Quiz 3 Take home sample
I expect that next week's quiz could have a take home portion.

Let's try to solve the heat problem on the unit disk: $$u_t = \Delta u, \: u(1,t) = 0, \: u(r,0) = 2r^2(1r^2),$$ where the problem is described in terms of polar coordinates. To do so:
 Express your solution as an infinite series involving Bessel functions with unknown coefficients $c_n$.
 Write out the first three terms of the series explicitly with numerically computed values of $c_1$, $c_2$, and $c_3$.
 Plot the graph of a higher precision approximation to the radial temperature profile at time $t=0.04$.
 Write down the temperature at the center of the disk at time $t=0.05$ to 4 digits of precision.
Solution: Let's write these out in order. 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.
 Reading the coefficients off from here and the roots from here, we find that $$u(t,r) \approx 0.1898 e^{2.4048^2t}F_1(r)  0.1665 e^{5.52^2} F_2(r) + 0.04864 e^{8.6537^2}F_3(r) + \cdots$$

I guess the solution looks like so:
 Finally, to 4 digits of precision, we have $u(0.05,0)=0.2423$.

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 $$u(r,\theta )=\frac{1}{2\pi }\int _{\pi }^{\pi }f(\varphi )\frac{R^2r^2}{R^2+r^22R r \cos (\theta \varphi )}d\phi$$ to express the resulting steady state temperature distribution.