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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