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\title{Midterm 2}
\date{}


\theoremstyle{definition} 
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\begin{document}
\maketitle
Please write your name on each page of your answer sheet and \textbf{do not fold the pages together}.
\begin{enumerate}
\item Show the uniqueness to the solution of
\begin{equation*}
\begin{cases}
u_t - ku_{xx} = f(x,t) & \text{for }0<x<L, \ \ t> 0\\
u(x,0) = \phi(x) \\
u(0,t) = g(t)\\
u(L,t) = h(t)
\end{cases}
\end{equation*}
for sufficiently nice functions $f,g,h,\phi$.  You can use whatever method you want.  A useful energy for the heat equation is the $L^2$ energy given by $E[u](t) = \int_0^L u^2(x,t)dx$.

\item Solve in terms of the error function
\begin{equation*}
\begin{cases}
u_t-ku_{xx} = 0 & \text{ on }0 < x <\infty, t> 0\\
u(x,0) = 0\\
u(0,t) = 1.
\end{cases}
\end{equation*}
Hint: First consider $v := u-1$.  What equation does $v$ satisfy? Then solve that equation, keeping in mind that we are solving this on the half-line.  The error function is given by
\begin{equation*}
\mathcal{E}\text{rf}(s) = \frac{2}{\sqrt{\pi}}\int_0^s e^{-x^2}dx.
\end{equation*}

\item Solve by finding an explicit formula.  Make sure to integrate out the solution of 
\begin{equation*}
\begin{cases}
u_{tt}-c^2u_{xx} = e^{2x} & \text{ on } (x,t) \in \mathbb{R}^2\\
u(x,0) = 0 \\
u_t(x,0) = 0.
\end{cases}
\end{equation*}
Note that $\sinh(x) = \frac{e^{x}-e^{-x}}{2}$, $\cosh(x) = \frac{e^x + e^{-x}}{2}$ and that $(\sinh(x))' = \cosh(x)$.

\item Let $L,T>0$.  Suppose $u$ is twice differentiable on the open rectangle $(0,L)\times (0,T)$ and satisfies the partial differential inequality
\begin{equation*}
u_t - u_{xx} + u \leq 0.
\end{equation*}
Suppose further that $u$ is continuous on $R=[0,L]\times [0,T]$.  If $M$ is the maximum on of $u$ on $R$ and $M\geq 0$, then show that $u$ attains the value $M$ on the sides $x=0$ or $x=L$ or on the bottom $t=0$ of $R$.  Hint: Consider the sign or value of each quantity in the partial differential inequality at a maximum point if it were to occur in the interior.  No $v=u+\eps$ trick is necessary for this problem.
\end{enumerate}


\end{document}


