Mètodes numèrics per a equacions diferencials ordinàries

Abstract

Mainly, the project studies some methods for numerical integration to approximate the solutions of ordinary differential equations, from now on they will be called ODE’s. Firstly, some concepts based on numerical methods for initial value problems studied during the degree subject Mètodes Numèrics 2 are reviewed and extended. We studied the Taylor methods, using automatic differentiation to compute the set of derivatives of a function, the Runge-Kutta methods and, as a variant of these, the Runge-Kutta-Fehlberg methods that lets us to control the step size. The next section of the project consists in the study of the stability of periodic orbits (Floquet’s theorem) and orbit’s continuation regarding to parameters. Finally, the last part studies a couple examples of classical mechanics where the tools learned on the first part of the project are used: pendulum periodically disturbed and Bicircular model. In both cases we explain the process to compute the periodic orbits and it’s stability. In this process, we use some concepts studied in Equacions Diferencials subject: variational equations and Poincaré map.