Solving numerically differential equations with distributed delays (Nicola Guglielmi, GSSI, L'Aquilla)

28.11.2023 14:00

Authors: Nicola Guglielmi and Ernst Hairer

Sinopsis:

In the early Nineties Ernst Hairer developed a famous code for the numerical approximation of stiff and differential algebraic problems, Radau5, which is based on the $3$-stage Radau IIa Runge Kutta method. The code became very popular and is nowadays one of the most important for the numerical approximation of implicit and stiff ODEs.

Later, between 2000 and 2005, Ernst and I developed a code, Radar5, which extended Radau5 to delay differential equations with discrete delays (in short DDEs).

Due to the increasing importance of models with distributed delays, for example in pharmacodynamics and pharmacokinetics (e.g Shuhua Hu et al., 2018), we have been recently addressed to certain kind of differential equations with distributed delays (in short DDDEs), where the dependence on the solution in the past appears through an integral term.

In this talk I will explain how these problems can indeed be approximated by suitable systems of ODEs or DDEs, according to the kind of distributed term.

The main ideas here are two:
(i) replacing the kernels by suitable quasi-polynomial expansions, that is sums of polynomials multiplied by exponentials, and then transforming the distributed (integral) delay term into a set of ODEs (or DDEs);
(ii) making the method efficient by exploiting the structure at the level of the linear algebra.

Numerical evidence of the effectiveness of the proposed approach is illustrated on a few illustrative examples.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-05, Séminaire d'analyse numérique

entrée libre