A partial differential equation (PDE) is a model of some physical process whose
variation over space and time is described in terms of a formula involving
partial derivatives. PDE's can model flow past an aircraft wing, the
beating of the heart, and the variation in atmospheric ozone. When augmented
with additional information such as initial and boundary conditions and
parameter values, we can forget about the original physical problem, and
concentrate on the PDE as a purely mathematical problem.
Given such a problem, the tools of functional analysis can determine whether
a given PDE actually has a solution, whether the solution is unique, how
smoothly the solution depends on conditions and parameters, if the solution
must be bounded by a maximum value, whether the solution must always
be positive, and whether the solution conserves some energy-like quantity.
Although functional analysis can give us so much insight into the properties
of the solution of a PDE, it usually cannot actually provide us with a
formula for that solution.
Numerical analysis is an alternative approach to PDE's, which suggests
techniques for representing, computing, and evaluating approximate solutions.
There is a strong emphasis on the quality of approximation of the
solution, and of any conserved quantities, as well as the long-time
stability.
Both functional and numerical analysis express many of their results in
terms of Sobolev spaces, which also appear in differential geometry, harmonic
analysis, optimal control, engineering, mechanics, physics. The difference
is that functional analysis results will be expressed in an infinite
dimensional space, whereas numerical analysis must work in a finite
dimensional space. However, there can be a natural link between these
approaches, so that if functional analysis can express a fact about
the solution to a PDE, this can be ``translated'' into a corresponding
numerical analysis result about solutions to a discretized version
of the PDE.
One of the main points of this class is to show how a researcher can
move easily between these two points of view, connecting
the infinite and finite domains, so that the power of functional analysis
can be joined to the expressiveness of numerical analysis.
After taking this class, you will know about
Hilbert and Banach spaces, maximal monotone and accretive operators, Lax-Milgram, Brower-Minty and Hille-Yosida theorems, Sobolev spaces, elliptic, parabolic and wave boundary value problems, bifurcation theory, maximum principles, Hamiltonian principle, modified equations, stability, accuracy and error estimates of numerical approximations.
The course is self-contained.
No computing skills are required, while some computer programs will be presented as demonstrations.
On homework, you may work with other students or use library resources, but each student must write up his or her solutions independently. Copying solutions from other students will be considered cheating, and handled accordingly.
This is especially notable during this
period.
Cheating/plagiarism will not be tolerated.
Students suspected of violating the University of Pittsburgh Policy on Academic Integrity will incur a minimum sanction of a
zero score for the quiz, exam or paper in question.
Additional sanctions may be imposed, depending on the severity of the infraction.
Please note, in particular, that Pitt has a data sharing arrangement with Chegg.com that enables us to identify in- stances in which Chegg.com has been used to cheat on assessments. Consequences of being caught in this academic integrity violation have included zero scores on assessments and F grades for the course.
Lectures could be recorded by the instructor, and this may include student participation. Students are not required to participate in the recorded conversation. The recorded lecture may be used by the faculty member and the registered students only for internal class purposes and only during the term in which the course is being offered. Recorded lectures will be uploaded and shared with students through Canvas.