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

Modeling and Simulation Fundamentals. Theory and Practice of Continuous Simulation and related Methodologies. Theory and Practice of Discrete Simulation and related Methodologies. Hybrid Simulation.

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.

PREREQUISITES

The course does not require any formal prerequisites, being self-consistent and including all the elements and references necessary for the study; while not formally required, we warmly recommend to brush up the basic knowledge of the theory of partial and ordinary differential equations.

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).

TEACHERS AND EXAM BOARD

Exam Board

ROBERTO CIANCI (President)

AGOSTINO BRUZZONE

LUCA FABBRI (President Substitute)

LESSONS

LESSONS START

https://corsi.unige.it/10728/p/studenti-orario

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

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

ASSESSMENT METHODS

The written or oral exam focuses on the learning of a few subjects from those discussed in class.

Exam schedule