## Matematica per l'ingegneria

__Curriculum in __

__MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING__

__PhD courses__

**Title: Partial Differential Equations of Parabolic Type**
**Instructor: Daniele Andreucci**
**Duration: 24 hours**
**Period: March-April 2021**
Abstract:** **Initial value and boundary problem for the heat equation and other,
also non-linear, equations of diffusion type.
We'll cover basic existence results, analysis of the asymptotic behavior for
large times, other qualitative properties of the solutions.

**Title: Lie algebras, vertex operators, and integer partitions.**

**Instructor: Stefano Capparelli**

**Duration: 15 hours**

**Period: Second semester**

web page of the course:

https://classroom.google.com/c/ODY4MDYzNDk1Njda?cjc=qz3vpwn

Abstract: This is a continuation of the course from last year.This course will be an introduction to the theory of vertex operators and the integer partition theory. The first part of

the course will develop integer partition theory and identities of the Rogers-Ramanujan type. The second part will deal with the relatively new concept of vertex operator starting from Lie algebra representations. Finally, we will show the connection between the two theories that at first appear to be very distant. The level of the course will be elementary/introductive, and I will try to keep it essentially self-contained.

**Title: Fractional Calculus and Probability**

**Instructor: Mirko D'Ovidio**

**Duration: 10 hours**

**Period: February-March**

Abstract: We introduce some basic aspects related to fractional calculus and probability. In particular, given a Markov process X and the random time T, we consider the time-changed process X(T) and show that the Cauchy problem for X(T) is written in terms of fractional operators (time and/or space non-local operators). Moreover, we introduce the boundary value problem in the framework of the time changes. That is, we show the connection between multiplicative functionals, semigroups and boundary conditions

Timetable

monday february 22 9:00-11:00

https://meet.google.com/avr-ryer-xrd

**Title: Introduction to fractals and boundary control problems in irregular domains)**

**Instructors: Anna Chiara Lai, Maria Rosaria Lancia, Raffaela Capitanelli**

**Duration: 3 CFU+3 CFU+3 CFU**

**Period: Second semester**

Abstract:

First module (Prof. Anna Chiara Lai)

This module presents an introduction to fractals and their mathematical foundations. Motivated by examples and applications, we introduce box counting dimensions, Hausdorff measure and dimension and some methods for their computation. In the second part of the module we focus on particular self-similar fractal structures: we introduce the Iterated Function Systems, we investigate the dimension of the related self-similar sets and we present a dynamical system perspective for this framework. The final part of the module is devoted to the implementation, with Wolfram Mathematica software, of simple visualization and fractal dimension algorithms.

Second module (Maria Rosaria Lancia)

PDES on fractal domains possibly with fractal interfaces.

It is crucial to introduce the trace spaces of Sobolev spaces on fractal sets. ù

We consider boundary value problems with dynamical boundary conditions these are

the most general boundary conditions which include, Dirichlet, Neumann or Robin bcs.

We will look for:

2) variational solutions for elliptic problems.

3) regularity properties,

4) asymptotic behaviour

5) numerical approximation by FEM

6) parabolic BVPs

Third module (Raffaela Capitanelli)

Nonlinear Analysis on Fractal Structures

This module presents an introduction to some nonlinear problems on fractal structures.

1 We present existence, uniqueness and approximation results for variational solutions for some quasilinear obstacle problems on domains with a fractal boundary. Our mail tools are suitable extension theorems, sharp quantitative trace results (on polygonal curves) in terms of the increasing numbers of sides and Poincaré type estimates adapted to the geometry.

2 In order to study mass transport problem on fractal structures, we study the limit of p-Laplace type problems with obstacles as p tends to infinity.

3 Finally, we present some similar asymptotic results on fractal domains like the Sierpinski gasket.

**Title: Introduction to the non local equations **
**Instructor: Maria Medina (Universidad Autonoma de Madrid)**
**Duration: 12 hours**
**Period: January**
Abstract:
1. The fractional Laplacian.
- Definitions, probabilistic motivation, limits in s.
- Basic properties: maximum principle, example of harmonic function.
2. Fractional Sobolev spaces.
- Sobolev inequality, compactness theorem, energy formulation.
- Problems in bounded domains. Neumann type boundary conditions.
3. Harmonic extension.
**Title: Sobolev Spaces and Applications to Partial Differential Equations,**
**Instructor: Prof. A. F. Tedeev (South Mathematical Institute of VSC RAS )**
**Duration: 3CFU**
**Planned period: Spring 2021**
Abstract:This course will cover the basic theory of Sobolev Spaces
in the Euclidean Space and in Riemannian manifolds. Some more advanced
topics will be also introduced, when necessary for the applications,
which will refer to problems in unbounded domains of the Euclideanspace
orin non-compact Riemannian manifolds.The proposed
duration of the course is 2 months, with 16 hours of lectures per month.
1) Sobolev spaces. Basic theory. Approximation by smooth functions.
Results of compactness. Embeddings. Poincaré and Hardy inequali-ties.
Isoperimetric and Faber-Krahn inequalities.
Interplay betweenthe geometry of the manifold and the embedding results.
2) Linear and non-linear diffusion equations; the concept ofsolutionsand
the variety of possible behaviors. The energy method. Regulariza-tion and
iterative techniques.
3) Asymptotics for large times: classical results in the Euclidean space.
The asymptotic profile in linear and nonlinear diffusion.
The case of the Neumann problem in subdomains of the Euclidean space.
Asymptotic behavior in manifolds.

__List of useful courses from the Master degree:__

**Title: Metodi Numerici per l’Ingegneria Biomedica (4 CFU) (Corso di Laurea in Ingegneria Biomedica) **

**Instructor: Francesca Pitolli**

**Planned period: November-December 2020**