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︰6/27 3:30~6:30
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試題 :
Choose 5 from the following 9 problems.
1. Let u satisfy u_t-u_xx=αexp(2t)x for 0 < x < ∞, 0 < t < ∞,u(x,0)=2x,
u(0,t)=0. (a) Find the solution u for α=0; (b) Find the solution u
for α=1.
2. Solve u_tt-u_xx=e^x for -∞ < x < ∞, -∞ < t < ∞, u(x,0)=0, u_t(x,0)=0.
3. Use the method of separation of variables to solve u_tt-u_xx=0 for 0<x<π
, u(0,t)=0, u_x(π,t)=0, u(x,0)=sin(x/2), u_t(x,0)=0.
4. Let g(x)=π-|x|, -π<=x<=π. Find its Fourier series.
π
5. (a) Let g(x) be a C^1 function on -π<=x<=π. Show that ∫sin(nx)g(x)dx->0
-π
∞
as n->∞.(b) Show that∫sin(nx)g(x)->0 as n->∞ still holds if g(x) is only
-∞
continous.
6. Let(r,θ) denote the polar coordinates in R^2. Solve △u=0 for r<1,
u=sin^2(θ) for r=1.
7. Let(r,θ) denote the polar coordinates in R^2 and v(r) be a function
depending on r only. Suppose △v=0 in Ω. (a) Show that v is a constant if
Ω=R^2. (b) Find all solutions v(r) for Ω=R^2/{0}.
8. Let u(x,y,z,t) be a bounded solution of the three-dimensional heat equation
u_t=△u in R^3 with u(x,y,z,0) = (1+x^2+y^2+z^2)^(-1). Show that
u(0,0,0,t)->0 as t->∞.
9.Assume that v(x,y,z,t) satisfies v_tt =△v and
_ 1
v(r,t)= ─── ∫∫ v(x,y,z,t) dσ .
4πr^2 x^2+y^2+z^2=r^2
_ _ _
Show that v_tt = v_rr + (2/r)v_r.
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