Physics 834: Problem Set #8

Recent changes to this page:

• 14-Nov-2011 --- Updates: please read!
• 11-Nov-2011 --- Original version.

1. 3-D wave equation.
   1. This part should be a direct generalization of examples we've considered or those in Lea in which we take the Fourier transform of an equation. Don't go to a particular coordinate system yet (i.e., keep x and k as vectors. The transform S is given in terms of the transform F (which is left as a function until parts c and d).
   2. Assume α > 0. Just find where the poles are! (What kind of equation do you need to solve?)
   3. First find F and then set up the inverse Fourier transform. You might find it simplest to do the ω integral first (but it is not essential). This is the same in parts c and d, and uses the result from part b. Consider t<0 and t>0 separately (one of them is very easy!). For the k integrals, you'll want to use spherical coordinates, as in problem 5 from the last problem set. Use the same trick of choosing the k_z axis so that the angle between k and x is the polar angle. Do the r integral last and you are encouraged to use Mathematica (but include a printout if you do).
   4. This is quite similar to the last part, right up to the r integral. But this one is easy, if you remember about Fourier transforms and delta functions. Also remember when getting your answer that t>0.
2. Helmhotz equation again.
   1. You need only consider k²>0 (I changed my mind). What is the general solution (with constants)? If you substitute it into the first boundary condition, you can eliminate one constant (and the other will multiply everything, so you can set it equal to one). To be complete, you will need to consider the possibility separately that a=0. How does the second boundary conditions determine the eigenvalue (by a transcendental equation you don't solve at this point!).
   2. You should find a closed-form solution for this case.
   3. If you have a transcendental equation, find the first few numerical values with Mathematica (try FindRoot).
3. Sturm-Liouville derivatives.
   1. The best way to do this is to show that the differential equation for the derivative can be put into the Sturm-Liouville form. The first step is to find this differential equation. Note that the original equation can be written in terms of u and y. A second equation involving u and y follows by differentiating the original equation. Then eliminate y. Now look to put this into the Sturm-Liouville form: d/dx F du/dx - G u + λ W u = 0, where we've introduced the new functions F, G, and W. We can always multiply the equation by a positive function p(x) to transform a non-self-adjoint operator to a self-adjoint one. (See section 10.1 of Arfken.) Determine p(x) by matching the equation for u with the new form (you will have a differential equation that you can directly integrate).
   2. If you've succeeded in the first part, you should just read off the weighting function W(x).
   3. If you rewrite the original boundary conditions in terms of u, you will find that either F must vanish at a and b or there is a condition on the α constants (note that you can't have the eigenvalue appearing in the boundary conditions).
   4. This last part should be immediate: just identify the Sturm-Liouville functions from the original Legendre differential equation to identify the new weight function W(x) for the derivatives. (It should agree with the example problem from class!)
4. Fourier-Legendre series. The idea is to relate I_l to the integral with a lower value of l, and then finally evaluating I₀ directly. What is I_l for odd l? The relevant recursion relations are given on pages 377 to 379 of Lea chapter 8. You will need more than one of them, plus a partial integration (as with the example from class). The Fourier-Legendre seres just means an expansion of the function in terms of Legendre polynomials. So how would you determined the coefficients of such an expansion (your answer should use I_l).
5. Euler-Bernoulli equation. This one is interesting because you don't often solve differential equations with fourth derivatives! Otherwise it is conceptually similar to other problems we've done. You are invited to use Mathematica here for any integrals.
6. Summing a series. Just follow the instructions. The Fourier transform is a much simpler sum than the original (think of the p dependence in the new sum as a power; then it is the Taylor expansion of a simple function). The Mathematica Sum function can do this sum, so be sure to check your answer against it.