課程名稱︰偏微分方程導論
課程性質︰數學系大三必修
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）︰2015/06/23
考試時限（分鐘）：110
試題 :
Choose 4 from the following 6 problems
1.
(a) Let r = |(x, y, z)|. Solve u + u + u = 0 in a < r < b with u = -1 on r
xx yy zz
= a and u = 3 on r = b.
(b) Let r = |(x, y)|. Solve u + u = 2 in a < r < b with u = 0 on r = a and u
xx yy
= 0 on r = b.
2. Use Fourier series method to solve u - u = 0 for 0 < x < 3 with u(0, t) =
tt xx
0, u(3, t) = 0, u(x, 0) = 0, u (x, 0) = x.
t
3. Find the harmonic function in the square {0 < x < 1, 0 < y < 1} with the
boundary conditions u(x, 0) = 0, u(x, 1) = 1 - x, u (0, y) = 0, u (1, y) =
x x
2
y .
4.
2
(a) Let u be a harmonic function in |R . Show that the value of u at the center
of a disk equals the average of u on the circumference of that disk.
(b) Use the maximum principle to show that the solution of the problem Δu = 0
∂u
in a bounded domain D with the Robin boundary condition ── + a(x)u = h(x)
∂ν
is unique if a(x) > 0.
3
5. Let D = {y ∈ |R | |y| < a}. By the Green representation theorem, the
solution of the problem Δu = f in D, u = h on ∂D is given by
∂G(y, z)
u(z) = ∫ h(y) ─────dS(y) + ∫ f(y)G(y, z) dy,
∂D ∂ν D
where G(y, z) is the Green function.
(a) Find the Green function G(y, z).
(b) Show that
2 2
a - |z| h(y)
u(z) = ─────∫ ──── dS(y)
4πa ∂D 3
|y - z|
if f = 0.
2 3
6. Let u be the solution of u - c Δu = 0 in |R with u(x, 0) = φ(x) and
tt
u (x, 0) = ψ(x). Prove the Kirchhoff formula
t
1 ∂ ┌ 1 ┐
u(x, t ) =────∫ ψ(y) dS(y) + ──│────∫ φ(y) dS(y)│.
0 2 |y-x|=ct ∂t └ 2 |y-x|=ct ┘
4πc t 0 0 4πc t 0
0 0
// ν指的是向外的法向量