On time Delay Differential Equations

The current final project belongs to a subject in the master’s degree in Advanced Mathematics at the University of Barcelona. It deals with time delays which usually are arisen in differential equations. Firstly, the project develops the main important known results of Delay Differential Equations, which are a specific case of Functional Differential Equations. In particular, we shall also focus on Delay Differential Equation with a constant delay. Secondly, different integrators of a general Delay Differential Equation with a constant delay are explained and their numerical results are exposed according to the made implementation scripts written in C and C++. By the way, an introduction to Automatic Differentiation Theory is also presented in order to be able to compute derivatives until a prefixed order of some suitable functions. Finally, a new method of computing periodic delayed orbits of a Delay Differential Equation with a constant delay is posed and a test of that new method is explained with some comments of the results. acknowledgements. Àngel Jorba by his supervision in whole this project and Carles Simó by his explanations in the master’s subject, Simulation Methods.