# Numerical bifurcation of equations with infinite delay via pseudospectral collocation

Delay equations (i.e., Delay Differential Equations, Renewal Equations and coupled Renewal and Delay Differential Equations) are fundamental in many scientific areas such as biology, medicine, engineering and physics as the introduction of the delay can provide a more realistic mathematical model incorporating the information from the past in the evolution law. Due to the complexity of the analytical investigation of the properties of delay equations with both finite and infinite delay, it is necessary to use suitable numerical methods such as the pseudospectral discretization approach. Whereas in the case of finite delay we have convergence results, in the case of infinite delay numerical evidences are achieved, but the proof of convergence is still ongoing.

In this seminar, the focus is on DDEs with infinite delay. Firstly, the theory of DDEs will be briefly presented, then the pseudospectral approach will be described, and finally the latter will be applied on a population model which describes the evolution of water droplets into a vapour. In particular, classical Laguerre–Gauss–Radau nodes and Laguerre-type nodes will be used as collocation nodes for the simulations.

This seminar concerns the results of Ilaria’s MSc thesis.