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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