# MATHEMATICAL METHODS

iten
Code
80136
2021/2022
CREDITS
3 credits during the 1st year of 9270 Mechanical Engineering - Energy and Aeronautics (LM-33) GENOVA
SCIENTIFIC DISCIPLINARY SECTOR
MAT/07
LANGUAGE
Italian
TEACHING LOCATION
GENOVA (Mechanical Engineering - Energy and Aeronautics)
semester
2° Semester
modules
This unit is a module of:
Teaching materials

## 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.

## AIMS AND CONTENT

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 techniques for elliptic, parabolic and hyperbolic cases.

AIMS AND LEARNING OUTCOMES

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.

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.

• 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).

## TEACHERS AND EXAM BOARD

Office hours: The teacher receives by appointment via email sent to roberto.cianci@unige.it.

Exam Board

ROBERTO CIANCI (President)

ANGELO ALESSANDRI

FRANCO BAMPI

STEFANO VIGNOLO

PATRIZIA BAGNERINI (President Substitute)

## LESSONS

TEACHING METHODS

The module is based on theoretical lessons.

Class schedule

MATHEMATICAL METHODS

## EXAMS

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

Date Time Location Type Notes
10/01/2022 14:00 GENOVA Orale
09/02/2022 14:00 GENOVA Orale
07/06/2022 14:00 GENOVA Orale
05/07/2022 14:00 GENOVA Orale
15/09/2022 14:00 GENOVA Orale

FURTHER INFORMATION