There is a final project instead of a final exam. In your project, you should consider a PDE (ideally in 2D; I would recommend against 3D for time reasons, and 1D problems may be too simple) or possibly a numerical method not treated in class, and write a 5–10 page academic-style paper that includes:

  • Review: Why is this PDE / method important, what is its history, and what are the important publications and references? (A comprehensive bibliography is expected: not just the sources you happened to consult, but a complete set of sources you would recommend that a reader consult to learn a fuller picture.)
  • Analysis: What are the important general analytical properties? e.g. conservation laws, algebraic structure, nature of solutions (oscillatory, decaying, etcetera). Analytical solution of a simple problem.
  • Numerics: What numerical method do you use, and what are its convergence properties (and stability, for time-stepping)? Implement the method (e.g. in Julia, Python, or Matlab) and demonstrate results for some test problems. Validate your solution (show that it converges in some known case).

Here are some final-project titles from previous semesters:

  • Conformal Mapping Methods for Solving Laplace's Equation in Two Dimensions
  • Electromagnetic Scattering From a Layered Dielectric Sphere
  • Convection-Diffusion and Burgers' Equations
  • The Black-Scholes Equation
  • Stokes Flow
  • Effects of Symmetry Groups on Eigenvalues and Eigenfunction of the Schrödinger Equation
  • Application of the Fokker-Planck equation to Brownian motion
  • The Schrödinger Equation with Time-Varying Potential
  • Reaction-Diffusion Equations
  • Bloch's Theorem and Bloch Waves
  • Linearized Elasticity with Lamé-Navier Equations
  • Inverse Scattering Problems
  • Methods for Solving Partial Differential Equations on Unbounded Domains
  • Chebyshev Spectral Methods