Laurea magistrale in Matematica

The aim of the course is to discuss a number of mathematical models yielding  the fundamental PDEs studied in other courses.  Attention will be given to problem arising from Physics, Finance, Biology and Computer graphics. Further, we will try to highlight the common features of the models, showing that mathematical modeling has a deductive and rigorous nature. We will focus on models of evolutionary phenomena.

Further, we will present some classical methods of solution for linear Ist and IInd order PDEs, in order to be able to investigate, in simple cases, the qualitative behavior of solutions and its relation with the mathematical structure of the corresponding PDE.  This is fundamental in order to analyze the results of more sophisticated analytical and numerical solution techniques.

Basic knowledge of the theory of partial differential equations from the classical point of view, as a basis for their numerical analysis. Knowledge of the most important models leading to PDEs in applied mathematics.

1) Conservation laws.
1.1) Conservation of mass. Flux and source terms. Advection, convection and diffusion. A crash course in continuum mechanics.
1.2) Ist order equations.
- the method of characteristics for the linear and quasilinear case;
- shock and rarefaction waves
- Burgers equation (with examples from fluid dynamics).
- the model of Lighthill-Witham-Richardson for highway traffic.
- the Lotka-Volterra- Von Foerster model for the growth of structured populations - the Gurtin- McCamy generalization;
- linear hyperbolic systems
-  quasilinear hyperbolic systems: shallow waters
- the Riemann problem for the shallow water equation: the Riemann solver.

1.3)  Parabolic equations.
- the heat equation, physical meaning of the boundary conditions (Neumann, Dirichlet,  Robin).
- solution techniques: the heat kernel, separation of variables and Fourier series. Examples. Ill and well-posed problems.
- the Black-Scholes equation; explicit solution for European Call Options

1.4) Reaction-diffusion systems and equations
- reaction-diffusion equations  and their origin in different contexts
- dependence from the domain size
- pattern formation in chemical and biological systems: Turing instability

2)Conservation of momentum and hyperbolic equations
2.1) The wave equation.
- elementary techniques for the solution of the wave equations: the D'Alembert formula, separation of variables and Fourier series.
- characteristics.
- vibrating string and membranes: harmonics.

3))Stochastic models: applications of partial differential equations to stochastic processes.

- Laplace equation- Green functions; and the harmonic measure.
- Applications to the computation of the expected first exit time and the first exit probability for the Brownian motion.

4) Some integral equations: Fredholm and Volterra.


5) Differential equations in computer graphics

- the eikonal equation: applications to shape offsetting and shape from shading

- diffusion filters: Perona Malik and backward parabolic filters.