課程名稱︰偏微分方程導論
課程性質︰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（年月日）︰2010.05.06
考試時限（分鐘）：120min
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.
Please write down your proof and answer carefully and clearly. Good Luck! :)
1. Solve
(a) yu_x + u_y = 0, u(x,0) = e^3x
(b) u_x^2 + 4u^y = 0 for y > 0, u(1,y) = 1/y
2. Solve u_xx + 2u_xy - 3u_yy = 0, u(x,0) = x, u_y(x,0) = sinx
3. Consider u_t = u_xx for -1<x<1, 0<t<∞; u(-1,t)=u(1,t)=0; u(x,0)=(1-x^2)x
(a) Show that |u(x,t)|≦(2/3√3) for t>0, -1<x<1.
(b) Show that u(x,t) = -u(-x,t) for t≧0, -1≦x≦1
1
(c) Show that∫ u^2 dx is a decreasing function in t.
-1
4. Prove that if ψ(x) is a bounded continuous function for -∞<x<∞, then
1 ∞ -(x-y)^2
lim --------∫ e^(----------) ψ(y) dy = ψ(x)
t->0+ √4πt -∞ 4t
5. Solve u_t = ku_xx, 0<x<∞, t>0; u(x,0)= e^x + e^-x; u_x(0,t) = 0
6. Solve u_tt = c^2 u_xx, 0<x<∞, t>0; u(0,t)=t^2; u(x,0)=0, u_t(x,0)=1
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