Homework 13 collaboration area
Question, Page 546, Problem 9:
Are the Boundary Conditions u(1/4,t) = 0 and u(3/4,t)=0 or u(0,t)=1/4 and u(L,t)=3/4? Also, can anyone help explain what f(x) is in this problem? I think g(x) = 0, but I'm not sure about f(x)?
Answer: Boundary Conditions: u(0,t) = 0, u(L,t) = 0. In this problem, L=1.
Initial Conditions: u(x,0)= f(x), which can be seen from the diagram as
f(x) = x - 1/4 for 1/4 < x < 1/2 and f(x) = -x + 3/4 for 1/2 < x < 3/4.
(Actually, the initial shape is supposed to be k=.01 times this, but that just puts the same k in front of the series for the solution.)
You'll have to split up the integral when calculating A_n. And yes, the last Initial Condition is
d(u)/dt(x,0) = g(x) = 0.
You'll get an ugly integral evaluation but most terms cancel and it leaves you with 3 sine terms that then go into the fourier series.
Question, Page 547, Problem 15:
How do we show that the constant is beta^4 without any boundary conditions to work with?
Answer. The beta to the fourth power is just a way to name the positive constant lambda to make the solutions easier to write. There will also be the cases lambda=0 and lambda negative (= minus beta to the fourth) to deal with. You won't use boundary conditions to eliminate solutions until problem 16.
Question, Page 552, Problem 5:
How do we show p_n? I think I understand that this is part of calculating lambdas using the Sturm-Louiville, but I haven't been able to figure it out.
Answer: The p_n come from the boundary conditions. (The problem is similar to the Sturm-Liouville problem on Exam 2.) The boundary conitions are
X(0)=0 and X'(L)=0.
Question, Page 548, Problem 16:
What about f(x) for this problem? I am really having a hard time identifying the f(x) for these problems. (Actually, this entire section in general) Does anybody know of a good reference for example problems?
Answer: You'll have to specify initial conditions
u(x,0)= f(x), and
$ \frac{\partial u}{\partial t}(x,0)=0 $
Question: Okay, so I have figured out how to solve for the c1, c2, etc. I can't figure out how to back-solve for the Lambda. Any advice.
Answer: The lambdas that are eigenvalues are just the values that allow non-zero solutions to the ODE with the given boundary conditions.
Question, Page 546, Problem 16:
Plugging in the boundary conditions, I get a set of four equations involving A, B, C, D, beta and L. I'm not sure what to do from here to solve for F(x).
Answer: The boundary conditions are used to eliminate many solutions. You'll end up with a pretty simple looking solution in the end.
Question: Is du/dt(x,0) = 0 the only given initial condition for this problem?
Answer: No, there is also an implied condition
u(x,0)= f(x).
Question: Also, I'm able to simplify it using the B.C.s and I.C. but I'm still not getting all of the constants, nor am I getting anything in terms of n.
Answer: The sum over n comes in when you take your solution to be a sum of
$ F_n(x)G_n(t) $
After that, you use the initial conditions to determine the two remaining constants that are part of the G_n's.
Question, Page 546, Problem 9:
Using the f(x) given in the problem and plugging into formula (17) on page 544 gives me plots that look like waves (using Maple). However the ends are not fixed. How do I incorporate this into the equation?
Answer: Sounds like you didn't take the odd periodic extension of f(x) properly (or perhaps not far enough out).
I am getting that u(x,t) = k*[x-(x^3)-(3*x*t*t)]. I plug in 4 different values of t, and the endpoints at -1 and 1 change. I know you did this problem in class, but I do not understand why this way of doing it is incorrect.
Question, Page 552, Problem 9:
a)Is anyone else getting a really horrific term for the bn of the odd periodic extension of f(x) = k*(x-x^3)? b)Should we do the integral WRT x and then later substitute x=(x-ct), is that allowable? c)How many N = 1,2,3,... terms should we include in the odd periodic extension? d)How is everyone plotting this?
Answer: It sounds like you are using Fourier series to get the odd periodic extension of f(x). That is definitely the hard way. You can use f(x) directly and the Heaviside function to piece together the odd periodic extension for a couple of periods to the left and right. That's all you need.
From what I understand, we are just supposed to use #13 from lesson 12.4, then make sure we graph it from -1 to 1, instead of from just 0 to 1.
Continued Question: The syllabus says to use #17 from Page 544 (12.3). How is this any different from using #13 from lesson 12.4? I'm unclear on how to graph a function of x and t since we haven't really done any examples in a very long time and the instructions "give 4 graphs" aren't really clear. 4 graphs of what? Are we supposed to select 4 random times? Is this going to be a 3-d graph of x in t?
Question: When you do this, don't you get that the endpoints (at x=-1 and x=1) move for t>0 (u no longer = 0)? If so, do you know how to fix it or what the problem might be?
Answer: No, the endpoints should not move. Sounds like you didn't take the odd periodic extension properly. The function is odd, so you can just use the formula to extend it to [-1,0]. Next, you extend it to the real line from [-1,1] as a function with period 2.
Question, Page 546, #15 -
How come when Prof. Bell worked on the case where lambda > 0 in class, he got the answer of exp(beta x) instead of cosh and sinh (given in the book)?
Answer from Bell: The linear span of exp(beta x) and exp(-beta x) is the same as the linear span of cosh(beta x) and sinh(beta x). The two ways of writing the general solution are equivalent.
