A stable numerical method for Volterra integral equations with discontinuous kernel

Abstract

Numerical methods for Volterra integral equations with discontinuous kernel need to be tuned to their peculiar form. Here we propose a version of the trapezoidal direct quadrature method adapted to such a type of equations. In order to delineate its stability properties, we first investigate about the behavior of the solution of a suitable (basic) test equation and then we find out under which hypotheses the trapezoidal direct quadrature method provides numerical solutions which inherit the properties of the continuous problem.

A numerical investigation on the asymptotic behavior of discrete Volterra equations with two delays

“…These assumptions include some that are significant from a biological point of view (see [Messina et al 2008a] and the bibliography therein) and guarantee that the solution y(t) is nonnegative for all t ≥ τ 2 . Define the positive function…”

“…In particular, it is strictly increasing for all x ≥ 0, if g(x) is a nonlinear function, while it is constant otherwise. From now on, we assume that g(x) is nonlinear, hence (d) and (e) are meant as strict inequalities and, in analogy with [Messina et al 2008a], we consider the following alternative formulation of (1):…”

“…Let a * = lim x→+∞ a(x). Theorem 2.1 [Iannelli 1994;Messina et al 2008a]. Let ρ be defined as in (4).…”

“…The interest of (1) in the applications is mainly in the modeling of age-structured population dynamics, as described in [Messina et al 2008a] and the references therein. Here, we continue those investigations with the aim of providing a more complete analysis of the dynamics of the solutions.…”

“…In Section 3, we consider a numerical method of direct quadrature type and look for conditions on the step size h of a direct quadrature method that lead to a numerical solution which mimics the behavior of the continuous one. The main novelty of this paper with respect to [Messina et al 2008a] is the compact form that we use to represent the method: this new form allows us to obtain some new results in the discrete case equivalent to those valid for the continuous case, and so to complete the parallelism between the behaviors of the analytical and the numerical solution. Finally, in Section 4 we report some numerical examples that show the nature of these behaviors.…”