INTRO TO PDE

Spring 2018，MIT Mathematics

Offce: 2-265

Offce hours: Tuesday 1:30-2:30

附：教材电子版。
Required Text: Partial Differential Equations in Action, 3rd Edition by Sandro Salsa
Course Description
The two primary goals of many pure and applied scientic disciplines can be summarized as
follows: i) formulate/devise a collection of mathematical laws (i.e., equations) that model the
phenomena of interest; ii) analyze solutions to these equations in order to extract information and
make predictions. The end result of i) is often a system of partial dierential equations (PDEs).
Thus, ii) often entails the analysis of a system of PDEs. This course will provide an application-
motivated introduction to some fundamental aspects of both i) and ii).
In order to provide a broad overview of PDEs, our introduction to i) will touch upon a diverse
array of equations including a) the Laplace and Poisson equations of electrostatics; b) the diusion
equation, which models e.g. the spreading out of heat energy and chemical diusion processes;
c) the Schrodinger equation, which governs the evolution of quantum-mechanical wave functions;
d) the wave equation, which models e.g. the propagation of sound waves in the linear acoustical
approximation; e) the Maxwell equations of electrodynamics; and f) other topics as time permits.
In our introduction to ii), we will study three important classes of PDEs that dier markedly
in their quantitative and qualitative properties: elliptic, diusive, and hyperbolic. In each case,
we will discuss some fundamental analytical tools that will allow us to probe the nature of the
corresponding solutions.