OVERVIEW

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.

LEARNING OUTCOMES

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

Teaching methods

The module is based on theoretical lessons.

SYLLABUS/CONTENT

1. Introduction to partial differential equations (PDE). The elastic string and the transition from discrete systems to continuous systems. Second order partial differential equations. Classification and normal form. Elliptic, hyperbolic and parabolic PDE.
2. Elliptic equations. The harmonic functions. Dirichlet and Neumann boundary conditions, the Poisson formula for the circle.
3. Separation of variables technique. Series and Fourier transform. The Gibbs effect, the analysis of normal modes, the delta Dirac "function”. Bessel functions and problems in polar coordinates.
4. Parabolic differential equations, diffusion and heat equations; descriptions in space and time domain.
5. Hyperbolic equations: the equation of D'Alembert. The method of characteristics, the elastic membrane, the mechanical interpretation of the normal modes.
6. Some concept on PDE of higher order: the biharmonic equation and its Cauchy problem. The vibration of bars and plates.
7. Non homogeneous PDE and Green functions.

RECOMMENDED READING/BIBLIOGRAPHY

• 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).

Exam Board

ROBERTO CIANCI (President)

STEFANO VIGNOLO

FRANCO BAMPI

ANGELO ALESSANDRI

PATRIZIA BAGNERINI (President Substitute)

Teaching methods

The module is based on theoretical lessons.

LESSONS START

Second semester.

ORARI

L'orario di tutti gli insegnamenti è consultabile su EasyAcademy.

Vedi anche:

MATHEMATICAL METHODS

Exam description

The examination mode consists of an oral test to ensure learning of the course content.

Assessment methods

The oral exam focuses on the learning of one or two subjects from those discussed in class.

Exam schedule