Class schedule

Date Topics
Th 1/7 Basic classification. Simple examples.
Fr 1/8 Linear advection equations with constant coefficients [§2.1].
Mo 1/11 More on constant coefficients [§2.1].
We 1/13 Variable coefficients [§2.2].
Fr 1/15 More on variable coefficients [§2.2].
Mo 1/18 Linear advection equations.
We 1/20 Wave equation in one dimension. Derivation [§5.1].
Fr 1/22 D'Alambert's solution [§5.2].
Mo 1/25 Wave equation on the half line. Method of images. Homogeneous Dirichlet boundary condition [§5.2].
We 1/27 Homogeneous Neumann boundary condition [§5.2].
Fr 1/29 Wave equation on an interval. Separation of variables [§5.1].
Mo 2/1 Dirichlet eigenfunctions of an interval. Standing waves and harmonics. Superposition principle [§5.1].
We 2/3 Fourier sine series [§4.3].
Fr 2/5 Solution of the wave equation in terms of Fourier sine series [§5.1].
Mo 2/8 Wave energy. Uniqueness. Neumann problem for the wave equation on an interval [§5.1].
We 2/10 Fourier cosine series [§4.3].
Fr 2/12 Heat equation. Derivation. Dirichlet problem on an interval [§3.1].
Mo 2/15 Neumann problem [§3.3]. Inhomogeneous boundary conditions [§3.3].
We 2/17 Periodic boundary condition [§3.1].
Fr 2/19 Fourier series [§4.1]. Thermal energy [§3.1].
Mo 2/22 Kinetic energy. Uniqueness [§3.2].
We 2/24 Maximum principle [§3.2].
Fr 2/26 Midterm exam
2/29–3/4 Reading week
Mo 3/7 Inhomogeneous heat equation. Duhamel's principle [§3.4].
We 3/9 The heat kernel of the line [§3.1].
Fr 3/11 Laplace equation. Harmonic functions [§6.1].
Mo 3/14 Dirichlet problem for a rectangle [§6.2].
We 3/16 Maximum principles. Uniqueness for the Dirichlet problem [§6.4]. Dirichlet energy.
Fr 3/18 Laplacian in polar coordinates. Dirichlet problem for disks [§6.3].
Mo 3/21 Dirichlet problem for annuli [§6.3]. Electrostatics and gravitation.
We 3/23 Lagrange, Laplace, Poisson.
Fr 3/25 Good Friday
Mo 3/28 Easter Monday
We 3/30 Multipole expansion in two dimensions.
Fr 4/1 Multipole moments. Examples.
Mo 4/4 Separation of variables for a rectangle [§9.1].
We 4/6 Eigenfunctions and eigenvalues of a rectangle. Heat equation on a rectangle [§9.2].
Fr 4/8 Wave equation on a rectangle. Laplace equation in a 3D box. Mixed boundary conditions [§9.2].
Mo 4/11 Radial symmetry in 3D [§9.3].
We 4/13 Wave equation in 3D. Kirchhoff's formula [§9.3].
Fr 4/15 Hadamard's method of descent. Poisson's formula.
We 4/27 Final exam (6pm, Currie Gym)

Reference books

Main reference

Additional references

Online resources

Course outline

Instructor: Dr. Gantumur Tsogtgerel

Prerequisite: MATH 223 (Linear Algebra) or MATH 236 (Algebra 2), MATH 314 (Advanced Calculus), MATH 315 (ODE)

Calendar description: First order equations, geometric theory; second order equations, classification; Laplace, wave and heat equations, Sturm-Liouville theory, Fourier series, boundary and initial value problems.

Topics to be covered

Homework: A few written assignments.

Exams: Midterm and final.

Grading: Homework 20% + MAX{ Midterm 20% + Final 60%, Final 80% }.

Midterm exam

The following is a list of skills and knowledge you are expected to master (There is no guarantee that this list is complete but it may help direct your study).

Final exam

  • The final exam is scheduled on Wednesday April 27, starting at 6pm, in the Currie Gym.
  • It is a closed book, closed note exam. No calculators will be allowed or needed.
  • The exam has 8 questions, including 6 simple questions (cf. Assignment 2, Problem 2), and 2 slightly more complicated questions (cf. Assignment 5, Problem 2). If you "know your stuff," the whole exam should take roughly 1 hour to complete.
  • The exam is intended to test how you recognize the relevant techniques for the particular problem at hand, whether you can apply the learned techniques to solve concrete problems, and whether you can effectively communicate your solution.
  • You are responsible for what is covered in class, in the reading assignments, and in the relevant sections of the textbook.
  • Please review the past assignments, practice problems, the midterm exam, and these problem suggested by students.
  • The final course grade will be computed as Homework 20% + MAX{ Final 80% , Final 60% + Midterm 20% } + Bonus mark.
  • Your single lowest assignment mark will be dropped, meaning that your grade for the homework will be determined by your 4 best scores (equally weighted).
  • The following is a list of skills and knowledge you are expected to master (There is no guarantee that this list is complete but it may help direct your study).