Homework Assignments

Contents

Assignment #1, due Feb 10

Due in Math Main Office at 2:55pm, Friday, Feb 10.

1.2

1.2.3 (derive heat equation for rod of varying cross-sectional area)

1.2.6 Grad students only

1.3

1.3.1, 1.3.2 (Show your work!*)

*Note that answers to starred exercises are in the back of the book.

1.4

1.4.3

1.5

1.5.3, 1.5.12

Assignment #2, due Feb 17

Due in Math Main Office at 2:55pm, Friday, Feb 17.

2.2

2.2.4 (Linearity) (a) [2pts] (b) [1pt]

2.3

2.3.2 (d) (Heat equation with prescribed temp at one end, no flux at the other) λ > 0 [4pts], λ = 0 [1pt ], λ < 0 [2pts].

2.3.5 (Show orthogonality of sines.) [4pts]

2.3.X Show that sinasinb = (1)/(2)[cos(a − b) − cos(a + b)] using Euler's formula. [3pts]

l_euler.jpg

2.3.7 (Heat equation IBVP with no-flux boundary condtions.) (a) [1pt], (b) λ > 0 [3pts], λ = 0 [2pts], λ < 0 [2pts], (c) [1pt], (d) [3pts], (e) [1pt].

2.3.8 Grad students only. (Heat equation IBVP with lateral loss of heat.) (a) [3pts] (b) [4pts] Conclude with a sentence describing the effect of the lateral heat loss on the eigenvalues. [2pts]

Assignment #3, due Feb 24

Due in Math Main Office at 3:50pm, Friday, Feb 24.

2.3

2.3.X

a. Make a plot of the solution of the heat equation for a laterally insulated copper bar of length 0.6 meters (2 ft), whose ends are maintained at 0 degrees, and whose entire interior is initially at 100 degrees. Show only snapshots for t = 10, 60, 360 seconds. For your approximation, include terms of the series for n up to M = 100.

Recall that the computation I did in class assumed k = 1, which is not the case here.

b. Run your code again for the case where the bar is stainless steel ("310") rather than copper. Comment on the difference.

c. How long does it take for the temperature at the middle of the bar drop "noticeably" below 100 degrees? (Both materials.) I intend that you do this simply by re-running the code you used for parts a and b for various t-values that you choose by trial and error. Extra credit: make a plot of the temperature at the mid-point versus time.

2.5

2.5.X

a. Solve Laplace's equation on the rectangle 0 ≤ x ≤ 2 0 ≤ y ≤ 3 with u = 0 on the boundary except for the side with x = 0 where u(0, y) = sin(πy).

b. Make a picture of your solution by slightly modifying the following code:

from numpy import *
from matplotlib.pyplot import imshow, savefig, colorbar

xmin,xmax = 0,2.5
ymin,ymax = 0,3.5
nx,ny = 250,350

x,y = meshgrid( linspace(xmin,xmax,nx), linspace(ymin,ymax,ny) )

u = 100*sin(x*y)

imshow(flipud(u),interpolation='bilinear',cmap='cool',extent=[xmin,xmax,ymin,ymax])
colorbar()
savefig('temp.png')
2.5.X_sample.png

2.5.Y

Solve Laplace's equation for u(r, θ) on the quarter-disk 0 < r < a, 0 < θ < (π)/(2) subject to boundary conditions u(a, θ) = g(θ) and no normal flux on the straight sides (θ = 0 and θ = (π)/(2)). I will show you a similar problem in class on Thursday.

Assignment #4, due March 10

Due in Math Main Office at 3:50pm, Friday, March 10.

IMPORTANT: Make it very clear which answer is for which question. And turn in the problems in the order they appear below.

3.2

At points of discontinuity, use open circles for a limit point that is not included, and filled-in circles for the actual value there.

3.2.1 b,d,f Sketch the Fourier series on the interval -3L to 3L.

3.2.2 e Sketch the Fourier series on the interval -3L to 3L.

3.2.4

3.3

3.3.1 c Sketch all the series on the interval -3L to 3L.

3.3.2 c. Additionally use Python to make a picture of the 100th partial sum of the sine series. (See Day 11 notes. You need only the delete key, I think!)

Assignment #5, due March 31

Due in Math Main Office at 3:00pm. NOTE EARLIER TIME (per request of grader)

IMPORTANT: Make it very clear which answer is for which question. And turn in the problems in the order they appear below.

4.2

4.2.1 (Wave equation for string sagging under gravity.)

4.4

4.4.1 Note that for part (b), the right end isn't "free" in the sense of being completely untethered. Think of it as being attached to a frictionless vertical rail.

4.4.3 (When you come up with a hulking collection of constants that will appear over and over again, give it a name so you don't have to write it more than once or twice!)

4.4.7 (Solution as two travelling waves.) Comment: I did not find it necessary (or helpful) to use either of the hints supplied in the book. You can just find out what properties of F are required for the given formula to satisfy all parts of the IBVP.

traveling_wave.png

4.4.9 (Conservation of energy.)

Assignment #6, due April 7

Due in Math Main Office at 3:00pm. NOTE EARLIER TIME (per request of grader)

Make it very clear which answer is for which question. And turn in the problems in the order they appear below.

5.4.X Use this code along with trial-and-error to find the third smallest eigenvalue of the non-uniform bar heat flow problem we have considered in class. Also make a plot of the corresponding eigenfunction. Determine λ3 accurately enough that |φ(20)| (which should be exactly 0) is no bigger than .001.