The two primary goals of many pure and applied scientific disciplines can be summarized as follows:
- Formulate/devise a collection of mathematical laws (i.e., equations) that model the phenomena of interest.
- 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 introduction to i) will touch upon a diverse array of equations including
- The Laplace and Poisson equations of electrostatics;
- The diffusion equation, which models e.g. the spreading out of heat energy and chemical diffusion processes;
- The Schrödinger equation, which governs the evolution of quantum-mechanical wave functions;
- The wave equation, which models e.g. the propagation of sound waves in the linear acoustical approximation;
- 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 quantitative and qualitative 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.