Math 331: Partial Differential Equations


May 10, 2026


Final Exam Review Topics


  1. Boundary value problems for ODEs; eigenvalue problems

  2. Singular boundary value problems for ODEs

  3. Review: Periodic functions; odd and even functions

  4. Orthogonality of functions and its applications

  5. Fourier series; derivation of coefficients

  6. Odd and even extensions of functions; Fourier sine series and Fourier cosine series

  7. Expanding functions in FS, FSS, and FCS

  8. Convergence of Fourier series: types of convergence; pointwise and uniform convergence theorems

  9. Applying Fourier series to find sums of various series

  10. Fourier integral: informal derivation

  11. Fourier integral representation theorem

  12. Applying Fourier integral to find definite integrals

  13. Fourier sine and cosine integral representations

  14. Complex form of Fourier series

  15. Operations on Fourier series: differentiation and integration

  16. Applying Fourier series to solve ODEs

  17. Introduction to PDEs: IBVPs

  18. Steady-state solutions of the heat equation; the transient component

  19. Solving the fixed end temperatures and insulated cases of heat equation IBVP: separation of variables; eigenvalues and eigenfunctions; superposition of solutions

  20. Sturm-Liouville problems: regular S-LP; Sturm-Liouville Theorem; orthogonality of eigenfunctions

  21. Generalized Fourier series: eigenfunction expansions

  22. Solving heat equation in unbounded cases with Fourier integral

  23. Solving the vibrating string problem (wave equation) with the use of separation of variables and Fourier series

  24. Solving the vibrating string problem with the use of d’Alembert’s method; understanding the d'Alembert's solution

  25. Demonstrating equivalence of solutions of the vibrating string problem obtained with the two methods (separation and d'Alembert's)

  26. Solving wave equation in an unbounded region

  27. Potential (Laplace’s) equation; harmonic functions; mean value property; the maximum principle; potential equation in non-rectangular coordinates

  28. Solving the basic potential in a rectangle problem

  29. Using superposition principle to solve more complicated potential in a rectangle problems

  30. Solving potential in unbounded regions problems

  31. Potential in a disk; solving Dirichlet’s problem (values of the solution function are defined on the whole boundary)

  32. Main steps in derivation of Poisson formula for the potential in a disk problem

  33. Solving two-dimensional heat and wave equation; double series solutions

  34. Introduction to Fourier transform; transform properties; using Fourier transform in solving PDEs

  35. Introduction to Legendre equation; Legendre polynomials and their orthogonality

  36. Solving potential on a sphere problem (Laplace’s equation in spherical coordinates); using Fourier-Legendre series; interior and exterior cases

  37. Introduction to the finite differences numerical method

  38. Using implicit finite difference method in solving Laplace’s equation

  39. Research presentations