課程名稱︰偏微分方程導論
課程性質︰大二必修
課程教師︰陳俊全
開課系所︰數學學系
考試時間︰5/2 3:30~5:20
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試題 :
偏微分方程導論 期中考
＜一＞ 25分
1. Solve
(1) u_x+u_y+3u=e^y-e^(-2x)
u(0,y)=-1
(2) yv_x-xv_y=0
v(x,0)=x^2 for x>=0 .
＜二＞ 以下6題選4題，各20分
2. Derive the equation of one-dimensional diffusion in a medium that is moving
along the axis to the right at const speed V.
10x
3. (1) Solve u_tt =4u_xx , u(x,0)= -log(1+x^2) , u_t(x,0)= ---------
(1+x^2)^2 .
(2) Find max{u(x,t) : xεR, tεR}.
∞
(3) Show that ∫((u_t)^2+4(u_x)^2)dx is independent of t.
-∞
4. Let u(x,t) be a C^2 function. Show that u satisfies the wave equation
u_xx=u_tt if and only if u(x+h,t+k) + u(x-h,t-k)= u(x+k,t+h) + u(x-k,t-h)
for all x,t,h, and k.
∞ 1 (x-y)^2
5. u(x,t)=∫ ─── exp(- ────)ψ(y)dy satisfies the initial value problem
-∞ √4πt 4t
u_t=u_xx , xεR, t>0
u(x,0) = ψ(x), xεR
Assume that ψ(x) is continous and |ψ(x)|<= 1/(1+|x|) for xεR.
Show that
(1) lim u(x,t)=0 for all t>0 and
|x|->∞
(2) lim u(x,t)=0 for all x.
t->∞
6. Let ψ(x) be a continous and bounded function and
∞ 1 (x-y)^2
u(x,t)= ∫ ─── exp(- ────)ψ(y)dy. Show that lim u(x,t)=ψ(x)
-∞ √4πt 4t t->0+
7. Solve u_xx-4u_xt-5u_tt =0 , u(x,0)=e^x, u_t(x,0)=x+1.
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