### Fall 2012

Web page: http://www.math.toronto.edu/ilia/MAT311.2012/.

Class Location & Time: Tue, 10:00 AM - 12:00 PM IB 260; Thu, 11:00 AM - 12:00 PM IB 380

Tutorials: Fr 12:00-13:00 NE174 (TUT0101); Fr 13:00-14:00 IB395 (TUT0201);

Instructor: Ilia Binder (ilia@math.toronto.edu), William G. Davis Bldg. 4038, Phone: (905) 569-4381.

Office Hours:  Th 10-11 and 1-2.

Teaching Assistant: Filip Ziaja (filip.ziaja@utoronto.ca).

Required Text: Partial Differential Equations: An Introduction. Walter A. Strauss. 2nd Edition (2007).Wiley (978-0470054567)

Prerequisites:  MAT102H5, 232H5/233H5, 212H5/242H5.

Topics.
This course introduces a range of mathematical concepts and techniques in the theory of partial differential equations.
Emphasis will be on specific equations and methods for solving them, rather than on general theory. Most of the course will be
devoted to studying the wave equation, diffusion equation and Laplace's equation. We will learn to pose and solve meaningful
boundary value and initial value problems for these types of equations and we will get acquainted with the basics of Fourier
analysis. By the end of the course students can expect to have a working understanding of the three main PDEs and techniques.

Students are expected to have a solid background in calculus, including all aspects of multivariable calculus. Familiarity with
basic linear algebra and ordinary differential equations is also required, but not as important. In particular students should

• how to compute partial derivatives of a given function
• that mixed partial derivatives are equal
• how to use the chain rule in one and multiple dimensions
• Green's theorem and the divergence theorem for computing integrals of derivatives
• Jacocbians (the multivariable change of variables formula)
• directional derivatives
• how to solve a few basic ODEs

Topics covered in class.
September 11: Ordinary and Partial Differential Equations; structure of solutions of linear ODE and PDE; introduction to diffusion, wave, and potential equations. Section 1.1.
September 13: Ordinary Differential Equations: review of the first and second order linear ODEs. Section 1.1.
September 18:
Ordinary Differential Equations: Boundary Value Problems. Solving first order linear homogeneous PDEs. Derivations of Simple Transport and Wave equations. Sections 1.2, 1.3.
September 20: Derivations of Wave and Heat equations. Initial and Boundary conditions for them. Sections 1.3, 1.4.
September 25:
Well-posed problems. General solution of the Wave equation. d'Alembert solution. Causality and Energy. Sections 1.5, 2.1, 2.2.
September 27: Energy and Uniqueness for Wave equation. Maximum Principle for Diffusion equation. Sections 2.2, 2.3.
October 2: Maximum Principle, Uniqueness, and Stability for Diffusion equation. Solution of Diffusion Equation on the real line. Sections 2.3, 2.4.
October 4:
Solution of Diffusion Equation on the real line.The Error function. Section 2.4.

October 9: Solution of Diffusion Equation on the real line - continuation. Comparison of Wave and Heat Equations. Separation of variables: the Dirichlet case. Sections 2.4, 2.5, 4.1.
October 11: Separation of variables: the Neumann case. Section 4.2.
October 16: Separation of variables: periodic, mixed, and Robin cases. Sections 4.2, 4.3.
October 18:
Midterm review.
October 23:
Midterm.
October 25:
Orthogonal and orthonormal basis. The mean-square convergence. Sections 5.3, 5.4.
October 30: Various notions of convergence. Series. General Fourier series. Sine and Cosine Fourier series. Sections 5.1, 5.3, 5.4, A.2.
November 1:
Parseval's identity, completness, Bessel inequality. Section 5.4.
November 6:
Periodic, odd, and even functions and extensions. Pointwise and uniform convergence of Fourier series. Sections 5.2, 5.5.
November 8:
Proof of pointwise convergence of Fourier series. Section 5.5.
November 13: Symmetric Boundary Conditions. Laplace equation. Harmonic functions. Sections 5.3, 6.1.
November 15: Laplace equation in polar coordinates. Harmonic functions. Section 6.1.
November 20: Dirichlet, Neumann, and Robin boundary conditions for the Laplace equation in the disk. Section 6.3
November 22:
Laplace equation in an annulus and a rectangle. Sections 6.2, 6.4.
November 29:
Laplace equation in a rectangle. Final review. Section 6.2.

Midterm Test. There will be an in-class midterm test on Tuesday, October 23. No aides are allowed for this test.
The midterm wil cover sections 1.1-1.5, 2.1-2.5, 4.1-4.3.
Recommended practice problems (do not turn in): 1.1.3, 1.1.10, 1.2.3, 1.2.7, 1.3.3, 1.4.1, 1.5.2, 1.5.5, 2.1.2, 2.1.7, 2.2.2, 2.2.4, 2.3.3, 2.3.4, 2.4.4, 2.4.9, 2.5.2, 2.5.3, 4.1.2, 4.1.4, 4.2.1, 4.2.2 (we did the last two problems in class, but try to re-derive them on your own).

Final exam. You will be allowed to use one one-sided letter-sized page of notes. Textbooks or calculators are not allowed for this exam.
Recommended practice problems (do not turn in): review all the midterm practice problems, and also: 4.3.2, 4.3.6, 5.1.4, 5.1.6, 5.2.1, 5.2.9, 5.3.3, 5.3.9, 5.4.8, 5.4.15, 5.5.4, 5.5.5, 6.1.5, 6.1.6, 6.2.3, 6.2.4, 6.3.2, 6.3.3, 6.4.1, 6.4.5.
Extra office hours for the final:
TA: Friday, December 14, 12-2. Location: DV2068B
Professor: Tuesday, December 18, 10-12. Location: DV2068B

Grading. Grades will be based on eight homework asignments (5% each), Midterm test (25%), and Final exam (35%). I will also occasionally assign bonus problems.

Late work. Extensions for homework deadlines will be considered only for medical reasons. Late assignments will lose 10% per day.Special consideration for late assignments or missed exams must be submitted via e-mail within a week of the original due date. There will be no make-up midterm tests or final. Justifiable absences must be declared on ROSI, undocumented absences will result in zero credit.