課程名稱︰偏微分方程式導論
課程性質︰系必修
課程教師︰黃信元
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰ 2012/6/19
考試時限（分鐘）：110 mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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)
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