5 02/12 F

1. Finish finding the derivatives of the unit vectors of the spherical coordinate system using the class formulae. Then finish 1st Ed p160 q47, 2nd Ed p183 q47, as started in class, by finding the acceleration. Note that the metric indices $h_i$ for spherical coordinates are in mathematical handbooks. Also,
   
   $$\frac{\partial {\hat \imath}_i}{\partial u_i} = \frac{1}{h... ...rac{1}{h_i} \frac{\partial h_j}{\partial u_i} {\hat \imath}_j$$
   
   

2. Express the acceleration in terms of the spherical velocity components $v_r,v_\theta,v_\phi$ and their first time derivatives, instead of time derivatives of position coordinates. Like $a_r = \dot v_r + \ldots$, etc. This is how you do it in fluid mechanics, where particle position coordinates are normally not used. (So, get rid of the position coordinates with dots on them in favor of the velocity components.)

3. The Laplace equation
   
   $$\mbox{PDE:}\quad u_{xx} + u_{yy} = 0$$
   
   
   where subscripts indicate derivatives, is an elliptic equation. Such a steady-state equation needs boundary conditions at all points of the boundary. For example, one properly posed problem on the unit square is
   
   $$\mbox{BC:}\quad u(x,0)=1\quad u_y(x,1)=0 \quad u(0,y)=0 \quad u_x(1,y)=0$$
   
   
   Identify $\Omega$, $\delta\Omega$, and the type of each boundary condition (Dirichlet, Neumann, Robin).

   The wave equation
   

   $$\mbox{PDE:}\quad u_{xx} - u_{yy} = 0$$
   
   
   is an hyperbolic equation. The above boundary conditions are not properly posed for the wave equation. (as you will see in a later homework.) For the wave equation, one of the coordinates must be time-like, and must have initial conditions instead of boundary conditions. The following initial and boundary conditions are properly posed for the wave equation,
   
   $$\mbox{BC and IC:}\quad u(x,0)=1\quad u_y(x,0)=0 \quad u(0,y)=0 \quad u_x(1,y)=0 \qquad y \le 1$$
   
   
   For each of the four, determine whether it is an IC or BC, and if so, what kind of BC. The time-like coordinate is not normally included in the domain $\Omega$. Under those conditions, identify $\Omega$ and $\delta\Omega$.

   Check that the following proposed solution satisfies the PDE and all BC/IC of both the Laplace and wave equation problems:
   

   $$u = \left\{\begin{array}{l}1\mbox{ for }y<x 0\mbox{ for }y>x\end{array}\right.$$
   
   
   However, it is a valid solution to only the wave equation. Explain for what qualitative reason is it not a valid solution to the Laplace equation.

4. The Laplace equation problem as written in the previous question does not have a simple solution. However, if you distort the domain into a quarter circle as in
   
   $$\mbox{BC:}\quad u(x,0)=1\quad \quad u(0,y)=0 \quad \quad \frac{\partial u}{\partial n}=0\mbox{ on }x^2+y^2=1$$
   
   
   then the solution is simple. Identify and draw $\Omega$ for the above problem. Identify the type of each boundary condition. Also draw $\Omega$ and the boundary conditions for the Laplace problem of the previous question, and then compare the two problems. Are they very similar?

   Now verify, by checking PDE and boundary conditions, that the correct solution to the modified problem is simply
   

   $$u = 1 -\frac{2\theta}{\pi} \qquad \theta=\arctan(y/x)$$
   
   
   You may want to switch to a different coordinate system to do so.

   Plot this valid solutions for the Laplace equation, as well as the valid solution for the wave equation of the previous question, on the circle $r=0.5$ against $\theta$. Comment on whether the change of a single sign between the wave equation and the Laplace equation makes any difference for the solution.

5. 4.18. You are to check that the given $u_N$ satisfies the given PDE, the heat equation, and the given boundary conditions. The given solution is the one you will find using the so-called method of separation of variables, and normally $N=\infty$. What is the type of the boundary conditions? What can you say about the initial condition that is satisfied?

6. The solution to the wave equation problem of question 3 that you would find using separation of variables is:
   
   $$u = \sum_{\textstyle{n=1\atop n {\rm odd}}}^\infty \frac{... ...xtstyle\frac{1}{2}}n\pi x)\cos({\textstyle\frac{1}{2}}n\pi y)$$
   
   
   Verify the PDE, BC, and IC for this solution. For the first IC, you will want to look up the sum you get in the “Fourier series” section of a mathematical handbook, with maybe a rescaled $x$-coordinate.

   Comment whether it would be easy to see the simple form of the solution from merely looking at the above sum. To make understanding the solution easier, use the fact that
   

   $$\sin\alpha\cos\beta = {\textstyle\frac{1}{2}} \sin(\alpha+\beta) + {\textstyle\frac{1}{2}} \sin(\alpha-\beta)$$
   
   
   Describe the two component solutions you get this way in physical terms.

7. The solution to the Laplace equation problem of question 3 that you would find using separation of variables is:
   
   $$u = \sum_{\textstyle{n=1\atop n {\rm odd}}}^\infty \frac{... ...le\frac{1}{2}}n\pi x)\cosh({\textstyle\frac{1}{2}}n\pi (1-y))$$
   
   
   Verify the PDE and BC for this solution.

   Next, shed some light on the question why this solution is smooth for $y>0$, while the previous solution of the wave equation has a jump for all $y$. To do so, first argue that a finite sum of sines of the form $\sin{\textstyle\frac{1}{2}}n\pi x$ is necessarily a smooth function of $x$ with no jumps or whatever other singularities.

   Then show that the coefficients of the sines go to zero much faster for $n\to\infty$ for the solution of the Laplace equation than for the one of the wave equation. In particular show that for any positive power $p$,
   

   $$\lim_{n\to\infty} n^p \frac{4}{\pi n} \frac{\cosh({\textst... ...tyle\frac{1}{2}} n \pi)} = 0 \quad\mbox{if}\quad 0 < y \le 1$$
   
   
   while
   
   $$\lim_{n\to\infty} n^p \frac{4}{\pi n} \cos({\textstyle\frac{1}{2}} n\pi y)$$
   
   
   does not exist for $p>1$.

   With such a fast decay of the coefficients of the Laplace equation solution, the sum is almost finite and singularities are not possible.

8. 2.19b, h. Show a picture of the different regions.