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Scheda Riassuntiva
Anno Accademico 2018/2019
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 096240 - MATHEMATICAL METHODS FOR MATERIALS ENGINEERING
Docente Arioli Gianni
Cfu 5.00 Tipo insegnamento Monodisciplinare

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - BV (478) NUCLEAR ENGINEERING - INGEGNERIA NUCLEARE*AZZZZ096240 - MATHEMATICAL METHODS FOR MATERIALS ENGINEERING
Ing Ind - Inf (Mag.)(ord. 270) - MI (491) MATERIALS ENGINEERING AND NANOTECHNOLOGY - INGEGNERIA DEI MATERIALI E DELLE NANOTECNOLOGIE*AZZZZ096240 - MATHEMATICAL METHODS FOR MATERIALS ENGINEERING

Obiettivi dell'insegnamento

The aim of the course is to provide a basic understanding of the theory of linear partial differential equations, their application to physical problems and mathematical modeling. In particular, the concept of well posed problem and more generally the relation between theory and applications is emphasised.


Risultati di apprendimento attesi

The lectures and the laboratories will provide students with:

i) knowledge and understanding of
* the main properties of the solutions of some classes of linear partial differential equations;
* the basic tools used for solving linear partial differential equations;
* modeling of some physical problems

ii) the ability to apply the previous knowledge to simple examples on the calculator. In particular the student is required to
* solve linear partial differential equations
* discuss the properties of mathematical models

The instructor expects a broad comprehension of the subjects which sould not be limited to the statement of theoretical results.
Instead, the acquired knowledge should enable students to express
critical judgment and make informed choices on analytical methods for partial differential equations. The students are expected to express their answers in a mathematically rigorous and clear way.


Argomenti trattati

Prerequisites: Exercises on Fourier series and linear ordinary differential equations.

Introduction: Mathematical modelling. Examples of partial differential equations. Well posed problems.

Conservation Laws: Transport equation. Traffic flow models. Method of characteristics. Rankine-Hugoniot relation. Shock and rarefaction waves. Entropy condition. 

Diffusion: Heat equation. Well-posed problems. Separation of variables. Maximum principles. Fundamental solution. Duhamel principle.

The Laplace and Poisson Equations: Harmonic functions. Mean value properties. Maximum principles. Poisson’s formula for the disk. Newtonian potentials. 

Waves and Vibrations: Wave equation. Well-posed problems and separation of variables. D’Alembert formula.

 


Prerequisiti

We recommend that students who attend this course have knowledge of linear algebra, calculus and numerical analysis, in particular:

* differential calculus for functions of several real variables;
* surfaces and integral surfaces, the divergence theorem;
* series of functions, Fourier series;
* linear second order ordinary differential equations;
* numerical solution of linear systems (by direct and iterative methods);
* polynomial interpolation;
* basic numerical methods for the approximation of ordinary
* differential equations;


Modalità di valutazione

The exam is written and consists in two questions on the theory (6 points each) and two exercises (10 points each). A positive evaluation requires at least 6 points on the theory, 10 points on the exercises and a total of 18.


Bibliografia
Risorsa bibliografica facoltativaS. Salsa, F. Vegni, A. Zaretti, P. Zunino, A Primer on PDEs: Models, Methods, Simulations, Editore: Springer, Anno edizione: 2013, ISBN: 978-88-470-2861-6
Risorsa bibliografica facoltativaFilippo Gazzola, Franco Tomarelli, Maurizio Zanotti, Analytic functions, integral transforms, differential equations, Editore: Esculapio, Anno edizione: 2015, ISBN: 978-88-7488-889-4

Software utilizzato
Nessun software richiesto

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
30:00
45:00
Esercitazione
20:00
30:00
Laboratorio Informatico
0:00
0:00
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 50:00 75:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
schedaincarico v. 1.8.0 / 1.8.0
Area Servizi ICT
05/12/2022