课程名称︰偏微分方程导论 课程性质︰必修 课程教师︰陈俊全 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010/6/24 考试时限（分钟）：120分钟 试题 : 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 --

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