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HW assignments (problem numbers followed by * will be graded):
HW 1 (due Mon 1/23): 1.4.7(c)*,2.3.2(d)*,2.3.6,2.3.7*,2.3.8*(solutions)
HW 2 (due Mon 2/6):Homework 2 (solution to wave problem,
rest of solutions)
HW 3 (due Mon 2/20): 5.3.5,5.3.6*,5.3.8*,5.5.1,5.5.3*,5.5.8*,5.5.9 (solutions)
HW 4 (due Wed 3/8 ): 5.8.1* (hint: read section 5.8), 5.8.5, 5.8.6*,5.8.8,9.3.5*,9.3.9*,9.3.10 (solutions)
HW5 (due Th 3/23. the day of the final exam; note: this HW is optional):9.3.7, 9.3.8,9.3.22*, 9.3.23, 9.3.26(c)*, rest of 9.3.26
- Notes on the Poisson kernel (Feb 6, 2017)
- The midterm on Fr 2/10 covers the material from the first 3 chapters of the text, as well as additional topics discussed in class. Specifically: Chap 1: modeling chapter focused on the heat equation, boundary conditions (physical interpretation) and the calculation of simple steady states of the heat equation. Chap 2: the method of separation of variables, applied to various problems arising from specific scenarios related to the heat equation. The Laplace equation (on rectangle and on disk), solved by separation of variables. Qualitative properties of solutions of the Laplace equation including the mean value property, the maximum principle and uniqueness of solutions. Chap 3: Fourier series, including Fourier sine and cosine series; know how to calculate Fourier coefficients. Understand the statement of Fourier's convergence theorem and be able to sketch the graphs of Fourier series of given functions. (skipped: justification of term-by-term differentiation and integration, and proof that Fourier series obtained from the method of separation of variables, are genuine solutions of PDEs). Wave equation on an interval (solved with separation of variables), and on the real line (solved using
d'Alembert's formula). Be able to sketch the solution to the wave equation based on d'Alembert's formula, given the initial displacement and velocity of the string. You need to be able to apply the main results and principles discussed in class, to problems which will be similar to those assigned as homework. Best preparation: solve the problems from the text in chap1-3.
Note: the midterm is closed book, and no formula sheets, calculators, electronic devices etc are allowed. Bring only white paper and pen/pencil. You will be provided with the eigenvalues and eigenfunctions of the boundary value problems on the 2nd page of your book (inside cover page), as well as the Laplacian in polar coordinates.
- The comprehensive final exam will be in our usual classroom (STAG 111) on Th March 23rd, 9:30-11:20am. You are allowed to bring a pre-approved
1 sided page containing not more than 15 formulas/theorem statements etc (but not worked-out solutions to problems). You should send this page to me via email by Sun March 19th. I will approve your list or send it back to you with edits. I will then print the approved sheet and bring it with me to the final exam, so you don't need to bring it yourself.
- The final is comprehensive and covers everything mentioned above for the midterm, as well as the following topics: Sturm-Liouville operators and boundary value problems, including Raleigh quotient, Lagrange identity and Green's formula, self-adjoint linear differential operators, and boundary value problems with boundary conditions of the third kind giving rise to graphical solution methods to find eigenvalues (all this roughly covers corresponding sections in chapter 5 of our text). Finding Green's function: using direct integration, variation of parameters method, eigenfunction expansion (including Fredholm's alternative when there is a zero eigenvalue), and using Dirac delta functions. You do not need to know infinite space Green's function, nor the method of images.
Note: the final exam is closed book, but a pre-approved formula sheet signed by me can be used; calculators, electronic devices etc are not allowed. Bring only white paper and pen/pencil. You will be provided with the formulas related to Sturm-Liouville problems on the 1st page of the inside of the cover of our text, and the eigenvalues and eigenfunctions of the boundary value problems on the 2nd page, as well as the Laplacian in polar coordinates.