PRESENTAZIONE

The course aims to provide a presentation of the most common partial differential equations (PDE) and their  solution techniques through an analysis of various applications. The emphasis is  devoted to  second order PDE and the understanding of the specific techniques  for  elliptic, parabolic and hyperbolic cases.

OBIETTIVI FORMATIVI

The unit deals with the most important partial differential equations through their most important mathematical physical in the pleasure of craft sector.

Modalità didattiche

The unit is organised with for hours a week: a part of this course takes place in the laboratory to teach Matlab (matrix laboratory)

PROGRAMMA/CONTENUTO

Introduction  to  partial differential equations (PDE). The elastic string  and the transition from discrete systems to continuous systems .
2 . The differential equations of the second order. The classification and the normal form; elliptic , hyperbolic and parabolic  PDE.
3 . Elliptic equations. The harmonic functions. The problems of Dirichelet and Neuman , The Poisson formula for the circle.
4 . The variable separation technique. The series and the Fourier transform . The Gibbs effect, the analysis of normal modes , the delta Dirac "function”.

5 . The Bessel functions, the problems in polar coordinates.
6 . The parabolic differential equations, diffusion and heat  equations; descriptions in the domain of space and time.
7 . The hyperbolic equations: the equation of D' Alembert. The method of characteristics, the elastic membrane, the mechanical interpretation of the normal modes
8 . PDE of higher order: the biharmonic equation, its Cauchy problem. The vibration of bars and plates.

TESTI/BIBLIOGRAFIA

• A.N.Tichonov, A.A.Samarskij: Equazioni della Fisica matematica, Problemi della fisica matematica, Mosca,1982;
• R. Courant, D. Hilbert, Methods of Mathematical Phisics vol I e II, Interscience, NY, 1973;
• R. Bracewell, The Fourier Transform and Its Applications, New York: McGraw-Hill, 1999;
• P. V. O’ Neil, Advanced engineering mathematica, Brooks Cole, 2003;
• H. Goldstein, Meccanica Classica, Zanichelli, Bologna, 1985;
• V. I. Smirnov. Corso di Matematica superiore, Vol. 3. MIR (1978).

Commissione d'esame

ROBERTO CIANCI (Presidente)

ROBERTA SBURLATI

Modalità didattiche

The unit is organised with for hours a week: a part of this course takes place in the laboratory to teach Matlab (matrix laboratory)

INIZIO LEZIONI

Second semester.

Modalità d'esame

Examinations are oral . Some mandatory exercizes can be requested.