課程名稱︰工程數學二
課程性質︰必修
課程教師︰謝傳璋/王昭男
開課學院：工學院
開課系所︰工程科學與海洋工程學系
考試日期（年月日）︰2008/5/20
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 :
1. Find the general solution of the following P.D.E for u(x,y):
(i) u = u = 0
yy xx
xy
(ii) u + u = e
y
2. The 2-D Laplacian operator in Cartesian coordinate is :
2 2
2 δ δ
▽≡( ─ + ─ )
2 2
δx δy
Transform this operator in Polar coordinate:
x = rcosθ ; y = rsinθ
3. Solve the following wave equation of u(x,t)
2 2
δ u 2 δ u
----- = c ------ , 0 < x < L 0≦t -----(1)
2 2
δ t δ x
u(0,t) = 0
B.C: 0≦t＜∞ -----(2)
u(L,t) = 0
3 πx
u(x,0) = sin (-----)
I.C: L ------(3)
u (x,0) = 0
t
Plot wave form at t = 1 and t = 2, assume c = 1; L = 1,
4. Use the method of Fourier transform, find the solution (you may express
the solution in double integral form) of the following heat equation:
2
δu 2δu
---- = c ---- -∞ < x < ∞ ; t>0 -----(1)
2
δt δx
-Uo when -1 < x < 0
I.C. u(x,0) = Uo when 0 < x < 1 -----(2)
0 otherwise
what is temperature at x = 0 i.e. u(0,t)?
5. Consider one dimensional wave equation
2 2
δu 2δu
---- = c ---- , -∞ < x < ∞ ; t > 0 -----(1)
2 2
δt δx
u(x,0) = f(x)
I.C. -∞ < x < ∞ -----(2a,b)
u (x,0) = g(x)
t
the D'Alembert's solution is:
1 x+ct
u(x,t) = 0.5 [ f(x+ct) + f(x-ct) ] + ---∫ g(s)ds
2c x-ct
If the initial conditions f(x) and g(x) are given as:
1 when -0.5≦x≦0.5
f(x) =
0 otherwise
1 when -1≦x≦1
g(x) =
0 otherwise
Plot the wave form u(x,t) at t = 1, and t = 2 , assume c = 1
每題20分，共100分
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