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Risorse bibliografiche
Risorsa bibliografica obbligatoria
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Scheda Riassuntiva
Anno Accademico 2015/2016
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 096119 - ADVANCED MATHEMATICAL ANALYSIS
Docente Tomarelli Franco
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) - MI (422) INGEGNERIA DELLA PREVENZIONE E DELLA SICUREZZA NELL'INDUSTRIA DI PROCESSO*AZZZZ096119 - ADVANCED MATHEMATICAL ANALYSIS
Ing Ind - Inf (Mag.)(ord. 270) - MI (472) CHEMICAL ENGINEERING - INGEGNERIA CHIMICA*AZZZZ096119 - ADVANCED MATHEMATICAL ANALYSIS

Programma dettagliato e risultati di apprendimento attesi

Aims and Scope - Problem solving for ordinary and partial differential equations.

Prerequisites - Multiple integrals. Integrals on surfaces. Divergence Theorem. Fourier series of periodic functions. Functions of complex variable. Cauchy-Lipschitz Theorem. Linear second order ordinary differential equations. Euler's equation. 

PROGRAM

Introduction to partial differential equations - Mathematical modeling. Examples of partial differential equations. Well posed problems. Boundary value problems: Dirichlet and Neuman boundary conditions. Initial value problems. Classification of second order linear PDE. 

Conservation Laws - Pollution in a channel. Linear transport equation. Distributed source. Decay and localized source. Characteristics. Inflow and outflow. First order quasi-linear equations. 

Diffusion - The diffusion equation. Heat conduction. Well posed problems. Separation of variables. Maximum principle and uniqueness. The fundamental solution. The Dirac distribution. The Cauchy problem for the heat equation. Energy estimates. An example of nonlinear diffusion: the porous medium equation.

Transforms - Fourier transform. Laplace transform. Fourier series of periodic signals. Function spaces: C^k, L^1, L^2. Distributions. Convolution. Transform-based methods for solving differential equations with initial and/or energy conditions.

The Laplace Equation - Well posed problems and uniqueness. Harmonic functions and their properties. Separation of variables. Eigenvalue problems. The fundamental solution: Newtonian potential. Solution of Laplace equation in R^n.

Waves and Vibrations - Types of waves. Transverse waves in a string. The one dimensional wave equation. D’Alembert formula. Domain of dependence and range of influence. Huygens principle. Energy conservation.

 


Note Sulla Modalità di valutazione

The final (written) exam consists in solving exercises and answering questions related with the topics of the course.

 

 


Bibliografia
Risorsa bibliografica obbligatoriaFilippo Gazzola, Franco Tomarelli & Maurizio Zanotti, Analytic functions, Integral transforms, Differential equations - Second Edition, Editore: Esculapio, Anno edizione: 2015, ISBN: 978-88-7488-889-4
Note:

English text is also available as an E-book. A printed version in Italian language is available too.


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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
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schedaincarico v. 1.6.8 / 1.6.8
Area Servizi ICT
22/09/2021