数理方程历年试题.pdf

2001—2002 (10 ) ℄ 1. G, k, c ρ G (x, y, z) t u = u(x, y, z, t) 2 ut − a ∆u = 0. k, c ρ a D ⊂ G G Æ Σ t1 t2 D ℄ Q = cρ[u(x, y, z, t ) − u(x, y, z, t )]dv = t2 cρ ∂u dvdt, 1 2 1 D t1 D ∂t Σ D Q2 = t2 k ∂u dSdt, t1 Σ ∂ν ν = ν (x, y, z) Σ (x, y, z) O − G Q2 = t t2 D k∆udvdt. 1 Q = Q , 1 2 t2 cρ ∂u dvdt = t2 k∆udvdt. t1 D ∂t t1 D t , t D 1 2 ∂u k = ∆u. ∂t cρ 1 a = k . cρ 2. 2 2 2 2 2 ∂u ∂ u ∂u ∂u ∂ u ∂u ∂ u 1 + − 2 + 1 + = 0 ∂y ∂x2 ∂x ∂y ∂x∂y ∂x ∂y2 Æ Æ u Æ 3. 2 2 uxx + (x + y )uyy = 2001, −∞ x +∞, −∞ y +∞.  u u   4. X′′ (x) + 2001X (x) = 0, x ∈ (0, 1), X (0) = X (1) = 0  X′′ (x) + λX (x) = 0, x ∈ (0, 1), X (0) = X (1) = 0. 2 λn = (nπ) , n = 1, 2, . 2001