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 reasonbable 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 enphasis on the elementary and fundamentally geometric aspects of analyticity.
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.
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 functios of one complex variable is also expected.
Calculus of functions of one and several real variables, linear algebra
Blackboard and computer illustrations
Fourier series for periodic functions and Fourier transform on R; main properties and applications to finding solutions of boundary value problems for the classial 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.
Class notes available online
Ricevimento: Weekly office hours will be communicated. Meetings upon email requests will also be considered.
FILIPPO DE MARI CASARETO DAL VERME (President)
MATTEO SANTACESARIA
GIOVANNI ALBERTI (President Substitute)
Written and oral examination
Students are required to work on standard problems in series expansions, Fourier transforms, appliactions to boundary value problems for classical PDE and basic properties of analytic functions.
