$6\mathrm{.}37.$ Find a bounded solution to Laplace's equation on the plane with a hole at the center

with radius $Q=1.$ This domain is infinitely large, and can be considered an annulus with

no outer boundary. The boundary condition is $u(Q,)=2+cos(6).$ Plot the solution

over a large enough range by slightly modifying the supplied code linked HERE.

Hint: use the boundedness requirement in order to zero out some of the coefficients.

$\%$ This script plots Laplace's solution in polar for Nesse's PDE text.

$r=$ linspace $(\mathrm{.}01,1,40)$;

phi $=$ linspace$(0,2^{**}$ pi$,70)$;

$[\frac{R}{P}HI]=$meshgrid$(r,)$;

$u\frac{?}{P}=HI**0$;

for $n=1$:$200$

$u=u-R{*}^{{?}^{?}}{?}^{**}*(\frac{2^{**}(-1{)}^n}{n}){*}^{**}sin(\frac{n^{**}}{P}HI)$;

end

figure$(1)$

$,$ R$.*sin(\frac{?}{\text{P}\text{H}\text{I}}),u$

set $($gca$,$ 'linewidth $\text{'},2)$

set $($gca$,$ 'fontsize', $18)$

xlabel$(\text{'}x\text{'})$

ylabel$(\text{'}$y$\text{'})$

zlabel$(\text{'}$u$\text{'})$