Analysis of Perturbed Volterra Integral Equations on Time Scales

Abstract

<jats:p>This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods.</jats:p>

Analysis of the Transient Behaviour in the Numerical Solution of Volterra Integral Equations

“…Starting from an idea developed in [3], here, we have introduced a technique for the analysis of the vanishing behaviour of the numerical solution to VIEs. This new approach, which is based on suitable splittings of the kernel function, allows one to preserve the character of the analytical solution even in the weighted sums that appear in the method, thus leading to unconditional stability results in many applications of interest.…”

“…The analysis of their dynamics allows one to describe the phenomena they represent. In [3], the two equations were analysed in the unifying notation of time scales, and some results were obtained under linear perturbation of the kernel. Here, we revise this approach to obtain results on classes of linear discrete equations whose kernel can be split into a well-behaving part (the unperturbed kernel) plus a term that acts as a perturbation.…”

“…The paper is organised as follows. In Section 2, we introduce the split kernel for Equation (2) and, using a new formulation of Theorem 2 in [3], we provide sufficient conditions for the above-mentioned splitting that imply that the solution vanishes. In Section 3, we propose a reformulation of the (ρ, σ) methods for (1) as discrete Volterra equations and exploit the theory developed in the previous section in order to investigate its numerical stability properties.…”