Department of Mathematics

University of Toronto

APM346 Partial Differential Equations

Spring of 2015; Section L5102 – Lecture notes

Current version :

Here current lecture notes will appear weekly--as soon as they are written. They will be based on lecture notes of the similar course. Usually each weekend (Sat or Sun) three new sections and a home assignment will be released covering the coming week and over next weekend (Fri or Sat) these lectures will be modified; further (hopefully minimal) modifications will be done as needed.
  1. Lectures and Home assignments
  2. Appendices
  3. Previous years home assignments
  4. Previous years tests and exams
  5. Extra Reading

Lectures and
Home assignments

    Week 1 (January 5—9)

  1. Introduction: PDE Motivations and Context
  2. First order PDEs
  3. Homogeneous 1D wave equation

    Week 2 (January 12—16)

  4. 1D Wave equation reloaded: characteristic coordinates
  5. Wave equation reloaded (continued)
  6. 1D Wave equation: IBVP

    Week 3 (January 19—23)

  7. Energy integral
  8. 1D Heat equation (method of self-similar solutions)
  9. Heat equation (Misc.) (misc.)

    Week 4 (January 26—30)

  10. Separation of variables: 1D wave equation Separation of variables (the first blood)
  11. Eigenvalue problems (examples)
  12. Ortogonal systems
  13. Week 5 (February 2—6)

  14. Ortogonal systems and Fourier series
  15. Other Fourier series
  16. Fourier transform, Fourier integral
  17. Week 6 (February 9—13)

  18. Properties of Fourier transform
  19. Applications of Fourier transform to PDEs
  20. Separation of variables for heat equation
  21. Week 7 (February 16—20)

    • Reading week; no lectures

    Week 8 (February 23—27)

  22. Separation of variables: Misc equations
  23. Laplacian in polar and spherical coordinates
  24. Laplacian: separation of variables in polar coordinates
  25. Week 9 (March 2—6)

  26. Laplace operator in the disk. II
  27. General properties of Laplace equation
  28. Potential theory and around
  29. Week 10 (March 9—13)

  30. Green function
  31. Wave equation in 3D, 2D: solution
  32. Wave equation: energy method

    Week 11 (March 16—20)

  1. Functionals, extremums and variations
  2. Functionals, extremums and variations. II
  3. Variational methods in physics
  4. Week 12 (March 23—27)

  5. Distributions
  6. Distributions: more
  7. Applications of distributions
  8. Week 13 (March 30—April 3)

  9. Burgers equation
  10. Review
  11. Review


Appendices are not required reading but you may find them helpful.

  1. Exploded (Incomplete) View of APM346
  2. Appendix A to Week 1--Analyzing the traffic flow
  3. Appendix B--Linear second order ODEs
  4. Appendix C--Fully nonlinear first order PDEs
  5. Appendix D--Maxwell equations
  6. Appendix E--Hyperbolic first order systems with one spatial variable
  7. Appendix F--Conservation laws
  8. Appendix G--Some classes of PDEs
  9. Appendix H--To Lecture 11 (analyzing some examples; variational principles)
  10. Appendix I--To Lecture 11 (analyzing some examples)
  11. Appendix J--Multidimensional Fourier series
  12. Appendix K--Quatum harmonic oscillator; Hermite functions, Hermite polynomials
  13. Appendix L--Spectrum: definitions
  14. Appendix M--Spectrum: examples
  15. Appendix N--Spectrum: explanations
  16. Appendix O--To Lecture 15: Fourier transform: kind of justification
  17. Appendix P--Weak solutions

  18. Appendix Y--Green, Gauss, Stokes formulae
  19. Appendix Z--Properties of ∇


(offered for reading course only)
  1. Intro into project: Random Walks

Previous years home assignments

  1. 2012F Home Assignment 1 (solutions are discussed on Forum#6.0)
  2. 2011F Home Assignment 2 (solutions are discussed on Forum 2011#8.0),
    2011F Home Assignment 3 (solutions are discussed on Forum 2011#9.0) and
    2012F Home Assignment 2 (solutions are discussed on Forum#9.0)
  3. 2011F Home Assignment 5 (solutions are discussed on Forum 2011#10.0) and
    2012F Home Assignment 4 (solutions are discussed on Forum#10.0)
  4. 2011F Home Assignment 4 (solutions are discussed on Forum 2011#15.0) and
    2012F Home Assignment 3 (solutions are discussed on Forum#11.0)

Previous years tests and exams
(with solutions and discussions)

2011F Test 1 2012F Test 1
2011F Test 1 2012F Test 2
2011F Final Exam 2012F Final Exam

Extra Reading

Not required but you may find useful:
  • Walter A. Strauss, Partial Differential Equations, Textbook and Student Solutions Manual: An Introduction; 2 edition Wiley (2007)
  • Richard Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems; Pearson; 5 edition (2012)