Final Exam Details: 9-DECEMBER-2008
The final examination for APM 346 is scheduled for Tuesday, 9 December, 2-5 p.m. Room assignments and other details are avaiable on the U.T. exam time table (http://www.artsci.utoronto.ca/current/undergraduate/exams/dec08) . Course material includes the starred sections in Strauss textbook, especially those emphasized and covered in class plus all material covered in lecture notes. Students are encouraged to review Homeworks:1- 9, Midterm Test 1 and Midterm Test 2 during the final exam preparation. Laplace equation (i.e Homework 10 IS NOT included into the final). Structure of the final exam: quiz part (40%) and problem solving part (60%). Quiz part combines some not all questions: from quiz parts of two midterm tests, max.-min. principle questions for the heat equation, energy method questions (heat and wave equations), Fourier series questions. Problem solving part consists of 5 problems: application of the max.-min principle (heat equation), application of the energy method (wave equation), inhomogeneous hyperbolic equation, application of the Fourier method, application of the method of characteristics for the inhomogenious transport equation with nonconstant coefficients. What would be usefull to remember: Fourier (sine, cosine, full) series representations, d’Alembert formula, classifications of PDEs, classifications of boundary conditions. Levels of difficulties for the problem solving part: application of the max.-min principle (heat equation) (easy), application of the energy method (wave equation) (easy), inhomogeneous hyperbolic equation (moderate), application of the Fourier method (moderate), application of the method of characteristics for the inhomogenious transport equation with nonconstant coefficients (difficult). Good luck !
General Information:
The midterm tests will last approximately 50 minutes and will be held during regularly scheduled lecture time.
The midterm tests and final exam will be closed book (i.e., NO aids allowed), and the midterm should be written in pen (remarking requests for tests written in pencil will not be accepted) – the test paper will have lots of room for rough work. Colour pencils should be used for pictures.
Second Midterm Test Details:
The second midterm test will be written during regularly scheduled lecture time on Monday, November 10, lasting about 50 minutes. The test will be written in room SF 3201 between 9:00am and 10:00am, NOT in the usual lecture room! SF is Sandford Fleming Building.
The test may cover
Lectures: 11 – 21, Sections from the textbook: 2.1 – 2.3; 3.4; 4.1 – 4.3, Homeworks: 4 – 7.
Some of the topics that may be covered by the test (obviously I can’t test everything) include:
- Homogeneous wave equation (infinite domain): general solution, IVP solution (d’Alembert solution), stability of the d’Alembert solution.
- Homogeneous hyperbolic equation (infinite domain, non-dispersive case):transformation by the substitution
u = exp(ax + bt)v, IVP solution. - Inhomogeneous wave equation (infinite domain): general solution, IVP solution (Green’s Theorem, method of characteristics).
- Homogeneous wave equation (finite domain): general solution of BVP by separation of variables, solution of BIVP by spectral decomposition (Fourier) method.
- Homogeneous heat equation (finite domain): general solution of BVP by separation of variables, solution of BIVP by spectral decomposition (Fourier) method.
- Application of the energy method to the wave equation.
- Maximum principle and application of the energy method to the heat equation.
First Midterm Test Details:
The first midterm test will be written during regularly scheduled lecture time on Monday, October 6, lasting about 50 minutes. The test will be written in room SF 3201 between 9:00am and 10:00am, NOT in the usual lecture room! SF is Sandford Fleming Building.
The test may cover
Lectures: 1 – 10, Sections from the textbook: 1.1; 1.2; 1.4; 1.5; 1.6; 2.1; 2.2; 3.4, Homeworks: 1-3.
Solutions for some practice problems.
Some of the topics that may be covered by the test (obviously I can’t test everything) include:
- General PDE classifications: order of an equation, linear, nonlinear, homogeneous, inhomogeneous
- Types of the second order PDE: elliptic, parabolic, hyperbolic.
- General solutions of the first-order linear PDE with constant and non-constant coefficients.
- Initial value problems for the first-order linear PDE with constant and non-constant coefficients. Characteristic curves. Application of the local well-posedness theorem.
- Domain of the well-posedness for the initial value problem. Uniqueness and non-uniqueness of the solution.
- General solution of the wave equation with constant coefficients
- Change of variables and transformation to the canonical form.