Paolo Ghelardoni

Approximations of Sturm-Liouville eigenvalues using boundary value methods

Abstract It is well known that the algebraic approximations to eigenvalues of a Sturm-Liouville problem by the central difference and Numerovs schemes provide only a few estimates restricted to the first element of the eigenvalue sequence. A correction technique, used first by Paine et al. (1981) for the central difference scheme and then by Andrew and Paine (1985) for Numerovs method, improves the results, giving acceptable estimates for a larger number of eigenvalues. In this paper some linear multistep methods, called Boundary Value Methods, are proposed for discretizing a Sturm-Liouville problem and the correction technique of Andrew-Paine and Paine et al. is extended to these new methods.

Spectral corrections for Sturm-Liouville problems

Abstract The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented.

Stability of some boundary value methods for IVPs

Abstract We investigate the stability properties of some classes of linear two-step methods of high order and involving derivatives of the solution, up to the order three. These methods are suitable to be used as boundary value methods (BVMs) for solving initial value problems (IVPs). The advantages with respect to other two-step BVMs are the following: (1) possibility of attaining high order of accuracy, without destroying the tridiagonality of the discretization matrix, and, (2) assurance of maintaining the accuracy of the complessive BVM of the same order as that of the basic method.

Boundary Value Methods for the reconstruction of Sturm-Liouville potentials

Abstract In this paper we present numerical procedures for solving the two inverse Sturm–Liouville problems known in the literature as the two-spectra and the half inverse problems. The method proposed looks for a continuous approximation of the unknown potential belonging to a suitable function space of finite dimension. In order to compute such an approximation a sequence of direct problems has to be solved. This is done by applying one of the Boundary Value Methods, generalizing the classical Numerov scheme, recently introduced by the authors. Numerical results confirming the effectiveness of the approach proposed are also reported.

An Algebraic Procedure for the Spectral Corrections Using the Miss-Distance Functions in Regular and Singular Sturm-Liouville Problems

A general method based on the evaluation of the zeros of a suitable polynomial is suggested in order to have an estimation of the spectral error in the numerical treatment of Sturm-Liouville problems. The method is strictly concerned with the miss-distance function arising in the shooting algorithm for eigenvalues. The error correcting procedure derived from the method is particularly helpful when difficulties arise in the numerical integration. Two kinds of Sturm-Liouville problems are considered: the standard regular problems on a closed interval and the problems where an eigenvalue is nonlinearly involved and embedded in an essential spectrum giving origin to an inner singularity. Numerical experiments clearly highlight the efficaciousness of the proposed method both in the regular and singular case.

Two-Step Multi-Derivative Boundary Value Methods for Linear IVPs ∗

It is known that linear k-step methods can be used for solving initial value problems by tranforming them to boundary value problems. They are known as boundary value methods.(BVMs). In this paper we obtain two-step BVMs with high order of accuracy and good stability properties. which can be competitive with other konwn BVMs with k > 2. Namely this aim has been reached by some classes fo two-step formulas involving derivatives of an higher order than the first. The proposed formulas. well known in literature as initial value methods. here are studied as boundary value methods and heir BV-stability properties are investigated. Relevant numerical experiments are quoted.

A quasi-extrapolation procedure for error estimation of numerical methods in Sturm-Liouville eigenproblems

This paper deals with a generalization of a technique already proposed by the authors for obtaining an effective estimation of the spectral accuracy in some regular and non regular Sturm-Liouville problems. The algorithm looks like a classical extrapolation process, but, unlike such a procedure, it does not require further approximations of the eigenvalues with different stepsize: for this reason it benefits from a moderate computational cost. Numerical experiments confirm the effectiveness of the suggested approach.

Error estimates for parallel shooting using initial or boundary value methods

Abstract In the first part of this work (Sections 2 and 3) we derive from previous papers an outline of a general method to estimate the global discretization error in the numerical solution of a linear boundary value problem when the parallel shooting technique is used. Then, in Sections 4 and 5, the proposed error estimation is shown to be well suited in the case that the involved initial value problems are solved either by traditional linear k -step initial value methods or by boundary value methods. As the estimated error follows carefully the behaviour of the true error it can be used to improve the numerical solution as shown in some numerical examples.

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems (SLPs) on bounded domains is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.

Tourist planning along the coast of Aquitaine, France

The Atlantic coast of south-west France, particularly that of Gironde and Landes, develops in a straight, sandy littoral stretch for over 200 km from the Gironde estuary to the mouth of the Adour: this is called the Cote d’Argent for the charm of its foaming waves. The coastal zone includes various types of environments: the beach, the dunes, the ‘lette’, the protective forest and the exploited forest. The beach (‘estran’) is subject to tides, which here have a maximum amplitude of five metres. High energy breakers and westerly winds, contribute to the formation of a series of dune-belts, which rise up towards the east. In the last century, the need to stop the displacement of the sand inland, gave way to a large-scale containment operation, consisting in the imposition of an artificial profile to the dunes, with a gradual slope on the ocean side, a flat section on top, and a steeper slope on the inland side. The special vegetation cover ensures an effective fixing of the dune complex and prevents its progress inland. Beyond these dune-belts, there is the ‘lette’, a corridor running parallel to the coast, sheltered from the wind, and 200 m to 2000 m wide. Protected from the sea winds, the ‘lette’ is often an area used for settlements.