課程名稱︰偏微分方程導論
課程性質︰必修
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2010/6/24
考試時限（分鐘）：120分鐘
試題 :
Choose 4 from the following 6 problems.
Please write down your answer carefully and clearly. Good luck. :)
1.Solve the problem by the method of separation of variables:
u_t = ku_xx in 0 < x < π/2
u(0,t) = u_x(π/2,0) = 0 , u(x,0) = sinx
2.Solve u_xx + u_yy = 0 in the disk {r < 2} with the boundary condition
u = 1 + 2cosθ on r = 2.
3.On 0 ≦ x ≦ 1 , consider the eigenvalue problem
-X'' = λX
X'(0) + X(0) = 0 , X(1) = 0
(a) Find an eigenfunction with eigenvalue zero.
(b) Find an equation for the positive eigenvalue λ = β^2
(c) Is there a negative eigenvalue ?
4.Let G(x,y) be the Green's function for -Δ and the domain D with the
the Dirichlet boundary condition. Show that G(x,y) = G(y,x) for x≠y ∈ D.
5.Solve the wave equation u_tt = c^2 Δu in three dimensions with the
following initial conditions
(a) u(x,y,z,0) = 0 , u_t(x,y,z,0) = 1
(b) u(x,y,z,0) = 0 , u_t(x,y,z,0) = z
Hint:p(x,y,z)
1 ∂ 1
u(p,t) = ───── ∫∫ ψ(q) + ──[─────∫∫ Φ(q)]
4π c^2 t |q-p|=ct ∂t 4π c^2 t |q-p|=ct
b
6.Let {X_n} be an orthogonal set of functions difined on [a,b],
b
i.e.∫X_n(x)X_m(x) dx = 0 for m≠n. Let ∫f^2(x) dx < ∞.
a ∞ b b
Prove the Bessel inequality : Σ(A_n)^2∫[|X_n(x)|^2]dx ≦ ∫[|f(x)|^2]dx
n=1 a a
b
∫f(x)X_n(x)dx
a
where A_n = ─────────
b
∫[|X_n(x)|^2]dx
a