課程名稱︰偏微分方程導論 課程性質︰數學系必修課 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）︰2014/04/29 考試時限（分鐘）：15:30~17:20 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Choose 4 from the following 6 problems. (1) Solve the following equations. (a) xu_x + u_y + u = 0, u(x,0) = x. 2 (b) 2u_x + u_y - u_z = x + 2z, u(x,x,0) = x . (2) Solve the equation: u_tt - 4u_xt - 5u_xx = 0, u(x,0)=Φ(x), u_t(x,0)=Ψ(x) 2 (3) Prove the maximum principle: If u(x,t) is a C function satisfying the diffusion equation in a rectangle {(x,t) : 0≦x≦l, 0≦t≦T}, then the maximum value of u(x,t) is assumed either on {(x,0)|0≦x≦l} or on {(x,t)|x=0 or x=l, 0≦t≦T}. 註：此處的l是英文字母 (4) Let ψ(x) be a bounded continuous function on R and ∞ u(x,t) = ∫ S(x-y,t)ψ(y) dy, -∞ 2 -1 -z /4t where S(z,t) = (√4πt) e . Show that lim u(x,t) = ψ(x). t→0＋ (5) Solve the inhomogeneous problem on a half line: ︴u_tt － u_xx = xt, 0 < x < ∞, t > 0, ︴ ︴ ︴u(x,0) = sinx, u_t(x,0) = 1 ＋ x ︴ ︴ ︴u(0,t) = t. (6) Let k > 0. Solve u_t = ku_xx, u(x,0) = 0, u(0,t) = 2 on the domain 0 < x < ∞, t > 0. --

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