Ing Ind - Inf (Mag.)(ord. 270) - BV (478) NUCLEAR ENGINEERING - INGEGNERIA NUCLEARE
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096240 - MATHEMATICAL METHODS FOR MATERIALS ENGINEERING
Ing Ind - Inf (Mag.)(ord. 270) - MI (491) MATERIALS ENGINEERING AND NANOTECHNOLOGY - INGEGNERIA DEI MATERIALI E DELLE NANOTECNOLOGIE
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096240 - MATHEMATICAL METHODS FOR MATERIALS ENGINEERING
Obiettivi dell'insegnamento
The aim of the course is to provide a basic understanding of some elements of functional analysis and some relevant examples of their application to the theory of partial differential equations (of elliptic, parabolic or hyperbolic type), in particular related to physical problems and mathematical modeling.
Risultati di apprendimento attesi
DD1: Knowledge and understanding - DD3: Making judgments
The students will have to know and understand some basic elements in functional analysis
DD2: Applying knowledge and understanding - DD3: Making judgments
The students will have to apply their knowledge to different kinds of differential problems, attacked through the mathematical methods learned in the first part of the course.
Argomenti trattati
Part I. Elements of functional analysis.
Norms and Banach spaces.
Spaces of continuous functions.
Lebesgue integral.
Spaces of summable and bounded functions.
Fourier transform in L ^ 1
Hilbert spaces.
Fourier series in Hilbert spaces.
Riesz and Lax Milgram's theorems.
Distributions.
Sobolev spaces.
Part II. Applications to partial differential equations
Parabolic equations: solution of the Cauchy problem for the heat equation through Fourier transform.
Hyperbolic equations: solution of the Cauchy problem for the wave equation through Fourier transform.
Elliptic equations: variational formulation of boundary value problems for the Poisson equation; fundamental solution for Laplace equation; Green's representation formulas for Dirichlet and Neumann problems.
Prerequisiti
The student is expected to have a solid knowledge of the program of the courses in Mathematical Analysis I and II.
Modalità di valutazione
The exams consists in a (written) test, divided into a theoretical and a practical parts.
The theoretical and the practical parts contain respectively questions and exercises on the topics of the course.
Both parts will contribute in a substantial way to the determination of the final mark (the amount of points for each correct answer will be specified on the text).
During the evaluation it is not possible to make use of books, phones, tablets or any other electronic devices.
Bibliografia
Sandro Salsa, Partial Differential Equations in Action, Editore: Springer Universitext
Filippo 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 libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
Inglese