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 -- 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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