課程名稱︰偏微分方程導論
課程性質︰數學系大三必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/04/24
考試時限（分鐘）：110
試題 :
Choose 4 from the following 6 problems.
1. Solve the following equations.
x
(a) 2u + u = u, u(x, 0) = e .
x y
(b) 2yuu + u = 0, u(x, 0) = x.
x y
x
2. Solve u - u - 2u = 0, u(x, 0) = e , u (x, 0) = x.
tt xt xx t
3. Let φ(x) be a bounded continuous function on |R and
∞
u(x, t) = ∫ S(x-y, t)φ(y) dy
-∞
2
＿＿ -1 -z /4t
where S(z, t) = (√4πt) e . Show that lim u(x, t) = φ(x).
t→0+
4. Solve
u - u = 0, -∞ < x < ∞, t > 0,
t xx
x -x
u(x, 0) = e - e , -∞ < x < ∞.
5. Consider the equations u = ku and v = kv in {(x, t)| -1 < x < 1, 0 < t
t xx t xx
2
< ∞} with u(-1, t) = 0 = v(-1, t), u(1, t) = 0 = v(1, t), u(x, 0) = 1 - x ,
2
and v(x, 0) = x (1 - x ).
(a) Show that 0 ≦ u ≦ 1 and -u ≦ v ≦ u.
(b) Show that u(x, t) = u(-x, t).
(c) Show that v(x, t) = -v(-x, t).
d 1 d 1
(d) Show that ─∫ u(x, t) dx ≦ 0 and ─∫ v(x, t) dx = 0.
dt -1 dt -1
6. Solve the problem on the half line:
u - u = 0, 0 < x < ∞, t > 0,
tt xx
2
u(x, 0) = x, u (x, 0) = x , x ≧ 0,
t
u(0, t) = 0, t > 0,
u(x, t)
and find lim ────.
t→∞ 2
t
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2 2 1 ψxavier13540
給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system
╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則
╰x(0) = x° t
ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三
次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。
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※ 編輯: xavier13540 (140.112.249.76), 04/24/2015 17:29:22