[試題] 106下 陳俊全 偏微分方程導論 期中考

作者: t0444564 (艾利歐)   2018-05-04 11:32:52
課程名稱︰偏微分方程式導論
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︰2018年05月04日(五)
考試時限:08:10-10:00,共計110分鐘
試題 :
Choose 4 from the following 6 problems
1. Solve the following equations.
2 2
(a) u + u + (y-x)u = y + 3xy - 4x .
x y
(b) uu + u = 0 for -∞<x<∞, t>0,
x t
         { 0, x≦0
u(x,0) = { x, 0<x<1.
         { 1, x≧1
2. Let ψ be a bounded continuous function on |R and
               ∞
          u(x,t) = ∫ W(x-y,t)ψ(y)dy,
               -∞
              -1 -z^2/4t
  where W(z,t) = (√(4πt)) e    . Show that lim u(x,t) = ψ(x).
                         t->0+
3. Solve the problem:
     2
  u - c u = 0, 0<x<3, t>0,
tt xx
  u(x,0) = 0, u (x,0) = 1, 0≦x≦3,
         t
  u(0,t) = u(3,t) = 0, t>0.
4. Solve the problem:
  u - u = 2 + x, 0<x<∞, t>0,
t xx
x -x
  u(x,0) = e + e , u (0,t) = t.
             x
5. Solve 2u - u - u = 0, u(x,0) = sin x, u (x,0) = x.
      tt  xt xx t
6. Assume u = ku for -1<x<1, 0<t<∞, u(-1,t) = 0 = u(1,t), u(x,0) = sin x.
t xx
  (a) Show that -1≦u≦1.
(b) Show that u(x,t) = -u(-x,t).
  (c) Let a(t) = min{u(x,t):-1<x<1}. Show that a(t) is nondecreasing
function of t.

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