Department of Mathematics

University of Toronto

APM346 Partial Differential Equations

Spring of 2018; Section L0201 – Lecture notes and Home assignments

Notes

Here current lecture notes will appear weekly. They will be based on Online textbook. So notes will be just references to sections of this textbook plus brief comments.

Usually each weekend (Fri, Sat or Sun) three new notes and a home assignment will be released covering the coming week and over next weekend (Fri or Sat) these notes will be finalized.

What is dimmed––to be updated as needed; sections will be moved, home assignments modified.

    Week 1 (January 04—05)
  1. Introduction: PDE Motivations and Context (Section 1.1)
  2. Introduction: Initial and Boundary Value Problems (Section 1.2)
    3. Week 2 (January 08—12)
  3. First order PDEs (Section 2.1)
  4. First order PDEs (Section 2.1 (end))
  5. Homogeneous 1D Wave equation (Section 2.3)
    6. Week 3 (January 15—19)
  6. Homogeneous 1D Wave equation--end (Section 2.3)
  7. 1D Wave equation reloaded: characteristic coordinates and more
    (Section 2.4 and Section 2.5)
  8. 1D Wave equation: IBVP (Section 2.6)
    9. Week 4 (January 22—26)
  9. Wave equation: Energy integral (Section 2.7)
  10. 1D Heat equation: method of self-similar solutions (Section 3.1)
  11. Heat equation (Misc.) (Section 3.2)
    12. Week 5 (Jan 29— Feb. 02)
  12. Heat equation (Misc.) -- end
  13. 1D wave equation: Separation of variables (the first blood) (Section 4.1)
  14. Eigenvalue problems (examples) (Section 4.2)
    15. Week 6 (February 05—09)
  15. Ortogonal systems (Section 4.3)
  16. Ortogonal systems (Section 4.3 (end))
  17. Ortogonal systems and Fourier series
    (Section 4.4)
    18. Week 7 (February 12—16)
  18. Other Fourier series
    (Section 4.5)
  19. Fourier transform, Fourier integral
    (Section 5.1)
  20. Properties of Fourier transform
    (Section 5.2) Term Test 1 (February 15).
    15:20 — 17:20 at EX 300,
    Covers Chapters 1--3.
    Solving HA 1—5 is a good preparation
Reading week (Feb. 19—23)
    Week 8 (Feb. 26— Mar. 02)
  1. Applications of Fourier transform to PDEs
    (Section 5.3)
  2. Applications of Fourier transform to PDEs
    (Section 5.3 (end))
  3. Separation of variables for heat equation
    (Section 6.1)
    Separation of variables: Misc equations
    (Section 6.2)
    4. Week 9 (March 05—09)
  4. Laplacian in polar and spherical coordinates
    (Section 6.3)
    Laplacian: separation of variables in polar coordinates (Section 6.4)
  5. Laplace operator in the disk. II
    (Section 6.5)
  6. General properties of Laplace equation
    (Section 7.1)
    7. Week 10 (March 12—16)
  7. Potential theory and around
    (Section 7.2)
    Green function (Section 7.3)
  8. Separation of variable in spherical coordinates (Section 8.1)
  9. Separation of variable in spherical coordinates (Section 8.1 (end))
    10. Week 11 (March 19—23)
  10. Wave equation in 3D and 2D
    (Section 9.1)
  11. Wave equation in 3D and 2D (end);
    Wave equation: energy method
    (Section 9.2)
  12. Functionals, extremums and variations
    (Section 10.1)
    13. Week 12—13(Mar. 26—Apr.04)
  13. Functionals, extremums and variations. II
    (Section 10.2)
    Variational methods in physics (Section 10.3)
  14. Distributions (Section 11.1)
  15. Distributions: more (Section 11.2)
  16. Applications of distributions (Section 11.3)
    Weak solutions (Section 11.4)
  • Before exam Review lecture (TBA)
  • Special office hours (TBA)
  • Final exam