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{ \large \textbf{\textsc{Math 5587 -- Homework 3 (Due Thursday Sept 22)}}}
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\begin{enumerate}
\item Solve $u_{tt} = c^2u_{xx}$ with $u(x,0) = x^2$ and $u_t(x,0) = \cos(x)$.
\item Find the solution of $u_{xx} - 3u_{xt} - 4u_{tt}=0$ with $u(x,0)=e^x$ and $u_t(x,0)=0$. [Hint: Factor the operator into the form $(\partial_x - 4\partial_t)(\partial_x + \partial_t)u = 0$, and proceed as we did in class to derive d'Alembert's formula.]
\item \textbf{The hammer blow:} Consider the wave equation $u_{tt}=u_{xx}$ on the entire real line $-\infty < x < \infty$ with zero initial position $u(x,0)=0$ and initial velocity $u_t(x,0)=g(x)$, where $g(x) = 1$ for $|x|<1$ and $g(x)=0$ for $|x|\geq 1$. Sketch the solution at time instants $t=1/2,1,3/2,2$ and $t=5/2$. What is the maximum displacement $\max_x u(x,t)$?
\item Suppose that $u(x,t)$ solves the wave equation $u_{tt} = u_{xx}$ for $0 < x < \ell$ and $t\geq 0$ with Robin boundary conditions
\[u(0,t) - u_x(0,t) = 0 \ \ \text{ and } \ \ u(\ell,t)+u_x(\ell,t) = 0.\]
\begin{enumerate}
\item Show that the energy
\[E(t) = \frac{1}{2}\int_0^{\ell} u_t(x,t)^2 + u_x(x,t)^2 \, dx + \frac{1}{2}u(0,t)^2 + \frac{1}{2}u(\ell,t)^2\]
is conserved (i.e., $E'(t)=0$). 
\item Give a physical explanation for the additional two terms in $E(t)$. [Recall the physical explanation of the Robin conditions for the wave equation from lecture 2.]
\end{enumerate}
\item  `Blow-up' for the heat equation.
\begin{enumerate}
\item Show that the function
\[u(x,t) = \frac{1}{\sqrt{1-t}} \exp\left(\frac{x^2}{4(1-t)}\right)\]
is a solution of the heat equation 
\[u_t - u_{xx} = 0 \ \ \text{ for } -\infty < x < \infty \text{ and } 0< t < 1.\]
\item Sketch the functions $t \mapsto u(0,t)$ and $x \mapsto u(x,1/2)$.
\item Can you give a physical explanation for the `blow-up' observed at $t=1$?
\end{enumerate}
\item 
\begin{enumerate}
\item Solve the diffusion equation $u_t = u_{xx}$ on the real line $-\infty < x < \infty$ with initial condition $u(x,0)=x^2$ \emph{without} using the fundamental solution. [Hint: Note that $u_{xxx}$ satisfies the same diffusion equation with zero initial condition. Therefore $u_{xxx}\equiv 0$ and we find that $u(x,t) = A(t)x^2 + B(t)x + C(t)$. Solve for $A(t)$, $B(t)$ and $C(t)$.]
\item Use your answer to part (a) to deduce the value of 
\[\int_{-\infty}^\infty x^2 e^{-x^2} \, dx.\]
[Hint: Write an expression for $u(0,1)$ using the fundamental solution and make a substitution in the integral.]
\end{enumerate}
\end{enumerate}


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