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Show all of your work. Full credit may not be given for an answer alone. You may use the backs of the pages for scratch work. Do not unstaple or remove pages.

This is a non-calculator exam. One side of one sheet of US letter-size paper on which definitions, theorems, and formulas are handwritten by you may be used. Worked-out examples are not allowed on the formula sheet. You will turn in your formula sheet with the exam.

By taking this exam you are agreeing to abide by these rules and Rutgers University's Academic Integrity Policy.

Problem Possible Points
Number Points Earned
$ \displaystyle\vphantom{\int}$ 1 10  
$ \displaystyle\vphantom{\int}$ 2 10  
$ \displaystyle\vphantom{\int}$ 3 15  
$ \displaystyle\vphantom{\int}$ 4 15  
$ \displaystyle\vphantom{\int}$ 5 15  
$ \displaystyle\vphantom{\int}$ 6 10  
$ \displaystyle\vphantom{\int}$ 7 25  
$ \displaystyle\vphantom{\int}$ Total 100  

  1. (10 points)

    Show that the function

    $\displaystyle u(x,y) = \arctan\left(\frac y x\right)
$

    is harmonic (i.e., $ \nabla^2 u = 0$) where it is defined.

  2. (10 points) Formulate a boundary value problem (partial differential equation, boundary conditions, and initial conditions) for the motion of an elastic string of length $ L$, fastened at the left end, and attached with a ring to a pole at the right end. The ring allows the string to slide up and down the pole, so that end is free. The initial position of the string is a function $ f(x)$.

    \includegraphics[height=1.5in]{mt2-2.eps}

    Just set up the problem and stop!

  3. (15 points) Solve the following initial value problem for the wave equation in an infinitely long string:

    $\displaystyle u_{tt}$ $\displaystyle = c^2 u_{xx};$ $\displaystyle (-\infty < x < \infty,\ t > 0)$    
    $\displaystyle u(x,0)$ $\displaystyle = \sin(x);$ $\displaystyle (-\infty < x < \infty)$    
    $\displaystyle u_t(x,0)$ $\displaystyle = x^2.$ $\displaystyle (-\infty < x < \infty)$    

  4. (15 points) Solve the following boundary-value problem for the wave equation:

    $\displaystyle u_{tt}$ $\displaystyle = c^2 u_{xx};$ $\displaystyle (0 < x < L,\ 0 < y < K \ t > 0)$    
    $\displaystyle u(x,0,t)$ $\displaystyle = u(x,K,t) = 0;$ $\displaystyle (0 \leq x \leq L,\ t > 0)$    
    $\displaystyle u(0,y,t)$ $\displaystyle = u(L,y,t) = 0;$ $\displaystyle (0 \leq y \leq K,\ t > 0)$    
    $\displaystyle u(x,y,0)$ $\displaystyle = x(L-x)y(K-y).$ $\displaystyle (0 \leq x \leq L,\ 0 \leq y \leq K)$    

  5. (15 points) Solve the following boundary-value problem for the potential equation:

    \begin{displaymath}\begin{array}{cr} u_{xx} + u_{yy} = 0; &(0 < x < \pi,\ 0 < y ...
...) \\ u(0,y) = u(\pi,y) = 0. &(0 \leq y \leq \pi) \\ \end{array}\end{displaymath}    

  6. (10 points) Consider heat conduction in a thin rod with insulated ends:

    \begin{displaymath}\begin{array}{cr} u_{t} = k u_{xx}; & (0 < x < L,\ t > 0) \\ u_x(0,t) = u_x(L,t) = 0. & (t > 0) \\ \end{array}\end{displaymath}    

    Remember $ k$ is a positive constant. Show that the integral

    $\displaystyle E(t) = \int_{0}^L u(x,t)^2 \,dx
$

    is a strictly decreasing function of $ t$ as long as $ u_x(x,t) \not\equiv
0$.

    Hint 1   Show $ E'(t) < 0$.

  7. (25 points) We will solve the following boundary-value problem for the heat equation:

    $\displaystyle u_{t}$ $\displaystyle = k u_{xx};$ $\displaystyle (0 < x < L,\ t > 0)$    
    $\displaystyle u_x(0,t)$ $\displaystyle = 0;$ $\displaystyle (t > 0)$    
    $\displaystyle u(L,t)$ $\displaystyle = T;$ $\displaystyle (t > 0)$    
    $\displaystyle u(x,0)$ $\displaystyle = f(x).$ $\displaystyle (0 \leq x \leq L)$    

    1. (5 points) Transform the problem into $ u(x,t) = U(x,t) + \psi(x)$, where $ U$ satisfies the heat equation with homogeneous boundary conditions:

      $\displaystyle U_{t}$ $\displaystyle = k U_{xx};$ $\displaystyle (0 < x < L,\ t > 0)$    
      $\displaystyle U_x(0,t)$ $\displaystyle = 0;$ $\displaystyle (t > 0)$    
      $\displaystyle U(L,t)$ $\displaystyle = 0.$ $\displaystyle (t > 0)$    

      What differential equation and boundary conditions must $ \psi$ satisfy? What initial conditions must $ U$ satisfy?

    2. (5 points) Solve for $ \psi(x)$.

    3. (10 points) Find the special solutions for $ U(x,t)$. Write down the general solution.

    4. (5 points) How would one find the coefficients in the general solution? You may find the following fact useful: If $ n$ and $ m$ are integers, then

      $\displaystyle \int_0^L \cos\left(\frac{(2n-1)\pi x}{2L}\right)
\cos\left(\frac{...
...2L}\right)\,dx
= \begin{cases}0 & n \neq m; \\ \frac{L}{2} & n = m.\end{cases}$



Matthew Leingang 2002-11-25