Stampa
Enrollment year
2021/2022
2021/2022
Regulations
DM270
MAT/05 (MATHEMATICAL ANALYSIS)
Department
DEPARTMENT OF ELECTRICAL,COMPUTER AND BIOMEDICAL ENGINEERING
Course
ELECTRONIC ENGINEERING
Curriculum
PERCORSO COMUNE
Year of study
Period
1st semester (27/09/2021 - 21/01/2022)
ECTS
9
Lesson hours
72 lesson hours
Language
English
Activity type
WRITTEN AND ORAL TEST
Teacher
ROCCA ELISABETTA (titolare) - 9 ECTS
ASPRI ANDREA - 0 ECTS
Prerequisites
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.
Learning outcomes
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.
Course contents
Ordinary differential equations

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

Basic tools of functional analysis

Lebesgue integral.
Functional spaces, norms and Hilbert spaces
Best approximation and projection theorem, orthonormal basis
Linear operators: boundedness and continuity, symmetry,
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
Teaching methods
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.
Assessment methods
Written and oral examination. The written examination consists of exercises on the topics of the course. The minimum score to pass the written examination is 18/30. Oral examination will follow the written one for the students who got at least 18/30. The oral examination will be based on questions aiming at understanding which are the concepts acquired by the student. The minimum score to pass the exam is 18/30, the maximum score is 30/30 cum laude. In case of online exams (due to Covid-19 emergency), the exam could be replaced by only oral exam. The rules will be communicated in due time by the professor.
Further information
A more detailed description of the course can be found on the web page at the URL
http://matematica.unipv.it/rocca/
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