• MOOC

Partial and ordinary differential equations are the main tool used to describe and study structures. This course introduces the main techniques used for understanding and solving differential equations, with emphasis on the applications. The main tools that are considered are functions series and basic functional analysis.

Students who pass the exam in differential equation can:
- know the basics of the theory of function series, and the application to the study of differential equations.
- know the different types of differential equations and the corresponding techniques

Part A
Sequences and series of functions: pointwise and uniform convergence, mean square convergence, convergence criteria.
Theorems regarding the computation of limits, derivatives and integrals of sequences and series.
Power series, Taylor series: radius of convergence. Abel's theorem. Regularity of a power series.
Exponential series, Euler's formula.
Fourier series: criteria for pointwise, uniform and mean square convergence. Parseval identity.

Ordinary differential equations.
Cauchy problem, local and global existence of solutions, dependence on the data. Well posedness. Qualitative study of differential equations.
Equations explicitly solvable (separation of variables, linear, Bernoulli, homogeneous, linear of second order).
Linear systems: superposition principle, wronskian matrix. Set of solutions of a system of differential equations. Systems with constant coefficients. Exponential of a matrix. Boundary value problems, eigenvalues and eigensolutions. The concept of stability and criteria for linear and nonlinear systems. 

Mod. B:
Functional analysis: scalar product, Hilbert spaces, L2, generalised Fourier expansions, Parseval identity.

Partial differential equations:
First order equations, solution with characteristic lines.
Classification of second order equations. The heat equation, the wave equation, Laplace's equation. The mean value principle and the maximum principle for elliptic equations. Range of influence and domain of dependence for the wave equation.

The program of the courses in Mathematical Analysis 1 and Geometry and Mathematical Analysis 2.

1) The exam is written: it consists of exercises and questions on the theory (both closed and open questions).
2) 12 points are assigned to the questions on the theory, 23 to the exercises. To pass the exam it is necessary to get at least 6 points on the theory and 12 on the exercises.
3) Students getting at least 32 points are graded with 30 lode

The exam checks that the students:
1)    understand the applications of differential equations.
2)    are able to explain and illustrate (in written form) a definition, a theorem, a proof.
3)    are able to solve exercises.