Quiz 3 Take home sample

I expect that next week's quiz could have a take home portion.

  1. Let's try to solve the heat problem on the unit disk: $$u_t = \Delta u, \: u(1,t) = 0, \: u(r,0) = 2r^2(1-r^2),$$ where the problem is described in terms of polar coordinates. To do so:
    1. Express your solution as an infinite series involving Bessel functions with unknown coefficients $c_n$.
    2. Write out the first three terms of the series explicitly with numerically computed values of $c_1$, $c_2$, and $c_3$.
    3. Plot the graph of a higher precision approximation to the radial temperature profile at time $t=0.04$.
    4. Write down the temperature at the center of the disk at time $t=0.05$ to 4 digits of precision.
    Of course, we're encouraged to use this web page.
    Solution: Let's write these out in order.
    1. According to the formula at the bottom of the web page, the solution is $$u(t,r) = \sum_{n=1}^{\infty} c_n e^{-z_n^2 t} F_n(r),$$ where $F_n(r) = J_0(z_nr)/\|J_0(z_nr)\|$ is is the normalized Bessel function scaled by its $n^{\text{th}}$ root.
    2. Reading the coefficients off from here and the roots from here, we find that $$u(t,r) \approx 0.1898 e^{-2.4048^2t}F_1(r) - 0.1665 e^{-5.52^2} F_2(r) + 0.04864 e^{-8.6537^2}F_3(r) + \cdots$$
    3. I guess the solution looks like so:
    4. Finally, to 4 digits of precision, we have $u(0.05,0)=0.2423$.
  2. Let $D$ denote the disk of radius 2 and let $C$ denote the circle of radius 2 - i.e. the boundary of $D$. Suppose we hold the temperature of the right half of $C$ (i.e.~the portion where $x>0$) at $3$ and the rest of $C$ at $0$ and allow the heat to diffuse through $D$. Use the Poisson integral formula $$u(r,\theta )=\frac{1}{2\pi }\int _{-\pi }^{\pi }f(\varphi )\frac{R^2-r^2}{R^2+r^2-2R r \cos (\theta -\varphi )}d\phi$$ to express the resulting steady state temperature distribution.