Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position.
For example, the heat equation can be used to describe the change in heat distribution along a metal rod over time. PDEs arise as part of the mathematical modeling of problems connected to different branches of science, such as physics, biology, and chemistry. In these fields, experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. These connections must be exploited to find an explicit way of calculating the unknown quantity, given the values of the independent variables – that is, to derive certain laws of nature. While we do not know why partial differential equations provide what has been termed the “unreasonable effectiveness of mathematics in the natural sciences” (the title of a 1960 paper by physicist Eugene Wigner[1]), they provide the foundation for a robust and important field concerned with applied mathematics.
A very large fraction of solvable PDEs are either linear first- or second-order PDEs, or are related to such PDEs by transformation or perturbation theory. Fortunately, these PDEs also make up the language for much of the mathematical description of nature. Most of this class will concentrate on those equations whose tremendous importance to real-world applications has been established.
Because methods for finding exact or approximate solutions to partial differential equations tend to be rather specialized, it is important to be able to classify these equations. Accordingly, notation, specialized terminology, and the classification scheme for partial differential equations will constitute the subject of Units 1 and 2. Subsequent units will examine some major solution methods: Fourier series and the Fourier transform, separation of variables, the method of characteristics, and impulse-response methods.
This course is a way of dipping your toes into the vast pool that is analysis and solution of PDEs, a place where some people spend their whole lives.
Course Requirements:
Have completed the following courses: Multivariable Calculus, Differential Equations, Linear Algebra II, and Real Analysis I. Complex Analysis provides useful background, but is not required.
