## final exam

The final exam is scheduled on Thursday April 19, at 14:00, in Trottier 2120. Office hours during the exam week: Monday April 16, 11:30-12:30 and Wednesday April 18, 14:00-16:00.

This is a list of topics you should master for the exam:
• Simple things: ODEs, change of variables, linearity, superposition principle
• Linear transport equations in 2 independent variables
• Heat equation: energy method, maximum/minimum/comparison principles
• Separation of variables for the heat equation
• Wave equation: energy method, D'Alambert's formula, method of images
• Separation of variables for the wave equation
• Fourier series: pointwise convergence, uniform convergence, Gibbs phenomenon

• Review homework and midterm problems, and do practice with the following.
§4.1: 7
§4.2: 12
§5.1: 5
§5.2: 3, 7
§6.1: 3
§6.2: 1
§7.1: 1, 8
(Do not forget Chapters 1-3)

Disclaimer: This is meant to be simply a guide to study for the exam. You are responsible for material from the textbook (§1.1-1.3, 2.1-2.2, 3.1-3.2, 4.1-4.3, 5.1-5.2, 6.1-6.2, half of 7.1), and the material covered in class.

## assignments

Assignment 1 due Th, Jan 19 [solutions to problems 8(b) and 9]
§1.1: 1(bdfh), 4(bdfh), 5, 7(bdf), 8, 9

Assignment 2 due Tu, Jan 31 [solutions]
§1.3: 2(bd), 3(bd),
§2.1: 1(bdf), 2(b), 3, 8, 9, 10

Assignment 3 due Th, Feb 16 [solutions]
§2.2: 1(bd), 2(bd), 3(bd), 4, 5, 8,
§3.1: 1

Assignment 4 not to be turned in [midterm solutions]
§3.1: 3, 4, 5, 6, 7

Assignment 5 due Th, Mar 15 [solutions]
§3.1: 6(bd), 8, 9,
§3.2: 1, 2, 3, 5

Assignment 6 due Th, Mar 29 [solutions]
§4.1: 5, 6
§4.2: 6, 15
§4.3: 9, 12
§5.1: 1, 2

Assignment 7 due Fr, Apr 13 [solutions]
§5.1: 4, 6, 9
§5.2: 1(bdf), 4
§6.1: 5
§6.2: 2(cd), 5

## class schedule

Note: This schedule is subject to revision during the term.

 Date Topic Mo 1/9 Ordinary differential equations [§1.1] Tu 1/10 Continuity and differentiability [§1.2] Th 1/12 Linear spaces, linear operators [§1.2] Mo 1/16 Superposition principle. Examples of PDE [§1.2, 1.3] Tu 1/17 Linear transport equations with constant coefficients [§2.1] Th 1/19 Examples [§2.1] Mo 1/23 Linear transport equations with variable coefficients [§2.2] Tu 1/24 Examples [§2.2] Th 1/26 Solution in terms of a parameterization [§2.2] Mo 1/30 Examples [§2.2] Tu 1/31 Global considerations [§2.2]. Th 2/2 Setup for rigorous treatment. Inverse function theorem. Mo 2/6 Local solvability of linear transport equations. Tu 2/7 Global solvability of linear transport equations. Th 2/9 Heat equation [§3.1]. Mo 2/13 Heaviside initial data. Gauss' error function. Tu 2/14 Heat kernel. Dirichlet boundary conditions and product solutions [§3.1] Th 2/16 Finite linear combination of product solutions [§3.1] 2/20–2/24 Study break Mo 2/27 Examples [§3.1] Tu 2/28 Neumann boundary conditions Th 3/1 Midterm exam [§1.1-1.3, 2.1-2.2, 3.1] Mo 3/5 Periodic boundary conditions [§3.1] Tu 3/6 Uniqueness by energy method. Maximum principle [§3.2] Th 3/8 Minimum and comparison principles. Continuous dependence [§3.2]. Mo 3/12 Maximum principle for periodic problems [§3.2]. Fourier sine series [§4.3]. Tu 3/13 Fourier cosine series [§4.3]. Th 3/15 Fourier series [§4.1]. Complex Fourier series [§7.1]. Mo 3/19 Bessel's inequality. Riemann-Lebesgue lemma [§4.2]. Tu 3/20 Dirichlet kernel. Pointwise convergence of Fourier series [§4.2]. Th 3/22 Decay of Fourier coefficients. Uniform convergence [§4.2]. Mo 3/26 Wave equation. Product solutions [§5.1]. Tu 3/27 Homogeneous Dirichlet and Neumann problems [§5.1]. Th 3/29 Uniqueness by energy method. D'Alambert's formula [§5.2]. Mo 4/2 Method of images [§5.2]. Tu 4/3 Laplace equation [§6.1]. Product solutions. Th 4/5 Dirichlet problem for a rectangle [§6.2]. Mo 4/9 Easter break Tu 4/10 Gibbs phenomenon [§4.2]. Th 4/12 Discussion on separation of variables. Mo 4/16 Review. Th 4/19 Final exam (at 2pm)

## course outline

Lectures
MTR 10:35–11:25, Burnside Hall 1B24

Office hours
TR 11:30–12:30, or by appointment

Instructor
Dr. Gantumur Tsogtgerel
Office: Burnside Hall 1123.
Email: gantumur -at- math.mcgill.ca.

Catalog 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
• Brief review of some relevant topics from linear algebra, calculus and ODE.
• First order equations. Method of characteristics.
• Fundamental solutions. Green's function.
• Cauchy problems for the heat and wave equations.
• Separation of variables.
• Fourier series. Orthonormal bases for function spaces. Integral transforms.
• Introduction to numerical methods.
• Derivations of some important equations of mathematical physics.
• Conservation laws (as time permits).

Prerequisites
MATH 223 (Linear algebra) or MATH 236 (Algebra 2),
MATH 315 (ODE).

Required book
• D. Bleecker and G. Csordas. Basic partial differential equations. International Press 1997

Recommended books
• R.P. Agarwal and D. O'Regan. Ordinary and partial differential equations. Springer 2008
• C.R. MacCluer. Boundary value problems and Fourier expansions. Dover 2004
• W.A. Strauss. Partial differential equations: An introduction. Wiley 2007
• S.J. Farlow. Partial differential equations for scientists and engineers. Dover 1993
• D. Colton. Partial differential equations: An introduction. Dover 2004
• J.D. Logan. Applied partial differential equations. Springer 2004
• E.C. Zachmanoglou and D.W. Thoe. Introduction to partial differential equations with applications. Dover 1986

Homework Assignments
Assigned and graded roughly every 2 weeks. All homework assignments will count towards the final grade. Late homeworks will not be accepted.

Exams
The midterm exam is on Thursday March 1, in class. The final exam is tentatively scheduled on Thursday April 19, at 2pm.

Homework 20% + max ( Midterm 30% + Final 50% , Midterm 10% + Final 70% )
• ## midterm

The midterm exam is on Thursday March 1, in class 10:35-11:25.

In the midterm week, we have
• Extra office hours on Tuesday 14:00-16:00
• Problem solving sessions on Monday and Tuesday, in class

• This is a list of the basic "skills" you should master for the midterm:
• 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.
• Recognize if a PDE is linear or nonlinear
• Apply the superposition principle
• Solving linear transport equations in 2 independent variables
• In particular, solve constant coefficient equations by a simple change of variables
• Solve variable coefficient equations by using the characteristic equation
• Solve variable coefficient equations in parametric form
• Assess the geometry of the characteristic curves and the side condition curve
• Familiarity with the solution of the heat equation with a step function initial condition
• Familiarity with the heat kernel
• Product solution with homogeneous Dirichlet boundary condition
• Solution by a finite linear combination of product solutions

• Disclaimer: This list is meant to be simply a guide to study for the exam. I have tried to make the list complete, but you are also responsible for items that were inadvertently omitted. You are responsible for material from the textbook (§1.1-1.3, 2.1-2.2, 3.1), and the material covered in class.

## online resources

Last year's Math 319 page

Robert Terrell's teaching page

Matthew Hutton's lecture notes

Paul Dawkins' online notes

Peter Olver's book, and his introduction to Matlab and example codes

Sung Lee's lecture notes