课程名称︰偏微分方程导论
课程性质︰数学系必修
课程教师︰林太家
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2017.4.18
考试时限（分钟）：110分钟
试题 :
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