## Announcements

• Reading assignment 2: Multipole expansions in the plane
• Assignment 5 [tex] due Friday April 15 (Some solutions)
• Practice problem set 5 (not to be handed in)
• §9.1: 1, 3, 5, 9, 12
• §9.2: 3, 4, 8(ab)
• §9.3: 1, 9, 10
• Reading assignment 1: §1.2 and §6.3
• Assignment 4 due Wednesday March 30
• §6.1: 1
• §6.2: 2(cd), 5, 7, 9
• §6.3: 1(be), 2, 4
• §6.4: 2, 7
• §1.2: 6, 12
• Practice problem set 4 (not to be handed in)
• §6.1: 2, 3
• §6.2: 1, 2, 3, 4, 6
• §6.3: 1, 3, 5, 6, 9
• §6.4: 1, 3, 4
• §1.2: 7, 8, 9, 10, 11
• Assignment 3 due Wednesday March 9
• §3.1: 3(bd), 9
• §3.2: 4, 5, 8
• §3.3: 3, 6
• §3.4: 6, 7, 10
• §4.3: 10, 11
• Practice problem set 3 (not to be handed in) [Some example solutions]
• §3.1: 1, 4, 5, 6
• §3.2: 1, 2, 3
• §3.3: 1, 2, 4, 5, 7
• §3.4: 3, 4, 5, 9
• §4.3: 9, 12, 13
• Assignment 2 [tex] due Wednesday February 10
• Practice problem set 2 (not to be handed in) [Some example solutions]
• §5.1: 1, 2, 4, 5, 8
• §5.2: 1, 2, 5, 6, 9
• Practice problem set 1 (not to be handed in) [Some example solutions]
• §1.1: 1, 4
• §1.2: 1, 5
• §1.3: 2, 3
• §2.1: 1, 2, 3, 4, 5, 6, 7, 8
• §2.2: 1, 2, 3, 4, 5, 6, 7
• Assignment 1 due Friday January 22 (Provide step-by-step detailed explanations. Formulas without explanations will not receive marks.)
• §1.3: 2(b), 3(b)
• §2.1: 1(bd), 4, 7
• §2.2: 1(b), 2(b), 3(b), 6
• If you want to have some chapters/sections of the book printed, my suggestion would be the following: §2.1, §2.2, Chapters 3, 5, 6 (except §6.5), and 9.
• Instructor's office hours are WF 10:30–11:30, Burnside Hall 1123

## Class schedule

• MWF 9:35–10:25, Burnside Hall 1B23
• Note: This schedule is subject to revision during the term.

 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)

Main reference

## 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

• First order equations. Method of characteristics.
• Cauchy problem for second order equations.
• Separation of variables. Fourier series. Sturm-Liouville theory.
• Fourier and Laplace transforms.
• Fundamental solutions. Green's function.

Homework: A few written assignments.

Exams: Midterm and final.

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

## Midterm exam

• The midterm exam is on Friday February 26, in class 09:35-10:25.
• It is a closed book, closed note exam. No calculators will be allowed or needed.
• The testable material is: Chapter 1, §2.1, §2.2, Chapter 3, §4.3, §5.1, §5.2.
• There will be 2-3 problems, similar to the problems from the assignments, practice problem sets, and this practice midterm (with some solutions).

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).

• Solving simple ODEs as in §1.1
• Change of variables in PDEs, in particular, changing the domain of the equation from (0,L) to (0,1) etc.
• Linear and nonlinear equations, the order of a PDE
• Solving linear transport equations in 2 dimensions by using the method of characteristics
• Assess the geometry of the characteristic curves and the side condition curve
• Wave equation: D'Alambert's formula, method of images, energy, uniqueness
• Separation of variables for the wave equation on an interval: Dirichlet and Neumann problems
• Fourier sine series, Fourier cosine series, Fourier series
• Heat equation: energy, maximum/minimum/comparison principles, uniqueness, the heat kernel of the line
• Separation of variables for the heat equation: Dirichlet, Neumann, mixed, and periodic boundary conditions, inhomogeneous boundary conditions, Duhamel's principle

## 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).

• Solving simple ODEs as in §1.1
• Change of variables in PDEs, in particular, changing the domain of the equation from (0,L) to (0,1) etc.
• Linear and nonlinear equations, the order of a PDE
• Solving linear transport equations in 2 dimensions by using the method of characteristics
• Assess the geometry of the characteristic curves and the side condition curve
• Wave equation: D'Alambert's formula, method of images, energy, uniqueness
• Heat equation: energy, maximum/minimum/comparison principles, uniqueness, the heat kernel of the line
• Separation of variables for the heat and wave equations: Dirichlet, Neumann, mixed, and periodic boundary conditions
• Fourier sine series, Fourier cosine series, Fourier series
• Inhomogeneous boundary conditions and Duhamel's principle for the heat equation
• Laplace equation: Dirichlet energy, maximum/minimum/comparison principles, uniqueness
• Heat, wave, and Laplace equations on a rectangle and in a 3D box
• Eigenfunctions and eigenvalues of the Laplacian on a rectangle and in a 3D box
• Laplace equation on a disk and an annulus
• Multipole expansions in two dimensions
• Radially symmetric heat and wave equations in 3D
• Kirchhoff's formula, Hadamard's method of descent, Poisson's formula