## Matematica per l'ingegneria

__Curriculum in __

__MATEMATICA PER L'INGEGNERIA /MATHEMATICS FOR ENGINEERING__

**Title: Representations and Number Theory**

**Instructors: Stefano Capparelli, Pietro Mercuri **

**Duration: 20 hours**

**Period: Second semester**

**Abstract: **In this course we will examine recent litarature that links representation theory of Lie algebras and Number Theory. In particular we shall examine:

*- Nandi,D: Partition identities arising from Standard A _{2}^{(2)}-modules of level 4, Rutgers PhD Thesis 2014*

*- Konan, I.: Identités de type Rogers-Ramanujan: preuves bijectives et approche à la théorie de Lie, PhD Thesis, Université de Paris, 2020.*

-* Capparelli, Meurman, Primc,Primc: New partition identities from C*_{l}^{(1)}*-modules, arXiv:2106.06262*

**Title: Variational problems for nonlinear Schrodinger equations on metric graphs**

**Instructor: Simone Dovetta **

**Duration: 15 hours**

**Period: January-Febraury 2022**

**Abstract: **Since their first appearance in physical chemistry in 1953, networks (or metric graphs) have been proposed to model almost one-dimensional ramified structures. Despite being more than sixty years old, it is within the last two decades that the theory of evolution on networks became popular, mainly driven by the ubiquity of networks in applications, from quantum mechanics to fluid dynamics, from nonlinear optics to traffic regulation. The aim of this course is to give an introductory overview of recent results for nonlinear Schrödinger equations on metric graphs. In particular, we will consider variational problems for the energy functional under the mass constraint on a given graph The focus will be set on the minimization problem: does the above energy admit global minimizers? What is the role of We will see that the answers to these questions are strongly sensitive to the specific properties of the graphs.

**Title: Fractional Calculus and Probability (1 ^{st }part)**

**Instructor: Mirko D'Ovidio, Raffaela Capitanelli **

**Duration: 15+5 hours**

**Period: Part 1: January 2022, Part 2: February 2022**

For information on the course, fill the form

https://forms.gle/k91mGmEVECFgs7CK6

**Abstract Part 1:** We introduce some basic aspects related to fractional calculus and probability. In particular, we consider time-changed Markov processes driven by Cauchy problems written in terms of non-local (time and/or space) operators. Moreover, we consider the boundary value problem in the framework of the time changes and in general, we show some connection between multiplicative functionals, semigroups and boundary value problems for PDEs and non-local PDEs. We introduce some basic notions about limit theorems, stochastic processes, PDEs connections, non-local operators with special attention for the case of fractional (Caputo) derivative and fractional Laplacian, additive and multiplicative functionals associated with boundary conditions, time changes associated with boundary conditions. We focus on Dirichlet, Neumann, Robin and Wentzell boundary conditions together with the probabilistic reading of killed, reflected, elastic and sticky processes. We also discuss some applications concerned with regular and irregular domains in the macroscopic analysis introduced by fractional equations.

**Abstract Part 2: **The aim of this module is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes.

**Title: Introduction to hyperbolic conservation laws and applications**

**Instructor: Elisa Iacomini (Aachen University)**

**Duration: 10 hours**

**Period: February ****7, 8, 11, 14, 16, h. 10-12, room 1.B1, building RM002**

**Abstract:** This course offers an overview of the theory of hyperbolic conservation laws and the related numerical methods. Conservation laws are essential for understanding the physical world around us, indeed they describe the conservation in time of a quantity in an isolated system. Starting from the linear case, we generalize the theory of characteristics analyzing the Riemann problem in details. Due to the nonlinearity of the problem, we will deal with the study of weak solutions, face the problem of non uniqueness of solutions and end up with the concept of entropy solution.

Then, we will focus on the numerical counterpart, in particular we will consider the so-called shock-capturing methods. We will investigate their main properties as consistency, stability and convergence.

As modelling example, we will focus mainly on vehicular traffic flow models.

**Title: Fractional Calculus and Probability (2 ^{nd} Part)**

**Instructors: Maria Rosaria Lancia, Anna Chiara Lai .**

**Duration: 4+4 hours**

**Period: March-May 2022**

**Abstract Part 1: **In this module we introduce the regional fractional Laplacian in extension domains, possibly with an irregular boundary. A fractional Green formula will be proved. We discuss some evolution BVPs with either Dirichlet, or Neumann or Robin type boundary conditions, possibly non local. Existence, uniqueness of the weak solution will be discussed as well as regularity properties of the associated semigroup. A comparison with the different definitions of fractional Laplacian will also be addressed.

References:

*- S.Creo, M.R.Lancia, P.Vernole, Convergence of fractional diffusion processes on extension domains, J. Evol. Equ. 20 (2020), 109 − 139.*

*- S.Creo, M.R.Lancia, Fractional (s,p)-Robin-Venttsel' problems on extension*

*domains, NoDEA Nonlinear Di erential Equations Appl., 28 (3), (2021), paper no. 31, 33pp. *

*- C.G. Gal, M. Warma, Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions. Evol. Equat. Control Theory 5, 61-103 (2016).*

**Abstract Part 2: **This module presents some results in the framework of fractional compartmental epidemic models, with particular attention to Caputo-type fractional SIS models.

We begin with a brief introduction to some compartmental epidemic models and with the derivation of the SIS model in the ordinary case. We then focus on the fractional SIS models based on the fractional Caputo derivative. Assuming the conservation of the total population, we discuss the equilibria of the system and we prove the existence and uniqueness of the solution. We finally present a local series representation for the solution.** **

References:

*- Balzotti, C.; D’Ovidio, M.; Loreti, P. Fractional SIS Epidemic Models. Fractal Fract. 2020, 4, 44.*

*- Diethelm, K., Ford, N. J. Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications, 2002, 265(2), 229-248.*

*-El-Saka H.A.A. Backward bifurcation in fractional-order vaccination models. J. Egy. Math Soc. 2015;23(1):49–55.*

**- **Hassouna M, Ouhadan A, El Kinani EH. On the solution of fractional order SIS epidemic model. Chaos Solitons Fractals. 2018 Dec;117:168-174.

**Title: Spectral geometry of the Laplace operator
Instructors: Luigi Provenzano, Alessandro Savo
Duration: 12 hours
Period: Second Semester**

**Abstract: **In the first part of the course, we will provide a brief introduction to the spectrum of the Laplacian with various boundary conditions, and to the spectrum of the Dirichlet-to-Neumann map. This includes a discussion of the functional setting of these problems, and a few basic examples. Then, we will focus on some classical problems in spectral geometry, such as eigenvalue bounds and isoperimetric inequalities for the eigenvalues. In particular, we will present a set of techniques for their study. In the final part, if time allows, we will consider the same kind of problems for the Laplacian with a magnetic field, and we will discuss some recent developments on this topic.

**Title: Inverse problems and time-fractional partial differential equations
Instructor: Masahiro Yamamoto (Tokyo University)
Duration: 16 hours
Period: May/June 2022**

**Abstract:**We consider an initial boundary value problem for time-fractional diffusion-wave equation:

The idea of fractional derivatives dates back to Leibnizand there have been many works including by Abel, Riemann, Liouville. Now the system (*) is widely recognized as more feasible model for various phenomena such as anomalous diffusion in heterogeneous media, where the anomaly cannot be well interpreted by the classical advection. diffusion equation and the conventional models often provide wrong simulation results. Thus we have to exploit more relevant models because the issues are serious, for example, for the protection of the environments , and the mathematical researches should support such practical applications.

For researches on inverse problems, we need also mathematical analyses for (*). Usually in practice, we are not a priori given coefficients and other quantities in (*). The inverse problems are concerned with parameter identification, and are essential for more accurate prediction or simulations of anomalous diffusion. Therefore mathematical researches should be done for both foundations of direct problems and applications to inverse problems for (*).

__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**