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.

Students with certification of Specific Learning Disabilities (SLD), disabilities, or other special educational needs are advised to contact the instructor at the beginning of the course to agree on teaching methods and examination procedures that, while respecting the teaching objectives, take into account individual learning styles and provide suitable compensatory tools.

SYLLABUS/CONTENT

Course Introduction

Part One: Review of Numerical Series and Series of Functions

• Numerical Series: definitions, convergence, basic properties, examples.
• Convergence criteria for series with non-negative terms and absolute convergence. Leibniz criterion for alternating series.
• Series of Functions: definition, pointwise and uniform convergence, examples.
• Conditions for exchanging limits, derivatives, and integrals with series. Absolute convergence.

Part Two: Fourier Series and Fourier Transform

• Genesis of Fourier Series: solving the heat equation.
• Fourier Series: definition and basic properties. Criteria for pointwise, uniform, and total convergence of Fourier series.
• Term-by-term differentiability. Examples. Fourier Series of even and odd functions. Complex exponential form.
• L^2 and L^1 spaces and associated metrics. Inner product in L^2, norm, distance, orthonormal bases. Parseval's identity, L^2 convergence of Fourier series.
• Fourier Transform in L^1: definition, basic properties, examples. Fourier Transform of the Gaussian function. Convolution product: definition and properties. Inverse Fourier Transform. Fourier Transform in L^2. Plancherel's theorem.

Part Three: Applications to the Solution of Partial Differential Equations

• Well-posed problems: Hadamard's definition.
• Heat equation: introduction and well-posed problems. Solving the homogeneous Dirichlet problem with Fourier series and the non-homogeneous problem.
• Heat equation: uniqueness using the energy method. Stability properties and the maximum principle.
• Heat equation: global Cauchy problem and solution using Fourier transform.
• Wave equation: introduction and well-posed problems. Solving the global Cauchy problem with d'Alembert's formula.
• Waves: solving the homogeneous Dirichlet problem with Fourier series. Uniqueness with energy estimates.
• Laplace/Poisson equations: introduction and well-posed problems. Existence theorem for the Dirichlet and Neumann problems.
• Harmonic functions: definition, mean value property, maximum principle, uniqueness, and stability.
• Solving the Laplace equation in the disk using Poisson's formula. Other properties of harmonic functions.

RECOMMENDED READING/BIBLIOGRAPHY

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

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

There will be two midterm exams, one in the middle and one at the end of the semester. They will be two-hour written exams, with two or three exercises on the topics already covered in the syllabus. Those who pass both exams with a score above 15 in each and an overall average of at least 18 can directly record their grade without an oral exam. The oral exam is optional for those who want to try to improve their grade. A selection of course topics will be requested during the oral exam, which will be defined at the end of the semester and made available on Aulaweb.

In the regular exams, there will be a two-hour written exam, with two or three exercises on solving partial differential equations using Fourier series or Fourier transform methods. The oral exam is optional.

Rules for the written exam. Students are allowed to use handwritten notes of any kind. Electronic devices (calculators, phones, computers) are prohibited, except for tablets, which can be used solely to consult notes.

Students with certification of Specific Learning Disabilities (SLD), disabilities, or other special educational needs are advised to contact the instructor at the beginning of the course to agree on teaching methods and examination procedures that, while respecting the teaching objectives, take into account individual learning styles and provide suitable compensatory tools.

ASSESSMENT METHODS

The written exam will assess the ability to solve partial differential equations using Fourier series or Fourier transform methods. Clarity and organization of the presentation, correctness of the solutions, quality, and rigor of the arguments and deductions will be evaluated. The oral exam will assess more specific knowledge on a selection of theoretical topics covered in the course. The ability to present and the correctness of logical-deductive arguments will be evaluated.

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.