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.
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.
G.SPIGA, Problemi matematici della Fisica e dell'Ingegneria, PITAGORA,
Bologna;
A.N.TICHONOV, A.A. SAMARSKIJ, , Equazioni della Fisica Matematica, MIR,
Mosca;
F.G.TRICOMI,Equazioni differenziali,EINAUDI, Torino;
F.G.TRICOMI, Istituzioni di Analisi Superiore, CEDAM,Padova.
Teaching methods
Lectures and exercises; lab of numerical simulation in Matlab.
Assessment methods and criteria
Oral exam and discussion of a project about a specific mathematical model in Applied Sciences.
