|Instructor: Elisabetta Rocca
Office: C4, Mathematical Department
Diary of lessons
Office hours: appointments by e-mail.
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).
Ordinary differential equations:
Basic definitions, examples and properties. First order linear equations and separation of variable method.
The Cauchy problem. Existence and uniqueness: the Peano's theorem, the Cauchy-Lipschitz theorem.
Linear systems, exponential matrix, higher linear orders ODEs with constant coefficients. Boundary problems.
The Bernoulli and homogeneous equations. Qualitative study of solutions of Cauchy problems.
Asymptotic behaviour and stability of dynamical systems. Examples. The linearization method.
Basic tools of functional analysis:
Basic tools for Lebesgue integration. Convergence properties.
Functional spaces, norms and Hilbert spaces.
Best approximation and projection theorem, orthonormal basis.
Linear operators: boundedness and continuity, symmetry, self-adjointness, eigenvalues and eigenfunctions. Applications to simple PDE's.
The Sturm-Liouville operator.
Introduction, examples and applications.
Operating on distributions: sum, products, shift, rescaling, derivatives.
Sequence and series of distributions: Fourier series.
Fourier transform, temperate distributions, convolutions.
Discrete signals and distributions.
Partial differential equations:
Examples and modelling.
Wave equations in 1 and 2D. The D'Alambert formula, characteristics and boundary value problems.
The method of separation of variables and the resolution by Fourier transform.
Uniqueness for the 3D wave equation. Stability properties.
Simple techniques for calculating explicit solutions; separation of variables.
Introduction to the heat equation. The separation of varianle methods for the associated Cauchy-Dirichlet boundary value problem.
Uniqueness by energy methods.
Suggested Reading Material:
M.W. Hirsch, S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, 1974.
C.D. Pagani, S. Salsa, Analisi Matematica, Volume 2, Zanichelli, 2006 (Italian).
M. Bramanti - Metodi di Analisi Matematica per l'Ingegneria, Esculapio (Italian).
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.
L. Perko, Differential Equations and Dynamical Systems, Springer.
Notes of Fabio Bagagiolo on ODEs.
The final exam consists in a written test and an oral exam.
At the written test the students will be allowed to bring all the texts they want,
but not text of exercises, and hand-written notes or exercises.
Written Test of January 24, 2020
Written Test of February 13, 2020
Results (only the sufficient ones) of February 13,2020
The vision of the written tests and the oral exam
(for the students who got a sufficient evaluation - score bigger or equal to 18)
will take place on February 27, at 9.30, in room Beltrami at the gound floor of the Mathematical Department.
For students who have another exam on that day/time: please send me an e-mail as soon as possible.
Dates of the tests of 2019/2020
Written tests 2018/2019:
18.03.2019 (appelo straordinario)
Written tests 2017/2018:
Written tests 2016/2017:
Previous years exams and material