臺灣大學數學系

八十九學年度第一學期碩博士班資格考試試題

微分方程式

[回上頁]

Choose $4$ out of the following $6$ problems

1.

Find the solution $u(x,y)$ in each case, defined at least near the initial line $y=0$:

(a)

$x u_y - y u_x = x, \quad u(x,0) = \vert x\vert$,

(b)

$u_y - u u_x = 0, \quad u(x,0) = -x$.

2.

Let Ω be a domain in $\mathbb{R}^3$. Suppose $u(X,t)$ is a smooth solution of the wave equation

$$u_{tt} = c^2 \Delta u , \quad X \in \Omega, \quad t > 0$$

with the boundary condition

$$u(X,t) = 0, \quad X \in \partial \Omega, \quad t > 0.$$

Show that the energy

$$\frac12 \int_{\Omega} [ (u^2)_t + c^2 \vert\nabla u \vert^2] \ dX$$

is conserved in time.

3.

Let $u(X) \in C^2(\Omega) \cap C^0(\bar{\Omega})$ be a solution of

$$\Delta u + \sum_{k=1}^n a_k(X) u_{X_k} + c(X) u = 0,$$

where $c(X) < 0$ in Ω. Show that $u=0$ on $\partial \Omega$ implies $u=0$ in Ω.

4.

Consider the Laplace equation in two space dimensions

$$u_{xx} + u_{yy} = 0$$

in the upper halfplane $y > 0$ with the Dirichlet boundary condition $u(x,0) = f(x)$.

(a)

Find the solution of this problem.

(b)

Show that the solution you find in (a) actually represents a bounded solution of the Dirichlet problem under study, if $f(x)$ is bounded and continuous.

5.

Let $u(x,t)$ be the solution of

$$u_t = u_{xx}, \quad -\infty < x < \infty, \quad u(x,0) = f(x),$$

where f is continuous and $f(x)=0$ for $\vert x\vert \ge 1$. Show that there is a number $M$, not depending on $f$, so that

$$\Big \vert \frac{\partial u }{\partial x} (x,t) \Big \vert \le M \int_{-1}^1 \vert f(s)\vert \ ds$$

for all $t \ge 1$ and all $x$.

6.

Solve the initial-boundary value problem of the heat equation in two space dimensions,

$$\begin{array}{rcll} u_t &=& u_{xx} + u_{yy} & \mbox{for }0 <... ... f(x,y) & \mbox{for } 0 \le x \le 1, 0 \le y \le 1. \end{array}$$

[回上頁]