- Question: Where is the exam?
The exam will be in 309 BH.
- Question: In problem 2, we are given the initial
condition u(x,0)=sin x, when x>0. Should this be the partial with
to x? I factored the wave equation into two first order PDEs involving
and v, where w=Ut - Ux and v=Ut + Ux. To determine w and v I need Ux.
Or should I assume U is differentiable at t=0 to get Ux(x,0)=cos x?
I believe that the problem is stated correctly. You only need to
use the factorization of the wave equation to conclude that the
solution can be written as u(x,t)=F(x-ct)+G(x+ct). You should
use this form as your starting point. You can differentiate
u(x,0)=sin(x) with respect to x, if needed.
- Question: Is there a misprint for problem 1?
Shouldn't the boundary
on x=0 be replaced with Wx=h(y) on x=0?
Yes there is a misprint in question 1. The boundary condition
should be h(y) since it is on the x-axis.
- Question: I have two doubts about the final questions:
In problem no.1, shouldn't the boundary condition h be a function of y
Also, problem no.6 part c is a bit vague for me. Shouldn't we construct
using eigenfunction expansion method first and then generate the
utilizing Lagrange identity? I guess it's not correct if we use
expansion method to solve for u directly since it's not homogeneous in
and therefore we can't differentiate Fourier series w.r.t. x.
Thanks for your considerations,
The typo in problem 1 is addressed above. In question 6, I wanted
you to construct the Green's function in part (a) by solving the
homogeneous ODE for x not equal to x_0 using the general solution.
This leads to solution defined in two parts. In part (c), I wanted
you to construct the solution using eigenfunction expansion and
hence get a series solution. Part (b) requires the Lagrange
- In problem 4, we first use Laplace transform to solve the problem,
need to use
Green's function. Would you please talk a little bit about how to use
solve that heat eqution? Does the result look the same as the one by
The second part of question 4 asks you to construct the solution
using the Green's function. The Green's function solves the
problem with homogeneous bc and ic, and 1 replaced by the product
of delta functions. The Green's function for the semi-infinite
interval problem can be constructed by the method of images. The
formula is given in (11.3.34). With the Green's function, you
then construct the solution u using the Lagrange-type identity for
the heat equation (11.3.21), which is summarized for the one
dimensional case on the top of page 530 (boundary terms only).
Showing the two solutions are the same is a bit difficult and
I am not expecting you to be able to that on the exam. Constructing
either form of the solution is fair game.
- In question 4 you ask us to use Laplace transforms to solve the
equation. However, in the book and in the notes that you did in class
only did it with Utt=Uxx and Ut=Ux. Is the way if solving them the
way or is it a typo? Also with the +1 at the end of the equation do you
make the transform and then us it to solve?
I believe I solved a heat equation problem 13.4.4 in class before the
wine cellar problem. You apply the Laplace tranform in t to the PDE
to get an ODE in x with a boundary condition at x=0 and boundedness
at x=infinity. You use L=1/s in the PDE.
- For number 3, I got the resulting ODE's using the method of
are a. x'=-tx and b. w'=-w. I solved for a. using the integrating
(1/2)t^2 and I solved for be by assuming the solutions was Ce^rt. and
r=-1. I used x(0)= n to solve for the constants but I don't know what
with the boundary condition. I solved for w without it and now I don't
if my solution is correct. Any suggestions?
You need to solve the characteristic equations a. and b. but the
initial conditions for a. and b. depend on whether the curves are
starting on the initial curve t=0 or x=0. You must parameterize
these curves differently, leading to different forms for the
characteristics. I worked an example in class similar to 12.2.4.
- On problem 6 part c), I'm a little confused. Do you want us to
directly via eigenfunction expansion or construct the Green's function
eigenfunction expansion and then construct u?
I guess that I wanted you to construct the Green's function first
and then construct u. By constructing u directly you should see
the Green's function.
- Question: Just another question... This time number 4. After you
apply the Laplace
transform and end up with the ODE Uxx-sU= 1/s, the only way I know to
this is solving the homog. problem and then using variation of
Unfortunately, the Wronskian is ghastly. Did I make some horrible
is this what you wanted us to do?
The independent variable in the transformed ODE is x and the
Laplace transform variable s is
treated as a constant with respect to x. So you can just use
undetermined coefficients. In other words, treat s as a
constant and it has an easy solution.
- Question: I am still having trouble with number 3. Unlike 12.2.4
these aren't linear
characteristics. I don't know how to find the characteristics.
looked at the solution for x. In this case, I got x=ne^(-t^2/2). This
the t=0 line. However, I have no idea what the characteristics look
got the corresponding w=xe^(t^2/2)e^-t. This is true if x> than what?
on the x=1 line, I got x(1)=r=ce^(-1/2), then I got x=re^1/2e^(-t^2/2)
how do I solve for w(1,t). w still equals ke^-t. If I continue to solve
the fact that w(1,t)= w(1,r)=r= ke^-r, I can't get rid of the
I get e raised to xe raised to a function of t. and r is also
The characteristics are the curves in the x-t plane and
the general solution is x=ne^(t^2/2), use separation of
variables. The characteristics from t=0 have an initial
condition like x(0)=xi while the characteristics starting on
x=1 have the initial condition x(eta)=1. The critical
characteristic that separates the two types starts at
t=0 and x=1. You need to develop the similar types of
initial conditions for the w equation.
The w solution depends on the characteristic
variable xi or eta which is a function of
x and t thru the x solution. Does this help.
To plot the characteristics, you can graph x=ce^(t^2/2)
in the t-x plane and then flip and rotate the graph to
get the x-t plane.
- I am signing off for the night. I will check again
in the morning.