Giovanni Gheri

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.

Estimation of the global discretization error in shooting methods for linear boundary value problems

Abstract In this paper we consider an error estimation procedure in the numerical solution of a linear boundary value problem for a system of m first-order equations when the simple or parallel shooting method is used. We show that the global discretization error can be estimated through the numerical solution of the principal error equation related to only one initial-value problem, rather than the m + 1 involved by shooting. Four significant numerical experiments are presented.

A significant example to test methods for solving systems of nonlinear equations

A system of nonlinear equations verifying certain conditions is considered. Some resolution methods are applied and the relevant results are displayed.

Parallel shooting with error estimate for increasing the accuracy

The paper is concerned with two-point boundary value problems for ordinary differential equations and some estimates of the global discretization error produced in the numerical solution by shooting methods. These error estimates, proposed in some previous papers, are shown to be useful for correcting the numerical results as they do not require much additional effort in computations both for linear and nonlinear problems. Special attention is devoted to the use of some new boundary value methods. Applications to known test problems are developed.

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.

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.

Un metodo di approssimazione bilaterale per equazioni differenziali ordinarie

In this paper we describe a method, based on predictor-corrector formulas, for the bilateral approximation of the solution of special initial value problems for ordinary differential equations.For given predictor-corrector formulas, conditions are stated in order to obtain, at any mesh-point, an interval containing the exact solution. The amplitude of the interval gives an error estimate according to the order of the method.Some numerical examples are considered and relevant results are displayed.

Collocation for initial value problems based on hermite interpolation

This paper proposes a technique to approximate the solutions of nonlinear initial value problems with Hermite interpolation polynomials, using collocation in a special way. Neverthless the method is shown to be equivalent with a classical collocation method involving a larger collocation system. The order of the method is investigated bringing out some connection with a class of hybrid multistep formulas. Sufficient conditions are given guarenteeing convergence of iterative method for the solution of the nonlinear collocation system.

Risoluzione numerica di un problema elastoplastico

RiassuntoSi presenta un metodo numerico per risolvere approssimativamente un problema di deformazione plastica contenuta per un corpo cilindrico indefinito con fori cilindrici. Si confrontano poi la soluzione esatta e quella approssimata per un cilindro circolare cavo.AbstractA numerical method is presented to solve approximately a problem of contained plastic deformation for an infinite cylindrical body with cylindrical holes. The exact and the approximate solutions for a tube are then compared.