## Math 442 Homework, Spring 2008

Other books

I recommend that you get a book on ordinary differential equations, such as C. H. Edwards and D. E. Penney Differential Equations and Boundary Value Problems or any similar text. And some students will benefit from browsing S. J. Farlow Partial Differential Equations for Scientists and Engineers, a clearly written (and inexpensive) book that is somewhat more "physical" than the text by Strauss, and somewhat less theoretical. A more comprehensive book is R. Haberman Elementary Applied Partial Differential Equations, which covers far more examples than Strauss, although with somewhat less theoretical insight. The books of Strauss, Farlow and Haberman are all on reserve at the Mathematics Library in Altgeld Hall, and I encourage you to get to know them.

Homework procedures

Homework should be turned in by 5pm on the due date. To be accepted, every homework must start with a cover sheet (print 11 copies of this sheet). All homework must be stapled.

I encourage you to work with a partner or in a small group, both when preparing for class (doing the reading) and for the homework.
You must write up the homework solutions yourself, even if you have worked with others on solving the problem.

Late homework will be accepted only if you have made prior arrangement with me. Exceptions will be made, for example, in case of serious illness or family emergency.

I will only count your 10 highest homework scores, over the semester. Some assignments might be announced as mandatory (i.e. they cannot be dropped).

Homework study session

The Monday office hour 4:30-5:30pm will be a collaborative homework study session, held in Altgeld 345. Come and work with your classmates on the homework.

Announcements

The median scores on the Homeworks so far are: 8.5, 9.5, 9, 10, 10, 10, 9.5, 13.3 (out of 15), 9, 9.5, 10 (most HW out of 10)
The median score on Test 1: 51/75 (top score 75/75)
The median score on Test 2: 85/100 (top score 103/100)
The median score on the Final Exam: 134/225 (top score 223/225)

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 Homework # 1 Section 1.1: 2abc (True/False), 3bcef, 10, 11, 12 (For #10, you may use definitions and results from an ODE textbook.) Section 1.2: 1, 3, 6, 7 Due Wednesday 23 January Homework # 2 Section 1.3: 2, 3, 4, 6 (#2: take a naive approach - determine T and re-use the wave equation) (#3: you may use a 1D idealization of the rod, analogous to Example 4 on page 14. Note Newton's Law handles a different situation from Fourier's Law.) (#4: "homogeneity in the horizontal directions" means that u=u(z,t) is independent of x and y) (#6: we are assuming that kappa, c and rho are all constant, and we write k=kappa/(c rho); also, for #6 we can use the formula for the 2-variable Laplacian in polar coordinates, which is formula (5) on page 151, in Section 6.1) Section 1.4: 1, 3, 4, 5 Terminology: "steady state" and "equilibrium solution" mean the same thing, namely that the solution depends only on position and not on time. In other words, u=u(x) is independent of t. (Hence ut=0.) (#3: note that D is a region in 3 dimensions, and x=(x1,x2,x3) is a point in D. In this problem, you can assume on physical grounds that the steady state is a constant, u(x)=C. Your task is to find C in terms of the given information.) (#4: the answer in the back of the book is off by a minus sign) Due Wednesday 30 January Homework # 3 Section 1.5: 4 (Notes. The divergence theorem is stated in Appendix A.3, on page 393. Apply it with f=gradient(u). For part (c) of the problem, I want you to interpret both (a) and (b) physically for diffusion - do not give a heat flow interpretation. Here the function f(x,y,z) represents the net rate of mass creation at the point (x,y,z), via sources and sinks, as indicated briefly on page 15.) Section 2.1: 1, 5, 7, 8, 9 (#5: in the Hint, you are taking the intersection of two intervals and then finding the length of that intersection - why does this evaluate the given integral?) (#7: recall a function is odd if h(-x) = -h(x) for all x.) Section 2.2: 1, 2, 3, 5 Due Wednesday 6 February Homework # 4 Section 2.1: 5, the Hammer Blow problem again. On this HW, I want you to use some differential equations software to illustrate the solution you found on the last HW. For example, you could use the Iode software that runs under Matlab; see `www.math.uiuc.edu/iode/`. You can illustrate your solution both with snapshots and with a solution plot in the xt-plane. Take a=c=1 for simplicity. The office hour on Tuesday 12 February 4-5pm will be held in 130 Altgeld Hall (a computer lab), where I can help you get Iode installed and running. (No preparation is required, on your part.) Section 2.2: Consider the wave equation in a bounded region D in three dimensions, with homogeneous Dirichlet boundary conditions. Question 1. Show that energy is conserved. (Here we use the definition: energy=(1/2) times the integral over D of ut2 + c2 |gradient u|2.) (Hint: |gradient u|2=gradient u dot gradient u. Another hint: divergence theorem.) Question 2. Is energy also conserved under homogeneous Neumann boundary conditions? Question 3. State and prove an energy dissipation result for the diffusion equation in three dimensions, under homogeneous Dirichlet or Neumann boundary conditions. Section 2.3: 3, 6, 8 (#3(a): Just show u(x,t) is greater than or equal to zero. #3(c): I encourage you to use Iode. Play around with the "Plot resolution" option, in order to get a reasonably accurate graph.) Section 2.4: 1 Due Wednesday 13 February HW 4 Feedback Homework # 5 Download the homework Due Wednesday 27 February Homework # 6 Download the homework Due Wednesday 5 March Homework # 7 Download the homework Due Thursday 13 March by 5pm Homework # 8 Download the homework. Part of the homework is the Iode Fourier Series Project. (Problem 2(d) in this project is optional.) Due Wednesday 26 March Worth 15 points total. Homework # 9 Section 5.4: 8, 18 Section 5.5: 10 Remarks: You are asked to prove Theorem 5.4.2 (as stated in class, not in the book) for the case of sine series and cosine series. But we have proved it already for the full Fourier series, in Section 5.5, and so you may use that result here to help you. Section 5.6: 6, 13 Section 6.1: 4, 5, 2, 3 (but in problems 2 and 3, change k2 to -k2; this changes the answers of course!) Hint for #3: see page 252 for Bessel's equation and functions Additional problem: Show by example that term-by-term differentiation of the Fourier series of a function f(x) can fail to give the series of the derivative df/dx, for a function f that jumps. You could consider the sine series of f(x)=1, for example. (Aside: this sort of problem is why the Fourier series of uxx does not just equal the second x-derivative of the Fourier series for u, in Section 5.6.) Due Wednesday 9 April Homework # 10 Download the homework. Due Wednesday 23 April Homework # 11 Section 10.3: 4 (you may use orthogonality of Bessel functions in the sense of page 261, and you may use (8) on page 270) Section 10.3: 5, 10 Section 9.2: 11 and interpret the two parts of your solution in terms of traveling waves (hint for solving: reduce to the 1 dim wave equation similar to formulas (5) and (8) in Section 9.2) Section 9.2: 7ab (Hint: proceed like in problem 11, by solving the associated one dimensional wave equation on the halfline r>0; should you extend evenly or oddly? Aside: the sketch of the support region in space-time should help you understand the answer intuitively.) Due Wednesday 30 April