臺灣大學數學系

 九十學年度第二學期碩博士班資格考試試題

微分方程式

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$1$.
($10$ points) The well-known Tricomi equation takes the form
\begin{displaymath} u_{yy} - y u_{xx} = 0, \qquad \mbox{for} \qquad (x,y) \in {\mathbb{R}}^{2}. \end{displaymath}                                     (1) (1)

If $y > 0$, find the characteristic equations and the associated characteristic curves for (1).
$2$.
($25$ points) Consider the initial-boundary value problem of the damp wave equation $\Big\{ \begin{array}{lll} u_{tt} + 2 k u_{t} = c^{2} u_{xx}, & \mbox{for} 0 <... ...(x,0) = 0,& \mbox{for} 0 \leq x \leq L,\\ u(0,t) = 0, u(L,t) = 0 \end{array}$               (2)

where $k$ and $c$ are positive real constants. If we define the total energy $E$ as

\begin{displaymath} E(t) = \frac{1}{2} \int_{0}^{L} \left [ \left (u_{t} \right)^{2} + \left ( c u_{x} \right)^{2} \right ] \; dx. \end{displaymath}

(a) Show that the energy is decreasing, i.e., $dE/dt \leq 0$.
(b) Use the fact from (a) to prove the uniqueness of the solution for (2).
$3$.
(25 points) Consider the Laplace equation in polar coordinates:
\begin{displaymath} \Delta u = u_{rr} + \frac{1}{r} u_{r} + \frac{1}{r^{2}} u_{\theta \theta} = 0. \end{displaymath}                                             (3) (2)

(a) Solve (3) in a circular region: $0 \leq r \leq R$, $0 \leq \theta \leq 2 \pi$ with the Neumann boundary condition $u_{r} = f(\theta)$ on the boundary of the circle.
(b) What condition should $f$ be satisfied in order to have a existence and uniqueness of the solution for the problem ?
$4$
($25$ points) Suppose that $u(X,t)$ is a smooth solution of the heat equation in space $u_{t} = k \Delta u$ for $X \in \Omega$, $t \geq 0$, where Ω is a bounded region, and $k$ is a positive constant.
(a) Suppose that we have Dirichlet boundary condition $u = 0$ on $\partial \Omega$. Prove
\begin{displaymath} \int_{\Omega} u(X,t)^{2} dX \qquad \mbox{decays exponentially in time.} \end{displaymath}

(b) Suppose that we have Neumann boundary condition $u_{N} = 0$ on $\partial \Omega$, where $N$ is the unit-outward normal to $\partial \Omega$. Prove
\begin{displaymath} \int_{\Omega} u(X,t) dX \qquad \mbox{is constant in time.} \end{displaymath}

$5$.
($25$ points) Let $u(x,t)$ be a solution of class $C^{2}$ of
\begin{displaymath} u_{t} = a(x,t) u_{xx} + 2 b(x,t) u_{x} + c(x,t) u \end{displaymath}

in the rectangle
\begin{displaymath} \Omega = \left \{ (x,t) \vert 0 \leq x \leq L, 0 \leq t \leq T \right \}. \end{displaymath}

Let $\partial^{'}\Omega$ denote the ``lower boundary'' of Ω consisting of the three segments
\begin{eqnarray*} x = 0, & 0 \leq t \leq T \\ 0 \leq x \leq L, & t = 0 \\ x = L, & 0 \leq t \leq T. \end{eqnarray*}
(a) Prove that in case $c < 0$ in Ω
\begin{displaymath} \vert u(x,t)\vert \leq \mbox{sup}_{\partial^{'} \Omega} \vert u\vert \qquad \mbox{for} \quad (x,t) \in \Omega. \end{displaymath}

(b) Prove that more generally
\begin{displaymath} \vert u(x,t)\vert \leq e^{CT} \mbox{max}_{\partial^{'} \Omega} \vert u\vert, \end{displaymath}

where
\begin{displaymath} C = \mbox{max}\left(0, \mbox{max}_{\Omega}\vert c\vert \right ). \end{displaymath}

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