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\section*{\centerline{Assignment 3  SIO203B/MAE294B, 2025}}

\medskip

\centerline{Due by mid-night Friday April 25th }
\centerline{Submit by email to \texttt{wryoung@ucsd.edu} }
\centerline{with subject line \texttt{Third asymptotology assignment} }

\bigskip
\bigskip

	
	
\subsubsection*{A  corrugated boundary problem}

Consider the partial differential equation
	\beq
	\kappa\left(C_{xx} + C_{zz} \right)- \mu C=0
	\eeq
	 in the region above $z = h(x)$, with $h(x)=  a \cos kx$.  The 	boundary conditions are  $C(x, a \cos kx)=C_*	$ and $C(x,z)\to 0$ as $z \to \infty$.  \textit{(i)}  Describe a 	physical situation governed by this boundary value problem (ten or twenty words). 	\textit{(ii)} Solve the problem with $a=0$. 	\textit{(iii)}  Based on your exact solution, non-dimensionalize 	the problem with non-zero $a$ and determine 	the non-dimensional control parameters. The small parameter, call it $\epsilon$, should be linearly proportional to $a$. \textit{(iv)} Use 		perturbation theory to find the first effects of small non-zero 	$a$ on the ``inventory''
\beq
	 A \defn \frac{k}{2 \pi}\int_{h(x)}^{\infty}\!  \int_0^{2 \pi/k} \!\! C(x,z)\, \dd x \, \dd z\per
\eeq 
	 (I think you'll have to go to second order in the relevant small parameter.)
	
	
	 
\subsubsection*{Problem 2.6 a quintic. Use RPS or iteration -- your choice.}
\textit{(i)} Find a two-term approximation to all five roots of
	\beq
	x^5- x + \ep=0 \per
	\eeq
	 \textit{(ii)} Suppose that $\eta = \ep^{-1}$. Find a two-term approximation to the five roots in the limit $\eta \to 0$ (which is the same as the limit $\ep \to \infty$).
	 
Optional: 	 Take $\ep=1/4$ and compare your $\ep \ll 1$ approximation to a numerical solution e.g. use the handy  \matlab{} command \texttt{roots}.
	
\subsubsection*{A gift from John Hinch}

Find the asymptotic behaviour  of the Bessel function
$$
K_{\nu}(z) \defn \frac{1}{2} \int_{-\infty}^{\infty} \!\!\! \ee^{\nu t - z \cosh t}\, \dd t\com
$$
with $z$ fixed and $\nu \to \infty$. Use MATLAB \texttt{besselk} to make a comparison between the approximation and MATLABs evaluation of the Bessel function e.g. take $z=1$ and $0<\nu<10$. Optional: play around with $z$ and see if the approximations degrades as you make $z$ larger and smaller.

Advice: Use the notation 
\beq
\phi = -\nu t + z \sinh t \per
\eeq
Following the examples in the lecture let $t_*(\nu,z)$ denote the location of the peak and use the notation  $\phi_*$ and  $\phi''_*$ for $\phi'$ and $\phi''$ evaluated at $t_*$. Don't write out the details in terms of $\nu$ and $z$ till the end of the calculation.


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