Numerical Mathematics and Advanced Applications, 2003
ABSTRACT Two classical methods to compute the numerical solution of a λ-rational Sturm-Liouville ... more ABSTRACT Two classical methods to compute the numerical solution of a λ-rational Sturm-Liouville problem are analyzed in connection with the shooting technique: the method of Magnus series and the boundary value methods. We show that these methods display a weakening of their performances in the presence of an eigenvalue embedded in the essential spectrum. Nevertheless we show that some boundary value methods used in a particular form preserve their high order of convergence. Lastly we examine the performance of some boundary value methods applied to a singular problem when the Niessen-Zettl transformation is used.
Boundary Value Methods generalizing the Numerov's method are here proposed for the numerical... more Boundary Value Methods generalizing the Numerov's method are here proposed for the numerical approximation of the eigenvalues of regular Sturm-Liouville problems subject to Dirichlet boundary conditions. Moreover, an analysis of the error in the approximation of the k-th eigenvalue provided by such methods is reported. Some numerical results showing the possible advantages that may arise from the use of the new schemes are also presented.
ABSTRACT A general method based on the evaluation of the zeros of a suitable polynomial is sugges... more ABSTRACT 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.
Journal of Difference Equations and Applications, 1997
ABSTRACT It is known that linear k-step methods can be used for solving initial value problems by... more ABSTRACT 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.
Numerical Mathematics and Advanced Applications, 2003
ABSTRACT Two classical methods to compute the numerical solution of a λ-rational Sturm-Liouville ... more ABSTRACT Two classical methods to compute the numerical solution of a λ-rational Sturm-Liouville problem are analyzed in connection with the shooting technique: the method of Magnus series and the boundary value methods. We show that these methods display a weakening of their performances in the presence of an eigenvalue embedded in the essential spectrum. Nevertheless we show that some boundary value methods used in a particular form preserve their high order of convergence. Lastly we examine the performance of some boundary value methods applied to a singular problem when the Niessen-Zettl transformation is used.
Boundary Value Methods generalizing the Numerov's method are here proposed for the numerical... more Boundary Value Methods generalizing the Numerov's method are here proposed for the numerical approximation of the eigenvalues of regular Sturm-Liouville problems subject to Dirichlet boundary conditions. Moreover, an analysis of the error in the approximation of the k-th eigenvalue provided by such methods is reported. Some numerical results showing the possible advantages that may arise from the use of the new schemes are also presented.
ABSTRACT A general method based on the evaluation of the zeros of a suitable polynomial is sugges... more ABSTRACT 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.
Journal of Difference Equations and Applications, 1997
ABSTRACT It is known that linear k-step methods can be used for solving initial value problems by... more ABSTRACT 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.
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Papers by Paolo Ghelardoni