課程名稱︰偏微分方程導論
課程性質︰數學系必修
課程教師︰林太家
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/5/26
考試時限（分鐘）：110分鐘
試題 :
Test 2 May 26th
1.(10%)
Consider the diffusion equation u = u in {0＜x＜1, 0＜t＜∞} with
t xx
u(0,t) = u(1,t) = 0 and u(x,0) = 4x(1－x).
(a) Show that 0＜u(x,t)＜1 for all t＞0 and 0＜x＜1.
(b) Show that u(x,t) = u(1－x,t) for all t≧0 and 0≦x≦1.
1 2
(c) Use the energy method to show that ∫ u dx is a strictly decreasing
0
function of t.
2.(20%)
Solve the initial value problem
╭ u － u ＋ 2u = 0 for x ∈ R , t＞0,
│ t xx
│
╰ u(x,0) = g(x) for x ∈ R.
3.(10%)
Show that there is no maximum principle for the wace equation.
4.(20%)
Solve u = 9u in 0＜x＜π/2, u(x,0) = cos(x), u (x,0) = 0, u (0,t) = 0,
tt xx t x
u(π/2,t) = 0.
5.(20%)
∞
Letψ ∈ C (R). Prove
0
2
∞ 1 －x /(4ε)
lim ∫ ————— e ψ(x) dx = ψ(0).
ε→0＋ －∞ √(4πε)
6.(20%)
Let u solve
3
╭ u = Δu in R ×(0,∞),
│ tt
│ 3
│ u = h on R ×{t = 0},
│ x
│ 3
╰ u = g on R ×{t = 0},
where g, h are smooth and have compact support.
Prove that sup | tu(x,t) | ＜∞
3
x∈R , t>0