→ alan7872 :第一題是 √(1-x^2)Ux+Uy=0 04/30 23:08
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※ 編輯: maplesky 來自: 140.112.250.24 (05/01 00:04)
課程名稱︰偏微分方程導論
課程性質︰數學系必修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰98/4/23
考試時限（分鐘）：8:20am-10:00am (100min)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(20 points) Solve
√(1-X^2)Ux+Uy=0
with U(0,y)=y.
2.(20 points) Suppose that U(x,t) is the solution of the wave equation
Utt=4Uxx in 0<x<3,U(x,0)=x,U(0,t)=U(3,t)=0. Find U(2,7).
3.(20 points) Solve the diffusion equation Ut-Uxx=x in ｛-∞ < x < ∞,
0 < t < ∞｝with U(x,0)=0.
4.(10 points) Suppose U(x,y,z) is a solution to the following differential
equation
｛△U = f(x,y,z) in B(0,2),
αU/αn = 2 on boundary of B(0,2),
where B(0,2) :=｛x=(x1,x2,x3) 屬於 R^3︱(x1^2+x2^2+x3^2)^0.5 ≦ 2 ｝
Calculate ∫∫∫B(0,2) f(x,y,z) dxdydz.
(註:α表偏微分符號)
5.(20 points) The linearized equations of gas dynamics(sound) are
｛αv/αt+c0^2/ρ0 = 0
αρ/αt+ρ0 div v = 0,
where v is the velocity, ρ is densuty, and ρ0 snd c0 are two constants.
Prove
(1) If curl v=0 at t=0, then curl v=0 at all times.
(2) Each component of v and ρ satisfies the wave equation.
(You need to derive such wave equation)
6.(20 points) Use Fourier serise method to solve the problem
｛Utt = 9Uxx
U(0,t) = U(π,t) = 0
U(x,0) = x, Ut(x,0) = 0
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◆ From: 140.112.250.24
※ 編輯: maplesky 來自: 140.112.250.24 (04/29 04:32)