Sturm-Liouville Systems Research Articles

A non‐standard multiparameter eigenvalue problem in ordinary differential equations

AbstractWe consider two simultaneous Sturm–Liouville systems involving three spectral parameters. However unlike the standard multiparameter problem, we now suppose that the principal part of each of the differential operators involved is multiplied by a different parameter. For such a non‐standard system we establish the oscillation theory and discuss the associated eigenfunction expansion. Some open problems concerning the eigenfunction expansion are also presented. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Positivity Representations for Non-Hermitian Hamiltonians

We show how 1-dimensional, non-Hermitian, Sturm-Liouville systems, with rational fraction potentials, define a Moment Problem positivity representation, enabling the generation of (numerical and algebraic) converging lower and upper bounds to the (real and imaginary parts of) complex eigenenergies.

Observability analysis of nonlinear tubular (bio)reactor models: a case study

This paper deals with the observability analysis of nonlinear tubular bioreactor models. Due to the lack of tools for the observability analysis of nonlinear infinite-dimensional systems, the analysis is performed on a linearized version of the model around some steady-state profile, in which coefficients can be functions of the spatial variable. The study starts from an example of tubular bioreactor that will serve as a case study in the present paper. It is shown that such linear models with coefficients dependent on the spatial variable are Sturm–Liouville systems and that the associated linear infinite-dimensional system dynamics are described by a Riesz-spectral operator that generates a C 0 (strongly continuous)-semigroup. The observability analysis based on infinite-dimensional system theory shows that any finite number of dominant modes of the system can be made observable by an approximate point measurement.

Extension of Ambarzumyan's Theorem to General Boundary Conditions

We extend the classical Ambarzumyan's theorem for the Sturm–Liouville equation (which is concerned only with Neumann boundary conditions) to the general boundary conditions, by imposing an additional condition on the potential function. Our result supplements the Pöschel–Trubowitz inverse spectral theory. We also have parallel results for vectorial Sturm–Liouville systems.

Sturm-Liouville systems with rational Weyl functions: Explicit formulas and applications

This paper solves explicitly the direct and inverse spectral problems for Sturm-Liouville systems with rational Weyl functions. As in the previous papers of the authors on canonical and pseudocanonical systems, the method employed in the present paper is based on state space techniques and uses the idea of realization from mathematical system theory. For a subclass of rational potentials a property of modified bispectrality is proven. A wide class of solutions of the matrix Korteweg-de Vries equation is derived.

The continuum Schrödinger-Coulomb and Dirac-Coulomb Sturmian functions

Spherical continuum Sturmian functions for the Schrodinger-Coulomb and Dirac-Coulomb problems are constructed by solving appropriate Sturm-Liouville systems. It is proved that in the non-relativistic case a spectrum of potential strengths is continuous and covers the whole real axis. In the relativistic case two Sturmian sets may be derived. For the relativistic Sturm-Liouville problems their eigenvalue spectra consist of the real axes with zero excluded plus circumferences in the complex plane centred at zero. It is shown that, as a consequence of a relationship existing between the two families of the continuum Dirac-Coulomb Sturmians, each family obeys two orthogonality and two closure relations.

D-optimal designs for weighted polynomial regression

Using an Equivalence Theorem and the theory of Sturm-Liouville systems, we determine the D-optimal designs for some classes of weighted polynomial regression of degree d on the interval [−1, 1]. We also show that the number of the optimal support points for such models is d + 1. and that the optimal support points can be computed through a specific eigenvalue of a certain band matrix. Some results of Karlin and Studden ( Ann. Math. Statist. 37 (1996), 783–815) turn out to be special cases of our results.

Density, spectral theory and homoclinics for singular Sturm-Liouville systems

We consider the singular system −( P( t) u′)′ + Q( t) u = λ R( t) u. We give boundary conditions corresponding to the weak formulation, density results and conditions under which the set of eigenfunctions is total and orthogonal. Our results include the classical special functions. Finally, we apply these results to the problem on the real line: −(P(t)u′)′ + Q(t)u = V u(t, u), t ∈ R, ∫ R[(Pu′·u′) + (Qu·u)] < ∞ , where V is superquadratic at infinity and subquadratic at the origin.

Convolution and hypergroup structures associated with a class of Sturm-Liouville systems

Product formulas of the type \[ u k ( θ ) u k ( ϕ ) = ∫ 0 π u k ( ξ ) D ( ξ , θ , ϕ ) d ξ {u_k}(\theta ){u_k}(\phi ) = \int _0^\pi {{u_k}(\xi )D(} \xi ,\theta ,\phi )\;d\xi \] are obtained for the eigenfunctions of a class of second order regular and regular singular Sturm-Liouville problems on [ 0 , π ] [0,\pi ] by using the Riemann integration method to solve a Cauchy problem for an associated hyperbolic differential equation. When D ( ξ , θ , ϕ ) D(\xi ,\theta ,\phi ) is nonnegative (which can be guaranteed by a simple restriction on the differential operator of the Sturm-Liouville problem), it is possible to define a convolution with respect to which M [ 0 , π ] M[0,\pi ] becomes a Banach algebra with the functions u k ( ξ ) / u 0 ( ξ ) {u_k}(\xi )/{u_0}(\xi ) as its characters. In fact this measure algebra is a Jacobi type hypergroup. It is possible to completely describe the maximal ideal space and idempotents of this measure algebra.

Inverse parabolic problems and discrete orthogonality

This paper considers a discrete sampling scheme for the approximate recovery of initial data for one dimensional parabolic initial boundary value problems on a bounded interval. To obtain a given approximate, data is sampled at a single time and at a finite number of spatial points. The significance of this inversion scheme is the ability to accurately predict the error in approximation subject to choice of sample time and spatial sensor locations. The method is based on a discrete analogy of the continuous orthogonality for Sturm-Liouville systems. This property, which is of independent mathematical interest, is the notion of discrete orthogonal systems, which loosely speaking provides an exact (or approximate) Gauss-type quadrature for the continuous biorthogonality conditions.

Sturm-Liouville systems with potentials 0.5 cos 2 nx

Sturm-Liouville systems with potentials 0.5 cos 2 nx

QUALITATIVE PROPERTIES OF FINITE-ELEMENT MODELS I: STURM-LIOUVILLE SYSTEMS

Many important qualitative properties of Sturn-Liouville systems centre around the oscillatory character of the Green's function. It is shown that many of these properties are shared also by the piecewise-linear finite-element models of Sturm-Liouville systems

Orthogonal polynomials and higher order singular Sturm-Liouville systems

In addition to the classic orthogonal polynomials which satisfy second order differential equations, there are a number of orthogonal polynomials which satisfy differential equations of orders four or six. Like the classic sets, they have distributional weight functions, are the eigenfunctions for certain self-adjoint boundary-value problems, and sometimes are involved with indefinite boundary-value problems. The purpose of this survey is to summarize the work of the last decade and to exhibit the state of the art as it now stands. Of particular interest is the development of the theory of singular Sturm-Liouville systems, which is so necessary in order to describe the boundary-value problems associated with these polynomials.

On the solution of periodic multidimensional diffusion problems

Analytical solutions are obtained for periodic multidimensional diffusion problems through extension of the ideas in the generalized integral transform technique, completely avoiding the consideration of complex Sturm-Liouville systems. Exact solutions are obtained for a class of problems as a special case and, for a more general formulation, a coupled system of linear ordinary differential equations is generated, that can be solved by conventional algorithms and provide a complete solution to the periodic problem. Also, approximate analytical solutions for practical purposes are readily obtained from appropriate simplification of the coupled, denumerable system.

Comments on eigenfunction expansions of discontinuous Sturm-Liouville systems

In earlier works [4],[5] we examined the eigenfunctions of Sturm-Liouville systems of the form − y″( x) + q( x) y( x) = λy( x) on a bounded interval having eigenfunctions with jump discontinuities. An eigenfunction expansion theorem for these discontinuous Sturm-Liouville problems along with applications of the theory were presented. In this note we review earlier results and examine the possibilities for extending our results to general, single and multiple discontinuity cases.

Self-adjoint operators of transport in interacting solid-fluid systems—II

This paper is a follow-up of an earlier one (Arce and Ramkrishna, 1986, Chem. Engng Sci. 41, 1539) on an operator-theoretic analysis of interaction between a fluid and a bed of particles by exchanging heat or mass. In this paper we account for internal gradients within individual particles. The analysis reveals several interesting mathematical properties not characteristic of Sturm-Liouville systems (which describe transport in single phase media) with strong implications for the physics of interaction between the two phases. Limiting cases have been studied using the properties of the operator. Regions are identified in the multi-dimensional parameter space in which various model simplifications are possible.

Solution of generalized non-singular Sturm–Liouville systems via an approximation method in a Hilbert space

The approximation method in a Hilbert space is applied to the analysis of a generalized non-singular Sturm-Liouville eigenvalue problem. The variable coefficients in the system equation and the solution are approximated by a suitable linear independent basis, the expansion coefficients are explicitly expressed as a backward recursive formula and the eigenvalues are obtained by solving an algebraical characteristic equation. The computation is simple and straightforward.

Efficient computation of the Prüfer phase function for determining eigenvalues of stur -liouville systems

Abstract It is shown that a combination of phase integral approximation and numerical integration of the Prufer phase function yields a computationally effective technique for numerical solution of Sturm-Liouville eigenvalues. The technique is tested with sample Sturm-Liouville systems and compared with other methods.

Error Control of Phase-Function Shooting Methods for Sturm-Liouville Problems

A popular method for computing eigenvalues and eigenfunctions of Sturm-Liouville systems is to convert to phase-amplitude variables (θ, r ) in the (y′, y)- plane (a Prüer transformation) and solve the resulting equations by a shooting method using an ODE initial-value code. We study the error control of such methods with specific reference to two published codes based on this technique, and make comparisons and recommendations. Improved error estimates are derived and evidence for their efficacy presented.

Abstract oscillation theorems for multiparameter eigenvalue problems

We prove abstract analogous of Klein's oscillation theorem by demonstrating the existence (and in some cases uniqueness) of eigenpairs with a given index for the multiparameter problem T mx m = ∑ n = 1 k λ nV mnx m , 0 ≠ x m ϵ H m , m = 1 … k. (∗) Here T m and V mn are self-adjoint operators on Hilbert spaces H m . The index is based on the number of negative eigenvalues of T m − ∑ n = 1 k λ n V mn and on the sign of the determinant δ 0 with ( m, n)th entry ( x m , V mn x m ). We assume that certain cofactors of δ 0 are positive, and we complement previous work of Sleeman on Sturm-Liouville systems, and of Binding and Browne on (∗) in the case where δ 0 is positive.