課程名稱︰偏微分方程導論 課程性質︰數學系必修 課程教師︰林太家 開課學院：理學院 開課系所︰數學系 考試日期（年月日）︰2017/6/20 考試時限（分鐘）：110分鐘+延長n分鐘 試題 : Final June 20th Total: 120 points. 1.(10%) Solve the diffusion problem u = ku in 0＜x＜L, with thw mixed boundary t xx conditions u(0,t) = u (L,t) = 0. x 2.(20%) Suppose u is a smooth solution of ╭ u － u ＋c(x)u = 0 for x ∈ (0,1), t＞0, │ t xx │ │ u(0,t) = u(1,t) = 0 for t＞0, │ ╰ u(x,0) = g(x) for x ∈ (0,1), and the function c = c(x) sarisfies c(x)≧γ＞0 for x ∈ (0,1), where γ＞0 －σt is a constant. Prove if |g(x)|≦1 for x ∈ (0,1), then ∥u∥ ≦ Ce for 2k k ∈ N, t＞0, where C, σ are positive constants independent of k and ∥‧∥ 2k 2k 1 2k is defined by ∥u∥ = ∫ u dx. 2k 0 3.(10%) 2 Solve u = c u for 0＜x＜π, with the boundary conditions u (0,t) = tt xx x 2 u (π,t) = 0 and the initial conditions u(x,0) = 0, u (x,0) = cos (x). x t 4.(20%) Solve u ＋ u = 1 in the annulus a＜r＜b with u(x,y) vanishing on both xx yy parts of the boundary r = a and r = b. 5.(20%) Find the general solution of 3u ＋ 10u ＋ 3u = sin(x＋t). tt xt xx 6.(20%) －1 3 ∞ 3 Let u(x) = ∫ |x－y| f(y) dy for x ∈ R , where f ∈ C ( R ). Prove 3 0 R 3 3 that Δu = cf in R , where Δ is the Laplace operator on R and c is a constant. Calculate the constant c. 7.(20%) 2 4 4 Assume u ∈ C ( R \ {0} ), and u≧0 is a harmonic in R \{0}. Prove that u －2 4 has the form that u(x) = C |x| ＋ C for x ∈ R \{0}, where C , C ≧0 1 2 1 2 are constants. --

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