## OVERVIEW

The course aims to provide general mathematical techniques for the implementation of a mathematical model, for its formalization, and for the study of its behavior.

## AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide students with an overview of the basic mathematical methods used for the solution and the qualitative study of certain types of ordinary and partial differential equations of interest in engineering. At the end of the course, the student acquires the ability to study the behavior of complex systems through the formulation of a simplified mathematical model capable of describing and predict the salient features of the phenomenon.

AIMS AND LEARNING OUTCOMES

The course introduces the use of differential equations for modelling of physical phenomena. We will introduce mathematical techniques for the construction of a differential mathematical model, its formalization, and, by means of appropriate mathematical methods, the analysis of its qualitative (and sometimes quantitative) behaviour. Natural phenomena will be scrutinised under the magnifying glass of rigorous mathematical analysis. By the end of the course, we will introduce and study several examples and applications of engineering interest (e.g., traffic flow, diffusion of a pollutant, population dynamics, heat conduction, dynamics of electrical circuits). Armed with mathematical methods, we will then either obtain explicit solutions or analyse qualitatively these phenomena, highlighting their properties and their emergent behaviours.

TEACHING METHODS

Traditional lectures, with both theory and exercises in class. Attendance (and active participation) in the course is strongly recommended.

SYLLABUS/CONTENT

Introduction to mathematical modelling: aspects of the modelling process; representations scales; dimensional analysis.

Ordinary differential equations (ODEs): ODEs classification; mathematical statement of ODEs problems; qualitative analysis of dynamical systems; regular and singular perturbation methods; introduction to the problem of bifurcation.

Partial differential equations (PDEs): elementary models of mathematical physics (wave propagation, thermal diffusion); analytic methods for linear problems; discretization of continuous models.

RECOMMENDED READING/BIBLIOGRAPHY

N.Bellomo, E. De Angelis, M. Delitala, Lecture Notes on Mathematical Modelling From Applied Sciences to Complex Systems, SIMAI Notes 2010

S Strogatz, Nolinear Dynamics and Chaos, CRC Press 2018

S Farlow, Partial Differential Equations for Scientists and Engineers, Dover 1982

E Beltrami, Mathematics for Dynamic Modeling, Academic Press 1987

Further references will be suggested, time by time, during the course

## TEACHERS AND EXAM BOARD

**Office hours:** Savona: Monday 4pm - 6pm
Genova: Friday 10am - 12am
It is strongly recommended to book in advance via email: vincenzo.vitagliano@unige.it

Exam Board

VINCENZO VITAGLIANO (President)

CLAUDIO CARMELI

## LESSONS

TEACHING METHODS

Traditional lectures, with both theory and exercises in class. Attendance (and active participation) in the course is strongly recommended.

LESSONS START

Class schedule

## EXAMS

EXAM DESCRIPTION

Students can choose among the following possibilities:

1) **traditional examination**: It consists of a written test and an oral examination. The written test (open questions and problems) has a duration of three hours and the oral one a maximum of 30 minutes. A student is admitted to the oral exam only if s/he has achieved at least 50% of the maximum score (15 marks).

2) **writing a report**: It consists in writing an essay on a topic of interest for the student (to be agreed in advance with the lecturer). It should be structured as follows: explanation of the process/phenomenon under consideration, equations modeling it, explanation and interpretation of the solution. Once the essay has been written, it must be submitted to the teacher in due time. If the report is approved, students must prepare a set of slides and present them. During the presentation, the students are required to prove their understanding of the topic by answering questions.

ASSESSMENT METHODS

The exam verifies the student's ability to write the equations that model simple phenomena, to set the solution and to analyze the salient qualitative aspects.

Exam schedule

Date | Time | Location | Type | Notes |
---|---|---|---|---|

14/01/2022 | 10:00 | GENOVA | Scritto + Orale | |

04/02/2022 | 10:00 | GENOVA | Scritto + Orale | |

09/06/2022 | 10:00 | GENOVA | Scritto + Orale | |

08/07/2022 | 10:00 | GENOVA | Scritto + Orale | |

08/09/2022 | 10:00 | GENOVA | Scritto + Orale |