Associate Professor of Mathematics
Rose-Hulman Institute of Technology

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