# MODELS OF PHYSICAL MATHEMATICS

## Learning outcomes of the course unit

Aim of the course is to describe the fundamental tools for the qualitative analysis of differential equations.

At the end, the student will be able to apply such tools to the formulation and the study of simple mathematical models.

The student will acquire the knowledge of foundations of Mathematical Physics, with a deep understanding of the basic applications of mathematical methods to the study of physical problems. Moreover, the student must become able to read and understand advanced text of Mathematical Physics.

Applying knowledge and understanding: the student must become able to produce formal proofs of results of Classical Mechanics and Mathematical Physics, and to expose, analyze and solve simple problems of Classical Mechanics with a clear mathematical formulation.

Moreover, the student will be able to apply computational tools to the mathematical modeling.

Making judgements: the student must become able to construct, develop and apply theoretical reasoning in the context of Mathematical Physics, with a deep ability to distinguish correct and wrong assumptions and methods.

Communication skills: the student must acquire the correct terminology and language of Mathematical Physics and the ability to expose their results and techniques to an audience, in both cases of qualified and unqualified audience.

Learning skills: the student must become able to autonomously continue the study of Mathematical Physics and in general to complete his preparation in Mathematics or in other scientific fields with an open minded approach, and must become able to gain knowledge from specialized text and journals.

## Prerequisites

Basic calculus of the first year courses; mandatory propedeuticities: Mathematical Analysis 1, Geometry 1A.

## Course contents summary

Introduction to mathematical modelling through differential equations. The first part of the lectures is relevant to the Liapunov’s stability theory for systems of ordinary differential equations, with applications to mathematical models in Mechanics, Population Dynamics and Epidemiology. The second part deals with the equations of Mathematical Physics (heat, wave, Laplace equations) and more generally with the basic theory of second order PDE’s.

## Course contents

FIRST PART:

Dynamical Systems. Equilibria and Stability. Lyapunov Methods.

Linear and nonlinear models in Mechanics.

Mathematical models in Population Dynamics.

Van der Pol equation.

Bifurcation theory, Hopf theorem, limit cycles.

Poincarè-Bendixon theorem.

Lorenz system and chaos.

Discrete dynamical systems. Feigenbaum map.

SECOND PART:

Sturm Liouville problems. Eigenvalues and eigenfunctions.

Introduction to Distributions Theory.

Non-homogeneous boundary problems and Green's function.

Classification of linear second order PDE's. Cauchy problems.

First order quasi-linear PDEs. Method of characteristics.

## Recommended readings

G.L. CARAFFINI, M. IORI, G. SPIGA, Proprietà elementari dei sistemi

dinamici, Appunti per il corso di Meccanica Razionale, UNIVERSITA' DEGLI

STUDI DI PARMA, a.a 1998-99;

G. BORGIOLI, Modelli Matematici di evoluzione ed equazioni differenziali,

Quaderni di Matematica per le Scienze Applicate/2, CELID, TORINO, 1996;

R. RIGANTI, Biforcazioni e Caos nei modelli matematici delle Scienze

applicate, LEVROTTO & BELLA TORINO, 2000;

M.W HIRSCH, S. SMALE, Differential Equations, Dynamical Systems and

Linear Algebra, ACADEMIC PRESS, NEW YORK, 1974;

J. GUCKENHEIMER, P. HOLMES, Nonlinear Oscillations, Dynamical Systems

and Bifurcations of Vectors Fields, SPRINGER-VERLAG, NEW YORK, 1983;

M. SQUASSINA, S. ZUCCHER, Introduzione all'analisi qualitativa delle

equazioni differenziali ordinarie (ebook), APOGEO, 2008.

## Teaching methods

Lectures and exercises; laboratory of Matlab numerical simulation.

## Assessment methods and criteria

Oral exam and discussion of a project about a specific mathematical model in Applied Sciences.