Instructor: Prof. Robert (Bob) Haslhofer
Website: http://www.math.toronto.edu/roberth/C46.html
Lectures: Tuesday 3--5 in SW143 and Thursday 4--5 in SY110
Office hours: Tuesday 11:30--12:30 in IC404
Grading Scheme: Homework 20%, Midterm 30%, Final exam 50%
There will be 5 homework assignments. Your lowest homework score will be dropped. There will be no makeup test! (If you miss the midterm for a valid reason, your grade will be reweighted as Homework 30%, Final 70%.)Homework: via crowdmark
Weekly lecture notes: via quercus
Midterm Exam (in class): Feb 13 from 3--5
Final Exam: Apr 22 from 9--11 in HLB101
Prerequisites: MATB41 and MATB44
Corequisite: MATB42
Textbook: W. Strauss: Partial Differential Equations - An Introduction, Wiley
Secondary reference: D. Logan: Applied Partial Differential Equations, Springer
Topics to be covered:
What is a PDE: linear and nonlinear equations, first and second
order equations, static equations and evolution equations, modelling processes by PDEs, initial and
boundary conditions, well-posed problems.
The transport equation: derivation of the equation in engineering and optimization, method of characteristics, conservation laws, propagation of singularities.
Laplace equation and Poisson equation: physical motivation,
boundary value problems, separation of variables, the fundamental
solution, mean value formula, maximum principle, smoothness of
solutions.
The diffusion equation: derivation as a model for diffusion in
physics and geometry, scaling properties, initial value
problem, fundamental solution, solution via Fourier analysis, maximum principle.
The wave equation: derivation of the equation, the Cauchy
problem, solution in one and three spatial dimensions, energy conservation,
causality principle.
Schedule:
Week 1
What is a PDE, First order linear PDEs (Strauss 1.1, 1.2)
Week 2
Where PDEs come from, Initial and Boundary conditions (Strauss 1.3, 1.4)
Week 3
The Wave Equation, Energy and Causality (Strauss 2.1, 2.2)
Week 4
The Diffusion Equation, Diffusion on the whole line (Strauss 2.3, 2.4)
Week 5
Catchup and review
Week 6
Midterm, Comparison of Waves and Diffusion (Strauss 2.5)
Week 7
Boundary value problems (Strauss 4.1, 4.2)
Week 8
Fourier series (Strauss 5.1, 5.2)
Week 9
Harmonic functions (Strauss 6.1, 6.2)
Week 10
Poisson's formula, mean value property, strong maximum principle (Strauss 6.3, 7.1)
Week 11
Green's identities and Green's functions (Strauss 7.1 -- 7.4)
Week 12
Maxwell's equations and Euler's equations (Strauss 13.1, 13.2)