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A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.

The discrete-time relaxation methods based on Volterra-Runge-Kutta methods for solving large system of second-kind Volterra integral equations are proposed. Convergence of the discrete-time iteration process with particular attention to parallel methods is investigated.

Investigation of the boundedness of the solutions of Volterra equations with discrete time by means of the direct Liapunov method and comparison theorems is performed. Boundedness conditions of solutions of some particular classes of equations are derived. Using such conditions, stability of Direct Quadrature methods for second kind Volterra equations is analyzed.

In this paper the waveform methods for solving systems of second kind Volterra integral equations are introduced. The convergence properties of continuous-time waveform methods on finite length and infinite length time intervals are investigated, with particular regard to parallel waveform relaxation methods.

This paper concerns parallel frontal predictor-corrector methods. Order and stability of these methods are investigated, when the corrector is solved both by the fixed point iteration method and by the Newton method.

Fully parallel Runge-Kutta-Nystrom methods are introduced and their properties of accuracy, stability, and dispersion are investigated. Fully parallel Runge-Kutta-Nystrom methods with two and three stages are constructed.

We investigate the stability of the numerical solutions resulting from applying very general classes of Runge-Kutta methods to Volterra integral and integro-differential equations with degenerate kernels. The results are generalizations of previous results obtained by the authors for exact collocation methods for these equations.

Stability conditions for Volterra equations with discrete time are obtained using the direct Lyapunov method, without the usual assumption of the summability of the series of the coefficients. Using such conditions, the stability of some numerical methods for second kind Volterra integral equations is analyzed.

In this paper necessary and sufficient conditions for the exponential stability of discrete linear Volterra systems are proved. Sufficient conditions, expressed directly in terms of the coefficients, are derived

In this paper, we analyse parallel iteration of Volterra-Runge-Kutta methods (PIVRK methods) for solving second-kind Volterra integral equations on parallel computers. We focus on the determination of the region of convergence C and on the stability region Sm of the iterated method obtained after m iterations. Results are presented for the convolution test equation. It turns out that the stability region Sm does not necessarily converge to the stability region S of the corrector. However, for finite m, Sm need not to be contained in C or S and may be much larger than C.

A class of methods depending on some parameters are introduced for the numerical solution of the Abel integral equations of the second kind. Some bounds on the parameters are determined so that the corresponding methods have infinite stability intervals.

This paper is concerned with the analysis of the stability of a class of one-step integration methods, originated by the Lanczos tau method and applicable to particular linear differential systems. It is proved that these methods are A-stable for every order.

The purpose of this paper is to analyze the stability properties of one-step collocation methods for the second kind Volterra integral equation through application to the basic test and the convolution test equation.

The tau method approximates the solution of a differential equation with a polynomial, which is the exact solution of a differential equation obtained by adding to the right hand side of the given equation a perturbation term, consisting of a suitably chosen linear combination of polynomials. Until now the Chebyshev and Legendre polynomials have been used to this purpose, but the determination of the best perturbation term is, still, an open problem. In this paper a perturbation term depending on the differential equation is chosen. For this formulation of the tau method, the existence of an infinite sequence of tau approximants and the convergence of the error to zero is proved. An estimate of the local truncation error is also given, and the stability properties are discussed. Numerical results are also reported.

The stability of the de Hoog and Weiss Runge-Kutta methods is analyzed for the Volterra integrodifferential equations with respect to the basic test equation and for Volterra integral equations with respect to the linear convolution equation. Stability regions are determined for some choices of the parameters.

Stability criteria are derived for difference equations of Volterra type with degenerate kernels. The main tool in this analysis is the use of the new representation formula which allows us to express the solution of discrete Volterra equation with degenerate kernel in terms of the fundamental matrix of the corresponding first-order system of the difference equations.

The local stability properties of the collocation method applied to a second kind Volterra integral equation with degenerate kernel are investigated.

Facendo ricorso al metodo τ, si determina un nuovo approssimante polinomiale diJλ(x), valido perx≥0 e λ reale, e di esso si da una stima dell'errore. Si individua, poi, sperimentalmente una regioe del piano (x,λ) in cui i risultati numerici sono esatti fino alla quinta cifra significativa.

Ortiz’ step-by-step recursive formulation of the Lanczos tau method is extended to the numerical solution of linear systems of differential equations with polynomial coefficients. Numerical comparisons are made with Gear’s and Enright’s methods.