Perturbative and qualitative methods: tools to solve nonlinear problems

Ph.D. in Information and Communication Technologies ICT

Ph.D. Course:

Perturbative and qualitative methods:
tools to solve nonlinear problems

Edition 2020 (24 hours, 4 credits)

Instructor: Sandra Carillo

Dipartimento di Scienze di Base e Applicate per l’Ingegneria

Sapienza Universit`a di Roma

Course Description: This Course provides some methods to study nonlinear problems focussing mainly on cases which are modelled via nonlinear ordinary differential equations. The aim is to construct solutions to nonlinear problems which arise in applications via qualitative and perturbation methods. The first ones study the analytical consequences of conservation laws and their implications as far as the solutions admitted by such problems are concerned. The second ones are concerned with ad hoc methods developed when in the nonlinear a small parameter appears.

Class Schedule

The lectures of the course are scheduled in Spring 2020, March the 3rd - April the 2nd, 2020


Seminar room at the second floor of the DIET department, 

Via Eudossiana 18, 00184

Rome, Italy, tentative timetable:

Tuesday March the 3rd, 15:00–18:00

Tuesday March the 10th, 14:00–17:00

Tuesday March the 17th, 14:00–17:00

Thursday March the 19th, 14:00–17:00

Tuesday March the 24th, 14:00–17:00

Thursday March the 26th, 14:00–17:00

Tuesday March the 31st, 14:00–17:00

Thursday April the 2nd, 14:00–17:00


The arguments can be schematically listed in:

a) Qualitative methods: an introduction.

b) Straightforward Perturbation Method.

c) Multiple Scale Method.

d) Singular Perturbation Method.

e) Boundary Layer Method.

f) Visualisation of the results via computer algebra manipulation (MATLAB Toolbox).




Some background notions on ordinary differential equations open the course. In particular, on qualitative and perturbation methods are introduced as tools to study nonlinear ordinary differential equations. Then, some ideas on qualitative methods (conservation of energy, phase plane) are briefly illustrated and applied to physically significative cases.

Then, various perturbation 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.e.s, such as Duffing equation, are studied. Also the Van der Pol equation, which can be used to model the cardiac cycle, is analysed. In most of the provided examples, various methods are applied and a comparison among the different approximations obtained and the 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 developing a personal project with applicative meaning.

If there is interest in the audience, an overview on how to apply Perturbation Methods in the case of partial differential equations closes the course.