First order PDE: characteristics..
Wave equation: one-dimensional problems. D'Alambert solution
Heat equation: Fundamental solution, energy methods
Laplace equation: maximum principle, mean value formula, Perron's method.
Lecture notes and exercises available on the course web page.
Learning Objectives
Knowledge acquired:
Introduction to mathematical modelling of natural phenomena described by partial differential equations and their treatment by means of advanced mathematical tools.
Competence acquired:
Knowledge of the classical solution of PDE
Skills acquired (at the end of the course):
To be able to set the basic mathematical frame for simple continuous systems and to discuss its consequences.
Prerequisites
Courses required: Mathematical Analysis I, Mathematical Analysis II, Geometry I.
Courses recommended: Physics I, Dynamical Systems
Teaching Methods
CFU: 6
Total hours of the course: 130
Hours reserved to private study and other indivual formative activities: 82
Office hours:
See professors’ web page
http://www.dma.unifi.it/~minguzzi/DidatticaMain.html
Type of Assessment
Oral examination. The students are asked to illustrate some proofs given in class. They might also be asked to apply the theorical results to simple exercises in order to verify the acquisition of the new concepts.
Course program
First order equations: characteristics, local solutions via characteristics, conservation laws, shock solutions.
Wave equations:one-dimensional equation, Cauchy problem, D'Alambert's solution. Fourier's method. Three-dimensional equation, Kirchoff's formula.
Heat equation: maximum principle, Dirichlet problem in a bounded domain, Cauchy problem, Poisson's formula. Existence and uniqueness of the solution.
Laplace equation, maximum principle, Dirichlet problems, Neumann problems. Hopf's Lemma. Mean value formula, harmonic functions, Liouville's Theorem.