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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