58527 - Mathematical Analysis L-D

Learning outcomes

To know and to be able to use some mathematical tools that are useful for the study of partial differential equations

Course contents

Distributions
Definition of distribution. Derivative of a distribution, product of a distribution and a function. Tempered distributions; Fourier transform of tempered distributions. Periodic distributions. Convolution of distributions.

Second order ordinary differential equations
Sturm-Liouville problem for selfadjoint second order differential operators. Eigenvalues and eigenvectors, ortogonality of eigenfunctions, completness of the system of eigenfunctions. Boundary problems for non-homogeneous second order differential equations, Green's function. Power series solutions of differential equations.

Bessel functions
Bessel equation; properties of Bessel functions, asymptotic behaviour.

Parabolic partial differential equations
Heat equation in one space variable. Solution of Cauchy-Dirichlet problem by separation of variables; solution of Cauchy problem by Fourier transform.

Hyperbolic partial differential equations
Wave equation in one space variable. Solution of Cauchy-Dirichlet problem by separation of variables; solution of Cauchy problem by Fourier transform.

Elliptic partial differential equations
Laplace equation in two variables; solution of Dirichlet problem and of Neumann problem by separation of variables; fundamental solution, Green's function. Maximum principle for harmonic functions.
Helmholtz equation in two variables; Dirichlet problem in a rectangle and in a circle, eigenvalues and eigenfunctions; Dirichlet problem in the exterior of a circle.

Readings/Bibliography

G. C. Barozzi: Matematica per l'ingegneria dell'informazione; Ed. Zanichelli

Teaching methods

Lectures and exercises in the classroom.

Assessment methods

Preliminary written examination and oral examination.

Links to further information

http://www.dm.unibo.it/~dore/Analisi_L-D/index.html

Office hours

See the website of Giovanni Dore