課程名稱︰偏微分方程式一
課程性質︰數學系選修
課程教師︰夏俊雄
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/10/27
考試時限（分鐘）：100
試題 :
1. (10%) Suppose u is a harmonic function defined on a smooth bounded domain Ω
n 2 ＿ ∂u
⊂ |R (u ∈ C (Ω)) such that on an open portion of ∂Ω, u = ── = 0. Show
∂ν
that u(x) = 0 for all x ∈ Ω.
＿＿＿ 3
2. Let Ω = B (0) \ B (0) ⊂ |R .
2 1
(1) (10%) Find the Green function for Ω.
(2) (10%) Solve the following equation
╭ Δu = 0
│
╯ u(x) = 5 for |x| = 1
│
╰ u(x) = 7 for |x| = 2.
3
3. (20%) Let Ω = B (0) \ {0} ⊂ |R . Find 10 different solutions for the
1
following equation
╭ Δu(x) = 0, for x ∈ Ω,
╯
╰ u(x) = 3, for |x| = 1.
4. Denote
╭ 0 1 0 0 0 ╮
│ 0 0 1 0 0 │
A = │ 0 0 0 1 0 │.
│ 0 0 0 0 1 │
╰ 12 -4 -15 5 3 ╯
(A) (10%) Find the inverse matrix of A.
(B) (15%) Express the inverse matrix of A in terms of polynomial of A with
degree less than 5.
(C) (10%) Find all the eigenvalues of A.
(D) (15%) Suppose x(t) is the solution of the linear system
x'(t) = Ax(t),
T
with the initial condition x(0) = (0, 2, 8, 26, 80) . (Here, the notation
T means transpose.). Fine the exact value of x(100).