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## MATHEMATICAL ANALYSIS 4

CODE 86902 2022/2023 6 cfu during the 1st year of 8738 INGEGNERIA NAVALE (LM-34) - GENOVA MAT/05 Italian GENOVA 1° Semester AULAWEB

## OVERVIEW

The course focuses on Fourier analysis methods as applied to the solutions of boundary value problems for classical partial differential equations. Thus, a mathematical presentation of Fourier series and transforms is presented, combining a reasonable amount of formal precision with applications to explicit problems, to be solved with workable formulae. The basic facts about analytic functions of one complex variable are also introduced because of their pervasive use in applications, with particular emphasis on the elementary and fundamentally geometric aspects of analyticity.

## AIMS AND CONTENT

### LEARNING OUTCOMES

The main objective is to achieve a solid basic operative knowledge of Fourier analysis techniques (Fourier series and Fourier transform) for functions of one real variable as applied to boundary value problems for the classical partial differential equations (heat, Poisson, waves), and to understand the main properties of analytic functions of one complex variable.

### AIMS AND LEARNING OUTCOMES

Students are expected to master the  basic Fourier analysis techniques (series and transforms) that are needed in order to solve standard boundary value problems for classical partial differential equations (heat, Laplace-Poisson, waves), both using series expansions and integral formulae. Basic operative knowledge concerning analytic functions of one complex variable is also expected.

### PREREQUISITES

Calculus of functions of one and several real variables, linear algebra

### TEACHING METHODS

Blackboard and computer illustrations

### SYLLABUS/CONTENT

Fourier series for periodic functions and Fourier transform on R; main properties and applications to finding solutions of boundary value problems for the classical PDE, essentially through separation of variables techniques or via Fourier transform methods. The notion of holomorphic map is introduced and the main properties of analytic functions are investigated.

S. Salsa - Partial differential equations in action: from modelling to theory - Springer 2016

Marco Codegone e Luca Lussardi,  Metodi Matematici per l'Ingegneria, Zanichelli, 2021.

## TEACHERS AND EXAM BOARD

### Exam Board

MATTEO SANTACESARIA (President)

FILIPPO DE MARI CASARETO DAL VERME (President Substitute)

ERNESTO DE VITO (President Substitute)

## LESSONS

### LESSONS START

The class will start according to the academic calendar.

### Class schedule

All class schedules are posted on the EasyAcademy portal.

## EXAMS

### EXAM DESCRIPTION

Written and (optional) oral examination

### ASSESSMENT METHODS

Students are required to work on standard problems in series expansions, Fourier transforms, applications to boundary value problems for classical PDE and basic properties of analytic functions.

### Exam schedule

Date Time Location Type Notes
27/01/2023 09:30 GENOVA Scritto
10/02/2023 09:30 GENOVA Scritto
16/06/2023 09:30 GENOVA Scritto
07/07/2023 09:30 GENOVA Scritto
11/09/2023 09:30 GENOVA Scritto

### FURTHER INFORMATION

Students with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.