Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

Abstract

The problem of computing periodic solutions can be expressed as a boundary value problem and solved numerically via piecewise collocation. Here, we extend to renewal equations the corresponding method for retarded functional differential equations in (K. Engelborghs et al., SIAM J Sci Comput., 22 (2001), pp. 1593–1609). The theoretical proof of the convergence of the method has been recently provided in (A. Andò and D. Breda, SIAM J Numer Anal., 58 (2020), pp. 3010–3039) for retarded functional differential equations and in (A. Andò and D. Breda, submitted in 2021) for renewal equations and consists in both cases in applying the abstract framework in (S. Maset, Numer Math., 133 (2016), pp. 525–555) to a reformulation of the boundary value problem featuring an infinite‐dimensional boundary condition. We show that, in the renewal case, the proof can also be carried out and even simplified when considering the standard formulation, defined by boundary conditions of finite dimension.

Piecewise discretization of monodromy operators of delay equations on adapted meshes

“…Very recently, this methodology has been extended and applied to REs for the first time in [8], but a systematic treatment for DDEs, REs and coupled equations appeared only in [1]. Concerning the convergence, the only available rigorous error analysis can be found in [3] for DDEs and in [2,4] for REs. Here we just summarize from [20] the main aspects of this numerical scheme in the case of DDEs, which corresponds to the one implemented in DDE-BIFTOOL.…”