OBIETTIVI FORMATIVI

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

OBIETTIVI FORMATIVI (DETTAGLIO) E RISULTATI DI APPRENDIMENTO

Active participation in lectures and individual study will enable the student to:

- be able to classify the main partial differential equations;

- calculate the analytical solution of partial differential equations of elliptic, parabolic and hyperbolic types;

- use the techniques of separation of variables, series and Fourier transform, special functions.

MODALITA' DIDATTICHE

The module is based on theoretical lessons.

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).