# MATHEMATICAL PHYSICS

## Learning outcomes of the course unit

We expect that at the end of the course the students will have to following abilities:

- to know useful tools to face problems of Mathematical Physics;

- to be able to present in a clear way, with a correct language from the mathematical and physical points of view, the main arguments of the course;

- to be able to solve different kinds of partial differential equations appearing in problems of Mathematical Physics, owing to methods studied in the course for the classical equations.

Specifically, the expected skills are:

- Knowledge and understanding: students will know in detail and will autonomously use mathematical tools in the frame of Mathematical Physics; furthermore they will acquire a good level of understanding of the most recent mathematical contents and theories on the course topics, which enables them to read and understand advanced texts and research articles, and to elaborate then original ideas in specific research contexts.

- Applying knowledge and understanding: students will be able to produce rigorous proofs of (even new) mathematical results and to face new problems in the frame of equations of Mathematical Physics, proposing new models and studying their properties owing also to both analytical and numerical methods.

- Making judgements: students will be able to build up non trivial logical proofs, and will be capable to recognize correct or rather fallacious steps of the proofs.

- Communication: students will have to present the main topics of the course in a clear and mathematically correct way.

- Lifelong learning skills: the course will help students to form a flexible mentality allowing them to be employed in work environments that require the ability to face ever-new problems, or to continue their studies in a second level Master or in a Ph.D. programme in Italy or abroad.

## Prerequisites

Knowledge of topics of mathematical courses of class L-35 (first degree).

## Course contents summary

We can describe the course as divided into three parts. Il corso può pensarsi suddiviso in tre parti.

1) Functions of a complex variable.

Fourier and Laplace transforms.

2) Green function and Sturm-Liouville problems for linear differential second order operators.

Quasi-linear first order PDEs.

Classification of second order PDEs.

3) Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation.

Conservation laws.

## Course contents

Functions of a complex variable: Cauchy-Riemann conditions, types of singularities, residues and integration, Laurent series.

Fourier and Laplace transforms: definitions and basic properties, transforms of the fundamental functions, transform of Dirac delta, inverse transforms.

Introduction to the Green function and to Sturm-Liouville problems for linear differential second order operators.

Quasi-linear first order PDEs.

Classification of linear second order PDEs with two independent variables; Cauchy problem.

Fundamental equations of mathematical physics: Laplace equation, heat equation, wave equation (physical derivation, mathematical properties, methods of solution).

Conservation laws.

## Recommended readings

F. Gazzola, F. Tomarelli, M. Zanotti, Analisi complessa - Trasformate - Equazioni Differenziali, Esculapio, Milano.

S. Salsa, Equazioni a derivate parziali, Springer, Milano.

G. Spiga, Problemi matematici della Fisica e dell'Ingegneria, Pitagora, Bologna.

A. N. Tichonov, A. A. Samarskij, Equazioni della fisica matematica, MIR, Mosca.

## Teaching methods

During class lectures, the topics will be proposed from a formal point of view, and equipped with meaningful examples and applications.

Video of all lectures and pdf files of the slides written during the lectures will be uploaded on the web-site Elly.

## Assessment methods and criteria

The examination is based on an oral discussion, where the level of knowledge and understanding of the topics is valued, as well as the mathematical accuracy of exposition.

Typically, the oral exam consists in three questions, one for each of the three parts of the course described in the section "Contents"