Order Linear Differential Operator Research Articles

Localization at hyperplane arrangements: Combinatorics and [formula omitted]-modules

We describe an algorithm deciding if the annihilating ideal of the meromorphic function 1 f , where f = 0 defines an arrangement of hyperplanes, is generated by linear differential operators of order 1. The algorithm is based on the comparison of two characteristic cycles and uses a combinatorial description of the characteristic cycle of the D -module of meromorphic functions with respect to f, due to Àlvarez Montaner, García López and Zarzuela.

Differential Equations for Jacobi-Pineiro Polynomials

For r ∈ ℕ0, we present a Fuchsian linear differential operator of order r + 1 with three singular points at 0,1, ∞. This operator annihilates the r-multiple Jacobi-Piñeiro polynomial.

Lagrange Interpolation and Boundary-Value Problems

It is known that some boundary-value problems give rise to Lagrange-type interpolation series that can be used to reconstruct entire functions from their samples at the eigenvalues of any such problem. Such boundary-value problems, of which regular Sturm Liouville boundary-value problems are prototype, are said to have the Lagrange-type interpolation property. It was conjectured that any regular, self-adjoint, eigenvalue problem associated with nth order linear differential operator with simple eigenvalues has the Lagrange-type interpolation property. In 1994, P.L. Butzer and G. Schottler proved that conjecture, but a year later Annaby pointed out an error in their paper and gave an alternative proof which is not only imprecise, but also deals with a very special case in which the problem is assumed to be one dimensional. The aim of this article is to fill the gaps in these papers by providing an alternative proof.

On real-analytic recurrence relations for cardinal exponential B-splines

Let L N + 1 be a linear differential operator of order N + 1 with constant coefficients and real eigenvalues λ 1 , … , λ N + 1 , let E ( Λ N + 1 ) be the space of all C ∞ -solutions of L N + 1 on the real line. We show that for N ⩾ 2 and n = 2 , … , N , there is a recurrence relation from suitable subspaces E n to E n + 1 involving real-analytic functions, and with E N + 1 = E ( Λ N + 1 ) if and only if contiguous eigenvalues are equally spaced.

On characterizations of classical polynomials

It is well known that the classical families of Jacobi, Laguerre, Hermite, and Bessel polynomials are characterized as eigenvectors of a second order linear differential operator with polynomial coefficients, Rodrigues formula, etc. In this paper we present a unified study of the classical discrete polynomials and q-polynomials of the q-Hahn tableau by using the difference calculus on linear-type lattices. We obtain in a straightforward way several characterization theorems for the classical discrete and q-polynomials of the “ q-Hahn tableau”. Finally, a detailed discussion of a characterization by Marcellán et al. is presented.

Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation.

Removable singularities of weak solutions to linear partial differential equations

Suppose that P(x, D) is a linear differential operator of order m > 0 with smooth coefficients whose derivatives up to order m are continuous functions in the domain G ⊂ ℝn (n ≥ 1), 1 ∞, s > 0, and q=p/(p − 1). In this paper, we show that if n, m, p, and s satisfy the two-sided bound 0 ≥ n − q(m − s)< n, then for a weak solution of the equation P(x, D)u=0 from the Sharpley-DeVore class C p s (G)loc, any closed set in G is removable if its Hausdorff measure of order n − q(m − s) is finite. This result strengthens the well-known result of Harvey and Polking on removable singularities of weak solutions to the equation P(x, D)u=0 from the Sobolev classes and extends it to the case of noninteger orders of smoothness.

Symmetries of Modules of Differential Operators

Let be the space of tensor densities of degree (or weight) λ on the circle S1. The space of k-th order linear differential operators from to is a natural module over Diff(S1), the diffeomorphism group of S 1. We determine the algebra of symmetries of the modules , i.e., the linear maps on commuting with the Diff(S 1)-action. We also solve the same problem in the case of straight line ℝ (instead of S 1) and compare the results.

The Hyers–Ulam stability constants of first order linear differential operators

Let X be a complex Banach space, h a complex-valued continuous function on the real line R and T h :C 1( R,X)→C( R,X) the linear differential operator defined by T h u= u′+ hu. We completely determine the Hyers–Ulam stability constant of T h .

Complete second order linear differential operator equations in Hilbert space and applications in hydrodynamics

We study the Cauchy problem for a complete second order linear differential operator equation in a Hilbert space H {\mathcal H} of the form \[ d 2 u d t 2 + ( F + i K ) d u d t + B u = f , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 . \frac {d^2u}{dt^2}+(F+\textrm {i}K)\frac {du}{dt}+Bu=f,\quad u(0)=u^0, \quad u’(0)=u^1. \] Problems of this kind arise, e.g., in hydrodynamics where the coefficients F F , K K , and B B are unbounded selfadjoint operators. It is assumed that F F is the dominating operator in the Cauchy problem above, i.e., \[ D ( F ) ⊂ D ( B ) , D ( F ) ⊂ D ( K ) . {\mathcal D}(F)\subset {\mathcal D}(B),\quad {\mathcal D}(F)\subset {\mathcal D}(K). \] We also suppose that F F and B B are bounded from below, but the operator coefficients are not assumed to commute. The main results concern the existence of strong solutions to the stated Cauchy problem and applications of these results to the Cauchy problem associated with small motions of some hydrodynamical systems.

Homogenization in perforated domains beyond the periodic setting

We study the homogenization of a second order linear elliptic differential operator in an open set in R N with isolated holes of size ε>0. The classical periodicity hypothesis on the coefficients of the operator is here substituted by an abstract assumption covering a variety of concrete behaviours such as the periodicity, the almost periodicity, and many more besides. Furthermore, instead of the usual “periodic perforation” we have here an abstract hypothesis characterizing the manner in which the holes are distributed. This is illustrated by practical examples ranging from the classical equidistribution of the holes to the more complex case in which the holes are concentrated in a neighbourhood of the hyperplane { x N =0}. Our main tool is the recent theory of homogenization structures and our basic approach follows the direct line of two-scale convergence.

Nonresonant smoothing in Sobolev spaces for coupled wave+transport equations

Consider a system consisting of a linear wave equation coupled to a transport equation: □ t,xu=f, (∂ t+v(ξ)·∇ x)f=P(t,x,ξ,D ξ)g, where P( t, x, ξ, D ξ ) is a linear differential operator of order m in ξ. Such a system is called nonresonant when the maximum speed in the transport equation is less than the propagation speed in the wave equation. Velocity averages of solutions to such nonresonant coupled systems are shown to be more regular than those of either the wave or the transport equation alone. This question was investigated first in terms of Sobolev spaces H s in the paper of F. Bouchut, F. Golse and C. Pallard, Non-resonant smoothing for coupled wave+transport equations and the Vlasov–Maxwell system, (Rev. Mat. Iberoamericana, 2003, in press.) The same authors also studied a related question in On classical solutions to the 3D relativistic Vlasov–Maxwell system: Glassey–Strauss' theorem revisited (Arch. Rational Mech. Anal., in press). Here we state a result in Sobolev spaces W s, p . More precisely, if f, g belong to L p loc ( R ∗ +× R N× R M) and with initial data for u regular enough, then for any test function χ∈ C m c( R M ξ) we show that ∫u(·,·,ξ)χ(ξ) dξ∈W 1+γ,p loc R ∗ +× R N , when γ=1−(N−1)| 1 2 − 1 p |⩾0 and 1< p<+∞. We also study the limit cases p=1 and p=+∞ when N=3.

A characterization of Hyers–Ulam stability of first order linear differential operators

Let X be a complex Banach space and h : R→ C a continuous function. Let T h :C 1( R,X)→C( R,X) be the linear differential operator defined by T h u= u′+ hu. We give a necessary and sufficient condition in order that the operator T h has the Hyers–Ulam stability.

Hyers–Ulam stability of linear differential operator with constant coefficients

AbstractLet P(z) be a polynomial of degree n with complex coefficients and consider the n–th order linear differential operator P(D). We show that the equation P(D)f = 0 has the Hyers–Ulam stability, if and only if the equation P(z) = 0 has no pure imaginary solution. (© 2003 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Monsters in Hardy and Bergman Spaces

A monster in the sense of Luh is a holomorphic function on a simply connected domain in the complex plane such that it and all its derivatives and antiderivatives exhibit an extremely wild behaviour near the boundary. In this paper, the Hardy spaces H p and the Bergman spaces B p (1 h p < X ) on the unit disk are considered, and it is shown that there are no Luh-monsters in them. Nevertheless, it is proved that T -monsters (as introduced by the authors in an earlier work) can be found in each of these spaces for any finite order linear differential operator T .

Finite stopping time problems and rheometry of Bingham fluids

It is shown that steady channel, simple shear and Couette flows of a Bingham fluid come to rest in a finite amount of time, if either the applied pressure falls below a critical value, or the moving boundaries are brought to rest. An explicit formula for a bound on the finite stopping time in each case is derived. This bound depends on the density, the viscosity, the yield stress, a new geometric constant, and the least eigenvalue of the second order linear differential operator for the interval under consideration.

Orthogonal Polynomial Solutions of Spectral Type Differential Equations: Magnus' Conjecture

Let τ=σ+ν be a point mass perturbation of a classical moment functional σ by a distribution ν with finite support. We find necessary conditions for the polynomials {Qn(x)}∞n=0, orthogonal relative to τ, to be a Bochner–Krall orthogonal polynomial system (BKOPS); that is, {Qn(x)}∞n=0 are eigenfunctions of a finite order linear differential operator of spectral type with polynomial coefficients: LN[y](x)=∑Ni=1ℓi(x)y(i)(x)=λny(x). In particular, when ν is of order 0 as a distribution, we find necessary and sufficient conditions for {Qn(x)}∞n=0 to be a BKOPS, which strongly support and clarify Magnus' conjecture which states that any BKOPS must be orthogonal relative to a classical moment functional plus one or two point masses at the end point(s) of the interval of orthogonality. This result explains not only why the Bessel-type orthogonal polynomials (found by Hendriksen) cannot be a BKOPS but also explains the phenomena for infinite-order differential equations (found by J. Koekoek and R. Koekoek), which have the generalized Jacobi polynomials and the generalized Laguerre polynomials as eigenfunctions.

Isospectral Flows of Third Order Operators

We consider the third order linear differential operator $L_{p,q}=i\frac{d^3}{dx^3}+i\frac{d}{dx}q+iq\frac{d}{dx}+p$ on the unit interval. The associated boundary conditions are $y(0)=y(1)=0$, $y'(0)=zy'(1)$ with $z\in\C,\ \vert z\vert=1$. For all but a finite number of parameters z, the eigenvalues $\lambda_{j}$ of $L_{p,q}$ are of multiplicity one. For fixed boundary conditions we make the formulae for isospectral flows induced by the Hamiltonian function $\lambda_{j}$ explicit. These flows commute.

Sequences of differential operators: exponentials, hypercyclicity and equicontinuity

An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on cert

Formal Gevrey Class of Formal Power Series Solution for Singular First Order Linear Partial Differential Operators

Formal Gevrey Class of Formal Power Series Solution for Singular First Order Linear Partial Differential Operators