[試題] 105-2 林太家 偏微分方程導論 Test1

[Mathmaster (^www^)]

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）︰2017.4.18
考試時限（分鐘）：110分鐘

試題 :

         Test 1                  April 18th

                              x+2y
1.(20%) Slove 2u  + u  + u = e      with u(x,0) = 0.
                x    y


2.(20%) Solve the initial value problem
                                   n
    u  + b‧grad  (u) + cu = 0 in R  ×(0,∞)
  /  t          x                  n
  \ u = g                      on R  ×{t = 0}
                       n
  Here c ∈ R and b ∈ R   are constants.

3.(20%) Solve using characteristics:

   xu  + 2yu  + 3u  = u , u(x,y,0) = g(x,y)
     x      y     z

                           2
4.(20%) Solve u   = u   + —— u   for r,t＞0 with
               tt    rr    r    r

   u(r,0) = φ(r) and u  (r,0) = ψ(r)  for r＞0.
                       t

5.(20%) Let u = u(x,t) and v = v(x,t) be smooth function satisfying

    u   = u    for x ∈ (-1,1),t＞0
  /  tt    xx
  \ u = 0      for x = ±1,t＞0

  and

    v  = v    for x ∈ (-1,1),t＞0
  /  t    xx
  \ v = 0     for x = ±1,t＞0

               1     2     2                     1    2
  Let E(t) = ∫   (u   + u   ) dx  and  F(t) = ∫   (v  ) dx  for t＞0.
               -1   t     x                      -1

         dE           dF
  Prove —— = 0 and —— ≦ 0 for t＞0  (10% each)
         dt           dt

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