Math 385 Spring 2006
Khalid Boushaba
Assistant Professor of Mathematics
Office: 446 Carver Hall
Office Phone/Voice Mail: 294-8147
Office hours:
Tuesday:9:20am-10.10am
Thursday:9:20am-10.10am
Friday:10am-11am
Or by appointment see my schedule.
Math 385, Introduction to Partial Differential Equations. (3-0) Cr. 3. Prereq: Math 265 and one of Math 266,267. Separation of variables methods for elliptic, parabolic, and hyperbolic partial differential equations. Fourier series, Sturm-Liouville theory, Bessel functions, and spherical harmonics.
Course grades will be determined from student performance on exams, and homework assignments, although the emphasis will be on the exams. There will be two 75 minute (in class) exams. These will be given (at least approximately) at the six and twelve week marks of the semester. There will be also be an exam during the week of final exams. The exams will concentrate on material covered in the previous five weeks, but may include material from earlier in the semester. Questions will be based on homework assignments, which will be made each class meeting. A straight scale will be used which may be shifted down if necessary to account for more challenging exams.
Your course grade will be determined from (percentages subject to adjustment):
[20 %] Written homework. A written assignment will be due roughly every 4 weeks. You may work alone or with partners (max 3) on homework. If you work with a partners, submit one paper with group names on it.
[40 %] Exams (2) (20% each ) .
[40 %] Final
| Weekly Syllabus |
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| Week
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Dates
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Tuesday
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Thursday
|
| 1 |
Jan. 10-12 |
0.1-0.2 |
0.1-0.2 |
| 2 |
Jan. 17-19 |
0.3 |
2.1 |
| 3 |
Jan. 24-26 |
2.2 |
2.3 Assigned Problems I Due |
| 4 |
Feb. --2 |
2.4 |
2.5 |
| 5 |
Feb. 7-9 |
2.1-2.5 |
1.1 |
| 6 |
Feb. 14-16 |
Exam 1 |
1.1-1.2 Assigned ProblemsII Due |
| 7 |
Feb. 21-23 |
1.3 |
1.3-1.10 |
| 8 |
Mar. --2 |
1.10-1.11 |
1.11 |
| 9 |
Mar. 7-9 |
2.6 |
2.7-2.9 Assigned ProblemsIII Due
|
| 10 |
Mar. 14-16 |
spring |
break |
| 11 |
Mar. 21-24 |
4.1-4.2
|
4.5 |
| 12 |
Mar. 28 - 30 |
Review |
Exam 2 |
| 13 |
Apr. 4-6 |
3.1-3.2 |
3.3-3.6 Assigned Problems IV Due |
| 14 |
Apr. 11-13 |
5.2-5.3 |
5.4-5.5 |
| 15 |
Apr. 18-20 |
5.5 |
5.6 |
| 16 |
Apr. 25-27 |
5.9 |
5.10 Assigned ProblemsV Due |
|
A written assignment will be due roughly every 3~4 weeks. You may work alone or with partners (max 3) on homework. If you work with a partners, submit one paper with group names on it
( Although I may choose only 4 problem to grade, you must turn in all assigned problems)
Assigned Problems I:
page 10 #6, 12, 18
page 20 #6, 8, 22
page 31 #1,4,10
Assigned Problems II:
Page 142 #2,6,8
Page 156 # 9
Page 167 # 4.6,16
Assigned Problems III
Matlab assignment.
Download the linked m-file part_sum_c.m . This can be used to plot trigonometric polynomials of the form
y(t) = a0 + a1cos(t)+ b1sin(t) + a2cos(2t)+ b2sin(2t) + ... + ancos(nt)+ bnsin(nt)
In the m-file (program), the arguments are as follows:
a=[a0, a1 ,..., an]
b=[b1, b2 ,..., bn]
t=row containing times at which y(t) is evaluated.
A=½ period. (Ignore this for now.)
Here is an example of how this is used. Suppose we want to plot y=1+2cos(t)+3sin(t)-sin(3t) on [0,4p]. To do this, we type the lines below, hitting return after each line.
a = [1,2,0,0];
b = [3,0,-1];
t = linspace(0,4*pi,500);
y = part_sum_c(a,b,t);
plot(t,y);grid
xlabel('t'), ylabel('y'), title('y=1+2cos(t)+3sin(t)-sin(3t)')
Be careful about the backquote ' . This needs to be in the code; it indicates the start and finish of a character string. Check the Matlab documentation for exporting the figure. Include the plot in your homework. Do the same for these trigonometric polynomials:
a) y=3cos(t)-2sin(t)+½cos(2t)+½sin(2t)
b) y=sin(t)+sin(2t)/2+sin(3t)/3+sin(4t)/4 (Note: just set a=0).
b) y=cos(t)+cos(3t)/32+cos(5t)/52 (Note: just set b=0).
In this assignment, we will learn how to make a plot of a periodic functions, given the function on one period. We begin with the Matlab function mod. Look up the Matlab help entry on mod. Execute the following code:
x=linspace(-4*pi,4*pi,1000);
y1=mod(x,2*pi); %x mod 2*pi; values, [0,2*pi]; breaks, 2*n*pi.
y2=mod(x+pi,2*pi); %Graph of y1 shifted pi units to left.
y3=mod(x+pi,2*pi)-pi; %x mod 2*pi; values, [-pi,pi]; breaks, (2*n+1)*pi.
subplot(3,1,1), plot(x,y1,'b'),grid, axis([-4*pi 4*pi -pi 2*pi])
subplot(3,1,2), plot(x,y2,'r',x,y1,'b'),grid, axis([-4*pi 4*pi -pi 2*pi])
subplot(3,1,3), plot(x,y3,'m',x,y1,'b'),grid, axis([-4*pi 4*pi -pi 2*pi])
The point is that the functions mod(x,2*pi) and mod(x+pi,2*pi)-pi return the angles corresponding to x with values in [0,2p] and [-p,p], respectively. In the following plots, take x to be the same as above. By the way, don't worry about the vertical line joining discontinuous pieces. We'll get rid of it later. Be sure to put titles on all of the plots.
a) If f(x)=exp(x/4) on the interval [0,2p], plot the 2p periodic extension of f(x).
b) Repeat part (a), except that f(x)=exp(x/4) on [-p,p].
c) If f(x)=exp(x/4) on the interval [0,pi;], plot the even, 2p periodic extension of f. (Hint: the even extension of f from [0,p] to [-p,p] is f(|x|). In Matlab, |x|=abs(x)).
d) Repeat part (c), but find the odd extension instead of
the even one. (Hint: you will need to use Matlab's function sign.)
Assigned Problems IV
page 125 #6
page 125 #10
page 132 #68
page 257 #6
page 273 #1,2,7
Assigned Problems V
page 250 #22
page 280 #10
page 289 # 24
page 308 #6
Review Problems for Exam I
Answers for Review Problems I
Review Problems for Exam II
Sketchy Answers for Review Problems II