作者harveyhs (Hango)
看板NTU-Exam
标题[试题] 100-2 黄信元 偏微分方程导论 期末考
时间Thu Jun 21 21:07:06 2012
课程名称︰偏微分方程式导论
课程性质︰系必修
课程教师︰黄信元
开课学院:理学院
开课系所︰数学系
考试日期(年月日)︰ 2012/6/19
考试时限(分钟):110 mins
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
A (20 pts) Solve the diffusion equation with variable disspation
2
u -u + bt u = 0 for -∞ < x < ∞
t xx
with u(x,0) = φ(x), where b > 0 is a constant. 3
bt /3
(Hint. You may derive an equation for v(x,t) = e u(x,t).)
B (20 pts) Separate variables for the equation
tu = u + 2u, 0 < x < l, t > 0
t xx
with the boundary conditions u(0,t) = u(l,t) = 0. Show that there are
infinite number of solutions that satisfy the initial condition u(x,0) = 0.
C (20 pts) Let f(x) = exp[-a|x|], a > 0. Compute the Fourier transform of f(x).
D (20 pts) Define
{ 0 x < 0
u(x) = {
{ sin x x ≧ 0
Show that u + u = δ in the sense of distributions, where δ denotes the
xx 0
delta function.
n _
E (20 pts) Let Ω be an open, bounded connected set in R . Let f be in C(Ω),
2
and suppose u in C (Ω╳[0,∞)) is a solution of
1
{ u = △u in Ω╳[0,∞)
{ t
{ u(x,t) = 0 on ∂Ω╳[0,∞)
{ u(x,0) = f(x) in Ω
Prove
lim u(x,t) = 0.
t->∞ 1 _
(Hint: You may use the energy method and the fact that, for g in C (Ω)
with g| = 0, there exists a constant C such that
∂Ω
||g|| 2 ≦ C ||▽g|| 2 .)
L (Ω) L (Ω)
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.249.241
1F:→ t0444564 : 已收录 11/25 16:42