
INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
Prerequisites:

This course will be completely selfcontained, and can be profitably followed by any student who has had a good exposure to the fundamentals of calculus of one and several variables. Some of these fundamentals will be recalled in detail during the lectures. 
Examination methods:

The students will be provided with take home written exams of increasing level of difficulty. By taking these exams each student pledges that he/she will work on the test without communicating with any of his/her classmates or
anybody else. Each student is only allowed to discuss the exam with Prof. Garofalo.
Infringement of these rules will be considered academic cheating
and adversely affect the final grade in this course. 
Course unit contents:

Partial differential equations (PDEs) are expressions involving an unknown function of two or more variables and a certain number of its partial derivatives. Such equations govern the phenomena of the physical world, and they play a preeminent role both in pure mathematics and in the applied sciences:
1. The small vibrations of the string of a violin are described by the wave equation, a PDE that is ubiquitous in the description of undulatory phenomena.
2. The potential of the gravitational field generated by a certain distribution of mass satisfies (away from the mass itself) a PDE that is known as Laplace equation.
3. The distribution of temperature in a conducting body is described (at least near the source) by yet another PDE known as the heat equation. These are instances of PDEs of linear type.
The principal aim of this course is to bring the audience to mastering some of the basic aspects of PDEs, beginning with the linear models described above. The second part of the course will be devoted to providing the audience with a glimpse into some of the fascinating aspects of nonlinear PDEs. 

