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 -- 2 2 1 ψxavier13540 給定一個二次元(|R )上的開集 G，設 f: G →|R ∈ C 。考慮一 autonomous system ╭dx/dt = f(x)，若 ∀t ≧ 0，有φ (x°) ∈ K ⊆ G，其中 K 在 G 上 compact，則 ╰x(0) = x° t ω(x°) 只能是一定點、一週期軌道或連接有限個 critical point 的連通路徑，不會像三 次元一樣可能出現混沌(chaos)。此即為 ODE 動力系統中的 Poincaré–Bendixson 定理。 --

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