73488 - Complementary Principles Of Mathematical Analysis M

Course contents

Ordinary differential equations

The main theorems of the ordinary differential equations. Boundary value problems for linear ordinary differential equations of second order. Linear system of ordinary differential equations.

Basics real analysis

Sequence and series of functions. Uniform  and pointwise convergence. Gauss theorem. Lebesgue integral (basics) .

Fourier series. 

Fourier transform.

Partial differential equations.

Main definitions of the first order the characteristic method. Second order partial differential equations of linear type: a classification. The Laplace equation. The wave equation. The heat equation. The maximum principle for the Laplace operator and for the heat operator. The classical problems associated with the partial differential equations. Main properties of the harmonic functions, the maximum principle and main consequence on the uniqueness of the solution for the Dirichlet problem. Separation of variables.  The Fourier transform applied to the PDE's in unbounded domains.

 

Basics of functional analysis.

Metric spaces.  Banach spaces.  Hilbert spaces. The L^2 space (basics). Linear operators . Dual space. Orthonormal basis. 

Laplace Transform.

Distributional spaces (basics).

Readings/Bibliography

Salsa S., Partial Differential Equations in Action. From Modelling to Theory, 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 books:

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

 

Exercises

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

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

Teaching methods

The lessons will be taught at the blackboard. During such lessons the basics of functional analysis theory and the classical problems related to the ordinary differential equations as well as the partial differential equations will be introduced. In order to clarify these arguments and to help the student to understand the mathematical methods explained some lessons will be dedicated to the solutions of exercises.

Assessment methods

At the end of the course there will be a preliminary written test and an oral test. The written test, lasting two hours, is given a score of thirty. In this test, the student will have to perform some exercises and answer some questions in writing. The candidate is admitted to the successive phase (the oral phase) achieving at least 17 points out of 30. During the first part of this last phase, the questions, mainly about theory, can also be made in writing. The score of the candidates admitted to the oral test may be reduced up to 6 points or possibly increased up to 6 points based on the assessment that the Board will decide to assign to the answers given by the candidate.

Teaching tools

Exercises of the exams of previous years will be available in the   unofficial professor web site.

Links to further information

http://www.unibo.it/SitoWebDocente/default.htm?UPN=fausto.ferrari@unibo.it

Office hours

See the website of Fausto Ferrari