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).


Programme:

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. 

Distributions:

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.