課程名稱︰偏微分方程導論 課程性質︰數學系必修 課程教師︰林紹雄 開課學院：理學院 開課系所︰數學系 考試日期︰2013年03月29日，08:10-09:10 考試時限：60分鐘 是否需發放獎勵金：是 試題 : 　　　　　　Math 2206 (Introduction to PDE) Quiz No.3 (3/29/2013) Solve the following problems. Please write down your computational or proof steps clearly on the answer sheets. A. Consider the initial boundary value problem for the 1-d wave equation: 　　　u_tt = t_xx　　　　　　　　　　　　for x > 0, t > 0, 　　　u(x,0) = f(x) , (u_t)(x,0) = g(x)　for x > 0, 　　　u(0,t) = h(t)　　　　　　　　　　　for t > 0, Where f∈C^2([0,∞)) with f(0) = f'(0) = f''(0) = 0, g∈C^1([0,∞)) with g(0) = g'(0) = 0 and h∈C^2([0,∞)) with h(0) = h'(0) = h''(0) = 0. 　(a) (15 points) When f(x) =g(x) = 0 for x≧0, find the unique solution 　　　　　2　 2　　 1 ＿2　　　　　2　　　　　　2 　　　u∈C (Ｒ) ∩ C (Ｒ), where Ｒ = {(x,t)∈Ｒ　｜x > 0, t > 0}. 　　　　　　　+　　　　 +　　　　　+ 　(b) (15 points) For general f,g and h, find the unique solution 　　　u∈C2((Ｒ_+)^2)∩ C1(bar(Ｒ_+)^2). B. (20 points) Let f∈C2(R3 x [0,∞)). Prove that the unique solution 　　u∈C2(R3 x [0,∞)) to the inhomogeneous wave equation 　　　u_tt = Δu + f(x,t)　　　　　for x∈R3, t > 0, 　　　u(x,0) = 0 , (u_t)(x,0) = 0　for x∈R3, 　　is given by u(x,t) = 1/(4π) * ∫_(B_t(x)) ( f(y,t-|y-x|)/|y-x| )dy for 　　x∈R3, t≧0, where B_t(x) is the ball with center x and radius t. C. (20 points) Consider the boundary value problem for the wave equation in 　　0≦x≦1: 　　　u_tt = u_xx　　　　　　　　　　　　　　　　 for 0 < x < 1, t > 0, 　　　u_x(0,t) + u(0,t) = u_x(1,t) + 2u(1,t) = 0　for t > 0. 　　Write down the associated eigenvalue problem. Show the existence of 　　infinitely many eigenvalues "graphically". Are all eigenvalues nonnegative? --

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