Nonlinear ordinary differential equations: perturbation methods & applications 
Prof.
Sandra Carillo 
Sandra.Carillo@sbai.uniroma1.it http://www.sbai.uniroma1.it/~sandra.carillo 
Course Description: This Course
provides some methods to study nonlinear ordinary differential
equations aiming to construct solutions to nonlinear problems
which arise in applications in all cases in which a small parameter appears. The
arguments can be schematically listed in:
Some background notions on Asymptotic Expansions and their application to study ordinary differential equations open the course. Then, the different listed methods are presented and illustrative examples are studied in detail. Critic aspects as well as advantages of each method are pointed to the students’ attention. In addition, via computer algebra methods, the solutions of the problems are constructed and plotted. Cauchy and boundary value problems are both treated. As a first toy problem, the Cauchy problem in the case of a linear weakly damped oscillator is studied. Then, nonlinear o.d.es, such as Duffing equation, are studied. Also the Van der Pol equation, which can be used to model the cardiac cycle, is analyzed. In most of the provided examples, various methods are applied and a comparison among the different approximations obtained and thew related region of validity (in time or space) is given. A variety of examples of application is provided and the students are invited to actively participate developping a personal project.
An overview on how to apply Perturbation Methods
in the case of partial differential equations closes the course.
Text(s): Selected Chapters from
ORARIO DELLE
LEZIONI (30 ORE TOTALE) 

