valued, scalar function specific to Sturm–Liouville differential inclusions is identically equal to one, we have immediately the optimality conditions for the second-order discrete and differential inclusions. Furthermore, practical applications of these results are demonstrated by optimization of some “linear” optimal control problems for which the Weierstrass–Pontryagin maximum condition is obtained.

The exceptional Bessel polynomials

Gomez-Ullate, Kamran and Milson have found polynomial eigenfunctions of a Sturm–Liouville problem. These polynomials, denoted by X1-Laguerre and X1-Jacobi and starting with degree one, are eigenfunctions of a second-order linear differential operator. In this paper, we investigate the X1-Bessel case which we denote by . We wrote these polynomials as explicit functions of n, decompose it for the basis (x−b)2xi, and we expand in terms of Bessel orthogonal polynomials , using generalized Carlitz formula. Finally, we give a non-hermitian orthogonality satisfied by these polynomials.

Type B 3-fold supersymmetry and non-polynomial invariant subspaces

We obtain the most general type B 3-fold supersymmetry by solving directly the intertwining relation. We then show that it is a necessary and sufficient condition for a second-order linear differential operator to have three linearly independent local analytic solutions. We find that there are eight linearly independent non-trivial linear differential operators of this kind. As a by-product, we find new quasi-solvable second-order operators preserving a monomial or polynomial subspace, one in type B, two in type C, and four in type X2, all of which have been missed in the existing literature. In addition, we show that type A, type B, and type C 3-fold supersymmetries are connected continuously via one parameter. A few new quasi-solvable models are also presented.

Chan–Chyan–Srivastava multivariable polynomials associated with a certain family of partial differential operators

In some recent investigations involving certain differential operators for the Chan–Chyan–Srivastava multivariable polynomials, Chan et al. encountered and proved a certain summation identity for the Chan–Chyan–Srivastava multivariable polynomials. The main object of this paper is to present a second-order linear partial differential operator L φ associated with the Chan–Chyan–Srivastava multivariable polynomials. We show that the Chan–Chyan–Srivastava multivariable polynomials are eigenfunctions of L φ. In addition, we find all eigenfunctions of L φ in the space of polynomials.

Cosmological perturbation theory revisited

Increasingly accurate observations are driving theoretical cosmology towards the use of more sophisticated descriptions of matter and the study of nonlinear perturbations of Friedmann–Lemaitre cosmologies, whose governing equations are notoriously complicated. Our goal in this paper is to formulate the governing equations for linear perturbation theory in a particularly simple and concise form in order to facilitate the extension to nonlinear perturbations. Our approach has several novel features. We show that the use of so-called intrinsic gauge invariants has two advantages. It naturally leads to (i) a physically motivated choice of a gauge invariant associated with the matter density, and (ii) two distinct and complementary ways of formulating the evolution equations for scalar perturbations, associated with the work of Bardeen and of Kodama and Sasaki. In the first case, the perturbed Einstein tensor gives rise to a second-order (in time) linear differential operator, and in the second case to a pair of coupled first-order (in time) linear differential operators. These operators are of fundamental importance in cosmological perturbation theory, since they provide the leading order terms in the governing equations for nonlinear perturbations.

Galoisian approach to integrability of Schrödinger equation

In this paper, we examine the nonrelativistic stationary Schrodinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second-order ordinary linear differential operators, so as to achieve rational function coefficients (“algebrization”), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schrodinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible.

Position dependent mass Schrödinger equation and isospectral potentials: Intertwining operator approach

Here, we have studied first- and second-order intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schrödinger equation. The second-order intertwiner is constructed directly by taking it as second-order linear differential operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A N-fold supersymmetry [T. Tanaka, J. Phys. A 39, 219 (2006); A. Gonzalez-Lopez and T. Tanaka, J. Phys. A 39, 3715 (2006)] for the particular cases N=1 and N=2, respectively.

Isospectral Euler–Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler–Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler–Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge–Kutta method.

Fuchs versus Painlevé

We, briefly, recall the Fuchs–Painlevé elliptic representation of Painlevé VI. We then show that the polynomiality of the expressions of the correlation functions (and form factors) in terms of the complete elliptic integral of the first and second kinds, K and E, is a straight consequence of the fact that the differential operators corresponding to the entries of Toeplitz-like determinants are equivalent to the second-order operator LE which has E as solution (or for off-diagonal correlations to the direct sum of LE and d/dt). We show that this can be generalized, mutatis mutandis, to the anisotropic Ising model. The singled-out second-order linear differential operator LE is replaced by an isomonodromic system of two third-order linear partial differential operators associated with Π1, the Jacobi's form of the complete elliptic integral of the third kind (or equivalently two second-order linear partial differential operators associated with Appell functions, where one of these operators can be seen as a deformation of LE). We finally explore the generalizations, to the anisotropic Ising models, of the links we made, in two previous papers, between Painlevé nonlinear ODEs, Fuchsian linear ODEs and elliptic curves. In particular, the elliptic representation of Painlevé VI has to be generalized to an ‘Appellian’ representation of Garnier systems.

Holonomy of the Ising model form factors

We study the Ising model two-point diagonal correlation function C(N, N) by presenting an exponential and form factor expansion in an integral representation which differs from the known expansion of Wu, McCoy, Tracy and Barouch. We extend this expansion, weighting, by powers of a variable λ, the j-particle contributions, f(j)N,N, in the form factor expansion. The corresponding λ-extension of the two-point diagonal correlation function, C(N, N; λ), is shown, for arbitrary λ, to be a solution of the sigma form of the Painlevé VI equation introduced by Jimbo and Miwa in their isomonodromic approach to the Ising model. Fuchsian linear differential equations for the form factors f(j)N,N are obtained for j ⩽ 9 and shown to have both a ‘Russian-doll’ nesting and a decomposition of the corresponding linear differential operators as a direct sum of operators equivalent to symmetric powers of the second-order linear differential operator associated with the elliptic integral E. From this, we show that each f(j)N,N is unexpectedly simple, being expressed polynomially in terms of the elliptic integrals E and K. In contrast, we exhibit some mathematical objects, built from these form factors f(j)N,N, which break the direct sum of symmetric powers decomposition, with its associated polynomial expressions. First we show that the scaling limit of these differential operators, and form factors, breaks the direct sum structure but not the ‘Russian-doll’ structure. Secondly, we show that the previous λ-extension of two-point diagonal correlation functions, C(N, N; λ), is, for singled-out values λ = cos(πm/n), (m, n integers), also solutions of Fuchsian linear differential equations. These solutions of Painlevé VI are not polynomial in E and K but are actually algebraic functions, being associated with modular curves.

Generation type inequalities for closed linear operators related to domains with conical points

Let be a second-order linear differential operator in divergence form. We prove that the operator , where λ ∈ C and I stands for the identity operator, is closed and injective when Re λ is large enough and the domain of consists of a special class of weighted Sobolev function spaces related to conical open bounded sets of R n , n ≥ 1.

A CHARACTERIZATION OF SECOND-ORDER DIFFERENTIAL OPERATORS ON FINITE ELEMENT SPACES

We describe all operators on scalar finite element spaces which appear as the restriction of a second-order (linear) differential operator. More precisely we provide a family of isomorphisms between this space of discrete differential operators and products of some exotic finite element spaces. This provides a unified framework for the Regge calculus of numerical relativity and the Nédélec edge elements of computational electromagnetism.

Normalized Solutions of Schrödinger Equations with Potentially Bounded Measures

Given a potentially bounded signed measure μ on a Brelot space (X,ℋ) with Green function G, it is well known that μ-harmonic functions (i.e., in the classical case, finely continuous versions of solutions to Δu−uμ=0) may be very discontinuous. In this paper it is shown that under very general assumptions on G (satisfied for large classes of elliptic second-order linear differential operators) normalized perturbation, however, leads to a Brelot space (X, $$\widetilde{\mathcal{H}}$$ μ) admitting a Green function T μ(G) which is locally (or even globally) comparable with G and has all properties required of G before. In particular, iterated perturbation is possible. Moreover, intrinsic Holder continuity of quotients of harmonic functions with respect to the local quasimetric ρ:=(G −1+* G −1)/2 yields ρ-Holder continuity for quotients of μ-harmonic functions as well.

Two-Dimensional Krall–Sheffer Polynomials andQuantum Systems on Spaces with Constant Curvature

Krall and Sheffer found in 1967 that there exists at most nine different types of two-dimensional orthogonal polynomials which are eigensolutions of a second-order linear differential operator with polynomial coefficients. We show that, for all these types, there correspond quantum mechanical systems on a Euclidean (pseudo-Eeuclidean) plane, two-dimensional sphere, or hyperboloid.

The asymptotic behavior of solutions to an inverse problem for differential operator equations

We show that, under some conditions, all solutions of the inverse problem for a class of first-order and second-order linear differential operator equations in a Hilbert space tend to zero when t → ∞. Applications to inverse problems for the heat equation and damped wave equations are given.