COURSE DESCRIPTION
The two primary goals of many pure and applied scientific 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 differential 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 investigation of
i) will touch upon a diverse array of equations including a) the Laplace and Poisson equations of electrostatics; b) the diffusion equation, which models e.g. the spreading out of heat energy and chemical diffusion processes; c) the Schrödinger 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 other topics as time permits.
In our introduction to ii), we will study three important classes of PDEs that differ markedly in their qualitative and quantitative properties: elliptic, diffusive, and hyperbolic. In each case, we will discuss some fundamental analytical tools that will allow us to probe the nature of the corresponding solutions.
Prerequisites: 18.100; and one of 18.06, 18.700, or 18.701
Textbook: Partial Differential Equations in Action by Sandro Salsa (3rd Edition)
RECENT UPDATE
[02.06.2018] Welcome to the start of the semester! Our first lecture falls on this date.
