Course 2024-2025 a.y.

1. Where PDEs Come From:
   - First-Order Linear Equations and transport
   - ODE's and characteristics
   - Flows, and Diffusions
   - Models for collective behaviour and social interactions
   - Initial and Boundary Conditions
   - Well-Posed Problems
    
2. Diffusions:
   - The Diffusion Equation
   - The Black-Scholes equation
   - Generative diffusion and simulated annealing
   - The Fokker-Planck equation
    
3. Reflections and Sources:
   - Diffusion on the Half-Line
   - Diffusion with a source
   - The Duhamel principle
    
4. Boundary Problems:
   - Separation of Variables
   - The Dirichlet and Neumann Boundary Conditions
   - Application of Fourier Series, orthogonality
5. Harmonic Functions:
   - Laplace’s Equation
   ​​​​​​​- Rectangles and Cubes
   ​​​​​​​- Poisson’s Formula
   - Fundamental solutions
   ​​​​​​​
6. Computation of Solutions:
   ​​​​​​​- Approximations of Diffusions and of Laplace's Equation
   ​​​​​​​- Finite differences and finite Element Methods

• Recoginize the main types of PDEs and know their basic features, in particular in the linear case (superposition principles, conservation/dissipation of energy/entropy, characteristics, diffusion and wave propagation).
• Recognize some of the links between PDEs and problems/models they describe (with an emphasis on financial models).
• Understand the role of boundary conditions.
• Know the basic properties of harmonic functions and of the eigenvalues-eigenfunctions of the Laplace operator.
• Know some distiguished class of special solutions.
• Know the main approximation schemes used for the difference classes of equations and their structural properties.

• Compute some explicit solutions of simple PDEs by the main methods: characteristics, separation of variables, Fourier series.
• Compute the eigenvalues and the eigenfunctions of simple operators in special geometry.
• Represent solutions of linear equations by convolution.
• Apply simple estimates involving basic principles. 
• Apply basic computational algorithm to approximate simple PDEs.

• Lectures
• Practical Exercises

Some exercises will be solved during the lectures, with the aim of applying the main theoretical results to different problems.

The assessment methods are the same for attending and non-attending students.

Students can choose one of the following two methods to be evaluated:

1. Method A: There will be three written individual assignments along the course. Each of them will count 20% of the final grade. There will also be an oral presentation of an in-depth collaborative work. The oral presentation will count 40% of the final grade. 
2. Method B: There will be a single written general exam, which covers the whole syllabus of the course and that counts 60% of the final grade, followed by an oral individual exam that counts 40% of the final grade.

The oral presentation and the oral exam mainly aim at evaluating: 

• The knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises.
• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.

The written individual assignments and the written general exam mainly aim at evaluating:

• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

• W.A. Strauss. Partial Differential Equations. An introduction. Wiley, 2022
• D. Logan. Applied Partial Differential Equations. Springer, 2014
• S. Salsa. Partial Differential Equations in Action. Springer, 2015
• S. Salsa, G. Verzini. Partial Differential Equations in Action. Complements and Exercises. Springer, 2015