73305 - Complementary Principles Of Mathematical Analysis M

Learning outcomes

The course will enable the students to learn the basic mathematical tools concerning with Fourier series and Fourier and Laplace transforms (via the knowledge of some basic functional and complex Analysis); applications to ODEs and PDEs will also be considered.

Course contents

Background material:

A brief review of the main theorems on Ordinary Differential Equations. A short review of complex numbers.

Real Analysis elements:

Sequances and series of functions. Pointwise and uniform convergence. Integral of Lebesgue (hints).

Functional Analysis elements:

Metric spaces. Banach spaces. Hilbert Spaces. Space L^2 (hints). Orthonormal bases.

Fourier Series:

Main definitions; Fourier series convergence theorems for a periodic function.

Holomorphic function elements:

The Cauchy Theorem; Laurent series; the Residue Theorem and the calculation of some integrals using the Residue Theorem.

Laplace transform:

Main definitions concerning the Laplace transform; Convergence domain. Calculation rules. Antitransform. Application to Linear ODEs.

Fourier Transform:

Main definitions relating to the Fourier transform of an L^1 function (and references to the L^2 case). Calculation rules. Antitransform.

Applications to PDEs (if time is left):

Linear first order PDEs and tha method of characteristics. Classification of equations for linear partial derivatives of the second order. Generalities and examples of PDEs: Laplace equation, wave equation and heat equation. Applications of the Fourier transform to selected PDEs.

Readings/Bibliography

For the exercises:
See the pdfs published on the teacher's website (amsCampus section); if the student carefully solves these exercises, he/she will not need further texts.

For the theory: it is enough that the students regularly follows ALL frontal lessons given in the classroom, and that he/she studies the theory on the notes taken during the lectures. The adopted textbook is an optional tool.

It is strongly recommended that non-attending students do obtain the notes taken by some regularly attending students. This will allow the non-attending student to save time and effort in preparing the written exam. Obviously, it is -however- a full right of the non-attending student to prepare the exam by also using the suggested texts.

Some reference texts are:

S. Salsa, Equazioni a Derivate Parziali, Springer

C.D. Pagani, S. Salsa, Serie di Funzioni ed equazioni differenziali (estratto da Analisi Matematica 2), Zanichelli

Fritz John, Partial differential equations, Springer

G.C. Barozzi, Matematica per l'informazione, Zanichelli

Other texts:

Mathematical Analysis, T.A. Apostol , Addison-Wesley Pubblishing Company

Method of Applied Mathematics with a MATLAB Overview, J. H. Davis, Birkhauser

Partial Differential Equations, V.P. Mikhailov, MIR Publishers

Partial Differential Equations for Scientist and Engineers, S.J. Farlow, Pubblications

Partial Differential Equations, L. C. Evans, GSM 19 of American Mathematical Society

Analisi di Fourier, M.R. Speigel, ETAS Libri

Equazioni a Derivate Parziali, S. Salsa. G. Verzini, Springer

Teaching methods

The course includes theoretical lessons (in which the first elements of complex and functional analysis, the Laplace and Fourier transforms and the Fourier series) will be introduced, together with exercises aimed to help the students gain familiarity with the tools and mathematical methods introduced during the lessons.

Assessment methods

ONLY written examination.

The written examination will check the knowledge of the topics presented in the exercises, regularly published on the AMScampus site; in the written part, some questions may be posed on theoretical topics as well.

Dates:

3 exams in January/February

1 in June: 1 in July, 1 in September

PLEASE, use Almaesami to book for the examination dates!

Teaching tools

See the dedicated AMScampus site.

Office hours

See the website of Andrea Bonfiglioli