2016/2017

2016/2017

DM270

MAT/05 (ANALISI MATEMATICA)

DIPARTIMENTO DI INGEGNERIA INDUSTRIALE E DELL'INFORMAZIONE

ELECTRONIC ENGINEERING

PERCORSO COMUNE

1°

Primo Semestre (26/09/2016 - 13/01/2017)

9

76 ore di attività frontale

ENGLISH

SCRITTO E ORALE CONGIUNTI

ROCCA ELISABETTA (titolare) - 9 CFU

Differential and integral calculus, complex functions, sequence and series of functions, linear algebra, differential operators, power and Fourier series, Laplace and Fourier transforms for classical signals, linear differential equations with constant coefficients.

The course is an introduction to some basic elements of linear functional analysis (Hilbert spaces and distributions), variational principles, ordinary differential equations and dynamical systems, with simple applications to basic partial differential equations.

Ordinary differential equations

Basic definitions, examples and properties
Existence and uniqueness, comparison
Linear systems, exponential matrix, Liouville Theorem

Basic tools of functional analysis

Functional spaces, norms and Hilbert spaces
Best approximation and projection theorem, orthonormal basis
Linear operators: boundedness and continuity, symmetry,
self-adjointness, eigenvalues and eigenfunctions. Sturm-Liouville
Problems.
Applications to simple PDE's

Partial differential equations

Examples and modelling
Wave equations, D'Alembert formula, characteristics and
boundary value problems, spherical waves, solutions in two
and three dimensions
The Laplace and heat equations
Simple techniques for calculating explicit solutions; separation of
variables.

Distributions

Introduction, examples and applications.
Operating on distributions: sum, products, shift, rescaling, derivatives.
Sequence and series of distributions: Fourier series.
Fourier transform, temeperate distributions, convolutions

Lectures (hours/year in lecture theatre): 54
Practical class (hours/year in lecture theatre): 22
Practicals / Workshops (hours/year in lecture theatre): 0

Ordinary Differential Equations and Systems
E.A. Coddington, An Introduction to Ordinary Differential Equations, Dover Publications, Inc., New York, 1961.
M.W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York, 1974.
V.V. Nemytskii and V.V. Stepanov, Qualitative Theory of Differential Equations, Dover Publications, Inc., New York, 1989.
W.T. Reid, Sturmian Theory for Ordinary Differential Equations, Applied Mathematics Series 31, Springer-Verlag, New York Heidelberg Berlin, 1980.

Basic Tools of Functional Analysis
B. D. Reddy, Introductory Functional Analysis, Texts in Applied Mathematics n. 27, Springer Verlag, New York, (1998).
W. Rudin, Functional Analysis, Mc Graw Hill, New York, (1973).
W. Rudin, Real and Complex Analysis, Mc Graw Hill, New York, (1966).

Distributions
E. DiBenedetto, Real Analysis, Birkhauser, Boston, (2002): Chapter VII.
F.G. Friedlander, Introduction to the theory of distributions, Cambridge University Press, Cambridge, (1998).
S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Springer-Verlag Italia, (2008): Chapter 7.

Partial Differential Equations
E. DiBenedetto, Partial Differential Equations, 2nd Edition, Birkhaüser, (2009): Chapter 6.
S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, Springer-Verlag Italia, (2008): Chapter 5.
W. Strauss. Partial Differential Equations: an introduction. Wiley.

Written and oral examination

A more detailed description of the course can be found on the web page at the URL
http://matematica.unipv.it/rocca/