Partial Differential Equations

Partial Differential Equations, University of Toronto 2019 (APM346)

These are the materials from a course on partial differential equations I coordinated at the University of Toronto in Winter semester of 2019.

Lecture Notes

  1. Administration / Introduction to PDEs / What does it mean to solve a PDE? (pdf, mathematica)
  2. Transport equation / Wave equation / Heat equation (pdf, mathematica)
  3. Solving the transport equation (pdf, mathematica)
  4. Homogeneous 1d wave equation (pdf, mathematica)
  5. Characteristic coordinates / Duhamel’s formula (pdf, mathematica)
  6. Wave equation with boundary conditions (pdf, mathematica)
  7. 1d heat equation (pdf, mathematica)
  8. Heat equation in higher dimensions / Maximum principle (pdf)
  9. Wave equation on a finite interval (pdf, mathematica)
  10. Wave equation on a finite interval (ctd.) (pdf)
  11. Orthogonality of eigenfunctions / Fourier series / Gibbs phenomenon (pdf, mathematica)
  12. Periodic extensions / Complex Fourier series (pdf, mathematica)
  13. The Fourier transform (pdf)
  14. Application of the Fourier transform to PDEs (pdf)
  15. Separation of variables (pdf)
  16. The Laplace operator in different coordinate systems (pdf)
  17. The Laplace equation in polar coordinates (pdf)
  18. Higher dimensional Laplace equation / Unicity for the Laplace equation (pdf)
  19. Green’s functions / Separation of variables in spherical coordinates (pdf)
  20. Applications of spherical harmonics (pdf)
  21. Wave equation in 3d and 2d (pdf)
  22. Dirichlet’s principle / Variation of functionals (pdf)
  23. Calculus of Variations / Euler-Lagrange equations / Functionals with boundary terms (pdf)
  24. Point particles and the strings they love… (pdf)

Selected Notes and Problems