**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

- 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

- D. Bleecker and G. Csordas, Basic partial differential equations, International Press 1997

Additional references

- G. Evans, J. Blackledge, and P. Yardley, Analytic methods for partial differential equations, Springer 1999.
- W.A. Strauss, Partial differential equations: An introduction, Wiley 2007

- Past Math 319 pages: Winter 2011, Winter 2012
- Robert Terrell's teaching page
- Paul Dawkins' online notes
- Peter Olver's book
- John Douglas Moore's lecture notes

**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% }.

- 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

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