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\section*{\centerline{Assignment 2  SIO203B/MAE294B, 2025}}

\medskip

\centerline{Due by mid-night Wednesday April 16th }
\centerline{Submit by email to \texttt{wryoung@ucsd.edu} }
\centerline{with subject line \texttt{Second asymptotology assignment} }

\bigskip
\bigskip


\subsection*{Belligerent drunks again}
Let's make a small change to the formulation of the belligerent-drunks example in section \textbf{3.2} of the notes. Recall that the steady state density equation  is
\beq
\kappa u_{xx} - \mu u^2 =0\per
\eeq
Suppose that we model the bars  using a Neumann boundary condition. This means that the flux of drunks, rather than the concentration, is prescribed at $x=0$ and $\ell$: 
\beq
\kappa u_x(0,t) = -F\com \qquad \text{and} \qquad \kappa u_x(\ell,t) = F\com
\eeq
where  $F$, with dimensions drunks per second, is the flux entering the domain from the bars. Try to repeat \textit{all calculations} in section \textbf{3.2}  including the analog of the weakly interacting limit $\alpha \ll 1$ perturbation expansion (find a ``reasonable'' number of terms). You'll find that it is not straightforward and that a certain amount of ingenuity is required to understand the weakly interacting limit with fixed-flux boundary conditions. 

Hint: the difficulty is not that the problem above is nonlinear. So  if you're absolutely stuck you can retreat to an easier linear problem that poses the same challenge:
\beq
\kappa v_{xx} - \nu v=0\com \label{lin}
\eeq
with Neumann BCs
\beq
\kappa v_x(0,t) = -F\com \qquad \text{and} \qquad \kappa v_x(\ell,t) = F\per
\eeq
Solve this $v$-problem exactly and then analyze the solution with  $ \ell^2 \nu/\kappa  \ll 1$ to understand the scaling. 

\clearpage
	
\subsubsection*{A logistic equation with time varying carrying capacity}	



%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%  FIGURE %%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[H]
\centerline{\epsfig{file=popFigure6,width=0.7\textwidth}}
\caption{Numerical solution of \eqref{logistic1} with various initial conditions. The black sinusoid is the periodic-in-time carrying capacity. At large time all initial conditions convergence to a periodic solution that lags the carrying capacity.}
\label{gregFig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%  FIGURE %%%%%%


Consider the logistic equation with a periodically varying carrying capacity:
	\beq
	\dot N = r N \left( 1 - \frac{N}{K} \right)\com \qquad \text{with}      \qquad K(t) =K_0 + K_1 \cos \omega t\per
	\label{logistic1}
	\eeq
	The initial condition is $N(0)=M$.
	\textit{(i)} Based on the $K_1=0$ solution, non-dimensionalize this problem. Show that there are three non-dimensional control parameters. \textit{(ii)} Suppose that $K_1$ is a perturbation i.e., $K_1/K_0 \ll 1$. The numerical solution in  Figure \ref{gregFig1} shows that eventually  the initial condition is ``forgotten'' and all solutions converge to a periodic oscillation about the mean carrying capacity $K_0$.   Use perturbation theory to determine the amplitude and phase of the long-term oscillation. (``Long-term'' means the large-time solution. You do not have to solve the complete initial value problem.) Show that peak population lags peak carrying capacity.
	
\subsubsection*{An integral}

Consider
\beq
J(x,p,q) \defn \int_x^\infty \!\!\!\ \ee^{- p t^q}\,  \dd t\per
\eeq
(a) Explain why this problem makes sense only if  $p$ and $q$ are both greater than zero (twelve words or less).
(b) With a change of variable express  $J(x,p,q)$ in terms of the simpler function $K(x,q) = J(x,1,q)$.  (c) Find the leading-order $x \to \infty$ approximation to $K(x,q)$. (There is no need to justify asymptoticness -- I'll assume that discussion in class, and in the recitation, has been sufficient.)
	
	




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