Math 5334 & 6333: Homework notes

Math 5334 & 6333: Homework Notes

Hint for 2.5.15

The eigenvalue problem should be in the y-direction using the Dirichlet BC's at y=0 and y=H.
The solution in the x-direction should be in terms of sinh and cosh.
Form the complete solution and apply the condition at x=0 to determine the coefficient in front of cosh.
To determine second coefficient use the physical argument that we expect the solution to remain bounded for x ->oo. Specifically,
lim_{x -> oo} |u(x,y)| < oo.
Note that cosh and sinh can be written in terms of exponentials where one decays and the other explodes as x -> oo. Must kill the unbounded terms.
Answer, A_n = - B_n

Suggestion for 3.3

Only two problems have been assigned but consider doing as many as you need to understand the difference between the different series. Also, for 3.3.3b where you are asked to sketch the first two terms of the series, you might use Mathematica or Maple to experiment. This is not necessary though.

Hint for 5.3

5.3.3: H(x) is derived just as you would an integrating factor for first order DE's.
5.3.7: This problem was solved in Section 2.4.2 so you don't need to do all the prelinary solution work over again. The point of this problem is to emphasize that even though the periodic BC's make this BVP NOT of Sturm-Liouville (SL) type, many of the (SL) properties hold.

Hints for 7.7.*

7.7.3: This problem is a small modification of our in-class example.
7.7.7: Now only the ode in time is modified
7.7.10: Now simply drop the theta dependene

Hints for 9.3.*

9.3.12: There is no need to reinvent the wheel in this problem. The appropriate integral relation is either in the book or can be determine from the general relation we discussed in class. The appropriate Green's functions have also been previously determined. The challenge is to identify which to use and how to put them together.
9.3.13: The problem asks for only outgoing waves referenced with respect to the pulse location at x = x_o. Thus, for x > x_o we want right going waves and for x < x_o we want left going waves. The time dependence is ~ e^{-iwt} while the space dependence is ~ e^{+- ikx}. Thus, there will be waves like e^{i(kx - wt)}, which is a right going wave, and waves like e^{-i(kx+wt)}, which is a left going wave. Choose the type appropriate for either x > x_o or x < x_o. In solving the problem this consideration takes the place of boundary conditions. You will still need to use continuity and a jump condition.

Assignment 3

This weeks homework is fairly light. Please use this "break" to study what we have been doing regarding Greens functions. It is important to understand where the adjoint operator comes from and how it is used. We start Greens functions applied to PDEs next week; it would be best to have a good handle on ODEs.

Hints for 9.5.*

9.5.5: You do not have to solve for u; just determine the appropriate integral relation.
9.5.9(b): The Dirac-d (d=delta) function in cylindrical coordinates is

d(x - x_0) = (1/r) d(r - r_0) d(theta - theta_0)

That is, there is a 1/r multiplying the usual delta functions. The BC at r = 0 is that the solution must be bounded. The algebra can be ugly but otherwise the method is the same as usual.
9.5.10(a,b): In Green's formula assume that the boundary contribution is 0. That is, J(u,v) is 0 due to solutions that decay at infinity.
The Dirac-d (d=delta) function in spherical coordinates is

d(x - x_0) = (1/p^2) d(p - p_0)

That is, there is a 1/p^2 multiplying the usual delta functions. I've left off the angular pieces since we assume symmetry in infinite space. To determine G let G = H/p and solve the resulting equation for H; this will produce the waves.
Please see me if you need additional clarification on any of these points.