Math 112 – Partial Differential Equations
Partial Differential Equations (PDEs for short) come up in most parts
of mathematics and in most sciences. For instance, complex analysis is the
study of the Cauchy-Riemann equations
ux=vy, uy=−vx.(1)
Another example is the recent resolution of the celebrated Poincar´e conjec-
ture in topology, which uses diffusion-type PDEs to analyze the singularities
of surfaces. Brownian motion is a random process that is described by a PDE.
Other examples of PDEs occur in the flow of fluids, diffusion of chemicals,
conduction of neural impulses along an axon, radiation of electromagnetic
waves, quantum mechanics, spread of heat, propagation of sound, spread of
epidemics, etc. PDE is a vast subject.
Math 112 is an introductory course in the subject. All you really have to
know is calculus and linear algebra. The key prerequisite is several-variable
calculus. However, it is not one of the easier 100-level courses, so it is recom-
mended (but not required) that you take at least one other 100-level course
before taking Math 112. Very good preparatory courses would be Math 101
(Introduction to Analysis), 126 (Complex Analysis), 113 (Real Analysis) or
111 (Ordinary Differential Equations). Math 111 is not a prerequisite.
A few PDEs are really easy to solve, such as
ut+ux= 0,(2)
which has the general solution u(x, t) = f(x−t) for any function fof one
variable. But most of them are difficult or notoriously difficult or impos-
sible to solve. (It’s something like indefinite integrals, most of which, like
Rexp(x2)dx, do not have explicit solutions.)
In Math 112 we will study the most important equations that are in
the easy-to-difficult category. The notoriously difficult ones are subjects of
current mathematical research or computational experimentation. Most of
the ones studied in this course have explicit solutions in the form of fairly
complicated formulas involving integrals or infinite series. Just about all
PDEs have infinitely many solutions, often an infinite-dimensional space of
solutions, as in the very simple example above. After a while, one realizes
that one has to rethink what one means by “solving” a PDE. Many PDEs
have no solution formula at all. Instead, one has to find properties of the
solutions.
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