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Lecturer:
Giuseppe Savarè
Course name: Modelli e metodi matematici I
Course code: 064071
Degree course: Ingegneria Elettronica, Ingegneria Informatica
Disciplinary field of science: MAT/05
L'insegnamento costituisce attività di base per: Ingegneria Elettronica e delle Telecomunicazioni, Ingegneria Informatica
University credits: CFU 5
Course website: n.d.
Specific course objectives
Learn the elementary facts of Hilbert spaces with applications to orthogonal decompositions of signals and the basic elements of the theory of dynamical systems. Introduce a few examples of linear partial differential equations.
Course programme
Ordinary differential equations
- Basic techniques to solve simple scalar equations
- Existence and uniqueness of solutions
- Solutions to linear systems of ODE, exponential of a matrix, Jordan decomposition
- Asymptotic behavior of solutions to linear systems; classification in two spatial dimensions
- Stationary points of nonlinear systems, stability
Elements of Hilbert spaces
- Functional spaces, norms and scalar products, L^2
- Hilbert spaces, orthogonal decompositions, best approximation, Gram-Shmidt orthonormalization, projection theorem.
- Applications: least squares, Fourier series, eigenvalues and eigenvectors of selfadjoint linear operators, orthogonal polynomials, Haar and sinc basis
Introduction to partial differential equations
- Transport equations, the methods of characteristics
- Linear scalar conservation laws
- The wave equation in one dimension
- Boundary value problems, separation of variables, Laplace equation
- Wave equations in three dimensions
Course entry requirements
All the mathematical notions developed in the first three years of degree.
Course structure and teaching
Lectures (hours/year in lecture theatre): 30
Practical class (hours/year in lecture theatre): 15
Practicals / Workshops (hours/year in lecture theatre): 0
Project work (hours/year in lecture theatre): 0
Suggested reading materials
M.W. Hirsch, S. Smale. Differential equations, dynamical systems, and linear algebra. Academic press, 1974.
S. Salsa. Equazioni a derivate parziali : metodi, modelli e applicazioni . Springer (Unitext), 2004..
U. Gianazza. Dispense. Disponibili online
http://www.imati.cnr.it/~gianazza/dispmodellimetodi1.html.
Testing and exams
Written and oral examination
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