Courses:

The purpose of the final project is to get each of you something interesting to think about on differential equations, to learn theories/methods for yourself or with your fellow classmates, and to exercise communicating with other people by a mathematical writing.

You will study about your own topic for about two weeks and submit a report of about five pages by the last class session. Although the project is optional, if you choose to do one, it will be graded. By completing the project work, a student may earn up to 5% in the final grade. You are encouraged to work in group.

The project report must be written in a manner that a fellow classmate in 18.034 could understand what you have done and reproduce the work on the basis of what you have written.

Some sample topics include:

• Green's functions for the Dirichlet boundary condition;
• Solution by a power series method and the method of majorants;
• Smoothness of the initial value problem;
• The stationary Schrodinger equation as a Sturm-Liouville system;
• Numerical methods of solving differential equations;
• Planetary motion and conservation laws;
• Bifurcation theory and elastic rods;
• Can one hear the shape of a drum?
• Linear instability of a free-surface shear flows due to boundary singularity, etc.

Please feel free to come up with your own idea!

Adaptive Stepsize Numerical Methods for SolvingOrdinary Differential Equations (PDF) (Courtesy of Oleg Golberg. Used with permission.)

Limitations of Euler's Method for Numerical Integration (PDF) (Courtesy of Laura Evans. Used with permission.)

Numerical Approximations in Differential Equations: The Runge-Kutta Method (PDF) (Courtesy of Ernest Ngaruiya. Used with permission.)

Higher Order Taylor Methods (PDF) (Courtesy of Marcelo Julio Alvisio and Lisa Marie Danz. Used with permission.)