Lecturer: Giuseppe Savarè  

Course name: Mathematical models and methods
Course code: 502949
Degree course: Ingegneria Elettronica
Disciplinary field of science: MAT/05
University credits: CFU 9
Course website: http://www.imati.cnr.it/~savare/didattica/modellimetodi

Specific course objectives

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 (Laplace, wave and transport).

Course programme

Ordinary differential equations

• Basic definitions, examples and properties
• Existence and uniqueness, comparison
• Linear systems, exponential matrix, Liouvill Theorem
• Asymptotic behaviour and stability of dynamical systems, Lyapunov functions (✵)

Basic tools of functional analysis

• Functional spaces, norms and Hilbert spaces
• Best approximation and projection theorem, orthonormal basis and applications to signal theory
• Linear operators: boundedness and continuity, symmetry, self-adjointness, eigenvalues and eigenfunctions. Applications to simple PDE's (✵)

Partial differential equations

• Examples and modelling
• Linear transport equation and scalar conservation laws
• Wave equations, D'Alambert formula, characteristics and boundary value problems, spherical waves and solutions in three dimensions
• Laplace equations, variational principles
• Simple techniques for calculating explicit solutions; separation of variables.
• A few examples of nonlinear equations (✵)

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
• Discrete signals and distributions, sampling theorem, filters and difference equations, Laplace and Z transform (✵)

Course entry requirements

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

Course structure and teaching

Lectures (hours/year in lecture theatre): 50
Practical class (hours/year in lecture theatre): 30
Practicals / Workshops (hours/year in lecture theatre): 0

Suggested reading materials

M.W.Hirsch, S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, 1974. .

H. Ricardo. A modern introduction to differential equations. Elsevier.

S. Salsa. Partial Differential Equations in Action. Springer.

C. Gasquet, P. Witomski. Fourier Analysis and Applications. Filtering, Numerical Computation, Wavelets. Springer.

W. Strauss. Partial Differential Equations: an introduction. Wiley.

Titolo del riferimento da modificare.

Testing and exams

Written and oral examination