E. Russo

A family of methods for Abel integral equations of the second kind

Abstract 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.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R0 1. Second we show that the model is permanent if and only if R0 > 1. Moreover, using a threshold parameter R0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R0 R0 and it loses stability as the length of the delay increases past a critical value for 1 < R0 < R0. Our result is an extension of the stability results in (J-J. Wang, J-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonl. Anal. RWA. 11 (2009) 2390-2402).

Stability of discrete volterra equations of hammerstein type

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

Stability analysis of discrete recurrence equations of Volterra type with degenerate kernels

Abstract 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.

Convergence of solutions for two delays Volterra integral equations in the critical case

Abstract In this paper, for the “critical case” with two delays, we establish two relations between any two solutions y ( t ) and y ∗ ( t ) for the Volterra integral equation of non-convolution type y ( t ) = f ( t ) + ∫ t − τ t − δ k ( t , s ) g ( y ( s ) ) d s and a solution z ( t ) of the first order differential equation z ( t ) = β ( t ) [ z ( t − δ ) − z ( t − τ ) ] , and offer a sufficient condition that lim t → + ∞ ( y ( t ) − y ∗ ( t ) ) = 0 .

On the exponential stability of discrete volterra systems

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

Discrete-time waveform relaxation Volterra-Runge-Kutta methods: convergence analysis

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.

Stability analysis of the de Hoog and Weiss implicit Runge-Kutta methods for the Volterra integral and integrodifferential equations

Abstract 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.

On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation

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.Stability regions are determined when the collocation parameters are symmetric and when they are zeros of ultraspherical polynomials.

On the stability of the one-step exact collocation method for the second kind Volterra integral equation with degenerate kernel

The local stability properties of the collocation method applied to a second kind Volterra integral equation with degenerate kernel are investigated.A finite length recurrence relation is derived and theorems for the local stability of the methods are proved.ZusammenfassungEs werden die lokalen Stabilitätseigenschaften der Kollokationsmethode, angewandt auf Volterrasche Integralgleichungen zweiter Art mit degeneriertem Kern, untersucht.Eine Rekursionsrelation endlicher Ordnung wird angegeben, und es werden Sätze über die lokale Stabilität der Methode bewiesen.