臺灣大學數學系

九十一學年度第一學期碩博士班資格考試題

偏微分方程(Differential)

Sept 11, 2002

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Total score: $100$ points

$1$.

($20$ points) Solve the following Cauchy problem for $u(x,y)$:

(a)

($10$ points) Equation $(x+2) u_{x} + 2y u_{y} = 2u$ with initial condition $u(-1,y) = \sqrt{y}$.

(b)

($10$ points) Equation $x^{2} u_{x} - y^{2} u_{y} = 0$ with initial condition $u(1,y) = f(y)$.

$2$.

($20$ points)(a) ($10$ points) Show that for $n=3$ the general solution of the wave equation:

$$u_{tt} = c^{2} \sum_{i=1}^{n} u_{x_{i} x_{i}},$$

with spherical symmetry about the origin has the form:

$$u = \frac{f(r+ct) + g(r-ct)}{r}, \qquad r^{2} = \sum_{i=1}^{n}x_{i}^{2},$$

with suitable $f$ and $g$. Here $c$ is a positive constant. (b) ($10$ points) Show that the solution of the above problem with initial data of the form:

$$u = 0, \quad u_{t} = \phi(r)$$

is given by

$$u = \frac{1}{2 c r} \int_{r-ct}^{r+ct} \xi \phi(\xi) \; d\xi.$$

Here $\phi$ is a even function of $r$.

$3$.

(20 points) Let $u$ be harmonic in a domain D. Show that $u$ has partial derivatives of all orders in $D$.

$4$.

($20$ points) Consider the following one-dimensional diffusion equation in the semi-infinite interval $0 \leq x < \infty$:

$$u_{t} - u_{xx} = 0, \qquad x \geq 0 , \quad t \geq 0$$

$$u(x,0) = 0, \qquad u(0,t) = a t^{n} \quad t > 0,$$

where $a$ is a positive constant and $n$ is a non-negative constant.

(a)

(10 points) Assume the solution of the problem take the form:

$$u(x,t) = a t^{n} f(\xi),$$   where$$\quad \xi = \frac{x}{2 \sqrt{t}}.$$

Show that $f$ satisfies the conditions:

$$f^{''}(\xi) + 2 \xi f^{'}(\xi) - 4 n f = 0, \\ f(0) = 1, \qquad f(\infty) = 0.$$

(b)

(10 points) Find the solution of the above ordinary differential equation, and hence the solution of the original diffusion problem.

$5$.

($20$ points) Solve the following one-dimensional diffusion equation in the unit interval $0 \leq x \leq 1$:

$$u_{t} - u_{xx} = x \sin{t}, \qquad 0 \leq x \leq 1, \quad t \geq 0$$

$$u(x,0) = x (1-x), \qquad u(0,t) = u_{x}(1,t) = 0 \quad t > 0.$$

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