MA 336 Boundary Value Problems
Instructor: David
L. Finn
Time:
MTHF 10:00-10:50 am
Prerequisites: MA221 and MA222
Are you interested in understanding the mathematical model behind
the animation of a drumhead (below)
then you should take Boundary Value Problems. Have you ever wondered
why, drum heads are not squares? Have you ever wondered what sound
would a square drum make? Yes, then you should take Boundary Value
Problems.
Have you ever been intrigued by the shape of a soap film
stretched across a glass? Or the shape of a tent being supported
by polls and rope stakes? Yes, then you should take boundary value
problems.
Have you ever watched a pot of water boiling, and wondered why
the first bubbles appear where they do? Yes, then you should take
boundary value problems.
Other examples of phenomena that will studied in boundary value problems
are
-
The motion of plucked guitar string.
-
The temperature at any point inside a pot of heated water.
-
The shape of a liquid drop resting on a table top.
The common themes that connect the mathematical models for these phenonena
are partial differential equations (PDEs) and boundary conditions,
that
is conditions given on the values the solution attains on the boundary
of the domain (the region in space) where the differential equation is
defined. For instance, in the examples given above the boundary conditions
are
-
The edge of a drum head does not move as the drum vibrates..
-
The temperature on the the surface of the pot or how the heat flows across
the pot surface.
-
The position and shape of the wire frame to which a soap film is attached.
In this course, we will examine the three main PDEs of mathematical physics
(the wave equation, the heat equation, and Laplace's equation) providing
applications of each equation and deriving the equation for at least one
application. The examples of the type of phenomena related to each
of the main PDEs are
-
The wave equation can be used to model the motion of a plucked guitar string or the motion of drum.
-
The heat equation can be used to model the temperature inside a pot of
water
-
Laplace's equation is a linearization of the equation that models the shape
of a soap film.
Time permitting we will examine other PDEs arising from models of phenomena
in fluid mechanics, geometry, electrostatics, quantum mechanics, chemical
reactions, traffic problems.
Our main tool in solving any PDE in this course will be separation of
variables and Fourier series. Once, we have a solution we will then
use animations and graphics to interpret the solution. Numerical
methods will also used to solve PDEs (i.e. numerical integration and approximations).
Questions, Comments, Queries? Send an email david.finn@rose-hulman.edu
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