## Teaching 2016-2017 Curriculum in Mathematics for Engineering

Problemi  a valori al bordo in domini con frontiere irregolari

Prof. Maria Rosaria Lancia, Prof. Maria Agostina Vivaldi

Inizio 28 febbraio / 4 ore a settimana dalle 14 alle 16 il martedì e il giovedì, per 5/6 settimane.

Aula 1B1 Pal. RM002 - Via A. Scarpa 16

Programma. Elenco (orientativo e non esaustivo) degli argomenti trattati:

• Soluzioni variazionali di problemi a valori al bordo stazionari
• risultati di regolarità
• omogeneizzazione e analisi asintotica
• approssimazione numerica con il metodo agli elementi finiti  -problemi  con condizioni dinamiche sul bordo
• problemi a valori al bordo evolutivi

Nonlinear ordinary differential equations: perturbation methods & applications

Prof. Sandra Carillo
S.B.A.I. Dept., Via A. Scarpa 16, 00161, Rome, Italy

Starting date :  8 novembre 2016

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:

a) Straightforward Perturbation Method;

b) Multiple Scale Method;

c) Singular Perturbation Method;

d) Boundary Layers Method.

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

• M.H.Holmes, Introduction to Perturbation Methods, Springer, New York, 1995;
ISBN-13: 978-0000000000
• further provided material.

ORARIO DELLE LEZIONI (30 ORE TOTALE)
 MARTEDI E MERCOLEDI ORE 14-15.30 + 15.45-16.30 (3 ORE) INIZIO DEL CORSO: 8 NOV. 2016 ORE 14 AULA 1B1 FINE DEL CORSO: 7 DIC. 2016

LUOGO DELLE LEZIONI: AULA 1B1, DIP. SBAI, PAL RM 002, VIA SCARPA 16 (TRANNE IL 9 E IL 16 NOV. IN CUI LA LEZIONE SI SVOLGERA` IN PAL. RM004)

Introduzione alle  equazioni alle derivate parziali

Referente: Lorenzo Giacomelli

1) dal 4.10.2016 al 21.12.2016, nell'edificio E di San Pietro in Vincoli, il mercoledi alle ore 14. Al bisogno gli incontri possono essere spostati al venerdi in via Scarpa.

2) Pagina web e dettagli : http://www.dmmm.uniroma1.it/~lorenzo.giacomelli/aero1617-pde/index.html

3A) An introduction to the four main linear PDEs (transport, wave, Laplace, heat) through constructive methods and classical solutions. Some digression on nonlinear PDEs (e.g. Burgers equation), enegy methods, and weak solutions.
3B) Introduzione alle quattro principali EDP lineari (trasporto, onde, Laplace, calore) attraverso metodi costruttivi e soluzioni classiche. Qualche digrssione su EDP non lineari (per esempio l'equazione di Burgers), sui metodi di energia e sui concetti di soluzione debole.

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