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

A matrix method for the solution of direct fractional Sturm-Liouville problems on bounded domain 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

A generalization of Gauss-Jacobi quadrature formulas

Si studiano formule di quadratura alle derivate dell’integrando negli estremi - 1 ed 1 dell’intervallo di integrazione ed aventi come nodi semplici gli zeri di un opportuno polinomio di Jacobi. Di tali formule si determina anche una espressione esplicita dell’integrale del nucleo di Peano, nucleo che si mantiene di segno costante sull’intervallo.The author studies quadrature formulae with the derivatives of the function which must be integrated calculated on boundary values - 1 and 1 of the integration interval and with the zeros of on opportune Jacobi polinomial as single nodes. On such formulae is also carried out an explicit expression for the integral of the respective Peano’s Kernel which is, on [ - 1,1], of constant sign

On a quadrature formula associated with the local calculus of elementary functions

Si stabilisce una formula di quadratura di tipo chiuso a nodi multipli evidenziando, con opportuni esempi, come essa possa efficacemente utilizzarsi per il calcolo puntuale di alcune funzioni elementari (1).The authors estabilish a quadrature formula of closed type with multiple nodes showing, with suitable examples, as this can be usefully applied to the local calculus of some elementary functions (1)

On some Toeplitz and Haenkel matrices with elements 0,1 and -1

Scopo di questo lavoro è dare un contributo alla raccolta di matrici test. Per questo motivo vengono considerate due classi di matrici, di Haenkel e di Toeplitz, delle quali si calcola l’inversa, inoltre, se n è l’ordine della matrice, vengono calcolati [n/2] autovalori, e i corrispondenti autovettori, esattamente.The aim of this paper is to give a contribution to the collection of test matrices. To this end, two classes of Toeplitz and Haenkel matrices are taken into account, of which inverse is calculated. Moreover, if n is the order, [n/2] eigenvalues and the corresponding eigenvectors are given exactly

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

Boundary Value Methods for the Reconstruction of Sturm-Liouville potentials

The paper deals with the numerical solution of the two-spectra and the half inverse Sturm-Liouville problems. The numerical procedure proposed provides a continuous approximation of the unknown potential and uses an approach similar to the one studied in [6] for solving the symmetric inverse problem. The results of numerical experiments confirm the effectiveness of the considered methods

P-stable boundary value methods for second order IVPs

We introduce a family of Linear Multistep Methods used as Boundary Value Methods for the numerical solution of initial value problems for second order ordinary differential equations of special type. The aim is to obtain P-stable methods with arbitrary order of accuracy. This result allows to overcome the order barrier established by Lambert and Watson which limited to p - 2 the maximum order of a P-stable Linear Multistep Method. In addition, an extension of the methods in the Exponential Fitting framework is also considered

Error estimates for parallel shooting using initial or boundary value methods

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