
Course unit
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
SCP3050960, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2018/19
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/05 
Mathematical Analysis 
8.0 
Course unit organization
Period 
First semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Examination board
Board 
From 
To 
Members of the board 
6 Introduzione alle Equazioni alle Derivate Parziali  a.a. 2018/2019 
01/10/2018 
30/09/2019 
ANCONA
FABIO
(Presidente)
ROSSI
FRANCESCO
(Membro Effettivo)
MARSON
ANDREA
(Supplente)
MONTI
ROBERTO
(Supplente)
SORAVIA
PIERPAOLO
(Supplente)

Prerequisites:

Differential and integral calculus.
Elementary theory of ordinary differential equations.
Basic theory of complex analysis (functions of complex variables, holomorphic and analytic functions). 
Target skills and knowledge:

Basic notions of the theory of linear partial differential equations. It's a fundamental course suggested to students with interests both in pure and in applied mathematics, and in particular to students with a curriculum in analysis. 
Examination methods:

The exam consists of a final oral examination on the topics treated in class. There will be both theoretical questions and the discussion of some exercise to solve. 
Assessment criteria:

The evaluation criteria will be the following:
 coherence and rigor in the exposure of statements and theorems
 thoroughness and adherence to the topics of discussion
 ability to use the acquired knowledge to solve problems and exercises. 
Course unit contents:

Didactic plan:
 First order PDEs: transport equation with constant coefficients, conservation laws (classical and weak solutions, RankineHugoniot conditions, Riemann problem).
 Wave equation: existence of solutions, D'alembert formnula, method of spherical means, Duhamel's principle, uniqueness, finite speed of propagation.
 Laplace equation: fundamental solution, harmonic functions and main properties, mean value formulas, Liouville's Theorem, Harnack's inequality, maximum principle. Poisson equation. Green's function and Poisson's representation formula of solutions. Basic notions of the theory of distributions.
Weak solutions of Laplace equations on bounded domains are harmonic functions.
 Heat equation: fundamental solution, existence of solutions for the Cauchy problem and representation formula. Uniqueness and stability of solutions. Mean value formulas, maximum principle, Hopf's maximum principle. 
Planned learning activities and teaching methods:

The methodology of teaching used will be the traditional lesson. 
Textbooks (and optional supplementary readings) 

Salsa, Sandro, Partial differential equations in actionfrom modelling to theorySandro Salsa. Cham [etc.]: Springer, 2015.

L.C. Evans, Partial Differential Equations, 2nd edition. Providence, Rhode Island: American Mathematical Society, 2010.

W. A. Strauss, Partial Differential Equations. An Introduction. New York: Wiley, 1992.


