課程名稱︰工程數學 下
課程性質︰系必修
課程教師︰林沛群
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2010/6/2
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(本文d均表示偏微分符號)
1.(35%) Solve the following boundary value problem
d^2 y/d t^2 = 4 (d^2 y/d x^2) + 2sin(x) 0 < x < π, t > 0
y(0,t) = y(π,t) = 0 for t ≧ 0
y(x,0) = 0
d y(x,0)/dt = 0 for 0 ≦ x ≦ π
2.(25%) Derive a solution of the following problem by Fourier Transform method
du d^2 u
─ = k ─── -∞ < x < ∞ , t > 0
dt d x^2
u(x,0) = f(x)
3.(40%) Solve the Dirichlet problem for a cube, with the given boundary
conditions
▽^2 u = 0 0 < x < 1 , 0 < y < 2 , 0 < z < π
u(0,y,z) = 0 ,u(1,y,z) = sin(πy)sin(z)
u(x,0,z) = u(x,2,z) = 0
u(x,y,0) = 0, u(x,y,π) = 4
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