作者t0444564 (艾利歐)
看板NTU-Exam
標題[試題] 106上 陳逸昆 偏微分方程式一 期中考二
時間Sun Sep 16 16:43:22 2018
課程名稱︰偏微分方程式一
課程性質︰數學研究所必選修
課程教師︰陳逸昆
開課學院：理學院
開課系所︰數學研究所
考試日期︰2017年12月
考試時限：110分鐘
試題 :
Partial Differential Equations, fall 2017
MIDTERM EXAM II
DEP.＿＿＿＿＿＿＿＿＿ NAME.＿＿＿＿＿＿＿＿＿＿ ID NUMBER. ＿＿＿＿＿＿＿＿＿
n
1. Suppose u is a smooth solution to u - △u = 0 in |R x (0,∞).
t
2
(a) Show u (x,t) := u(λx,λt) also solves the heat equation for each λ∈|R
λ
(b) Show that v(x,t) = x．▽u(x,t) + 2tu (x,t) also solves the heat equation
t
2. Write down an explicit formula for a solution to
n
u - △u = f in |R × (0,∞),
t
n
u = g on |R × {t=0}.
3. Write down an explicit formula for a solution to the initial-boundary-value
problem
u - u = 0 in (0,∞) × (0,∞),
tt xx
u = g, u = f on (0,∞) × {t=0},
t
u = 0 on {x=0} × (0,∞),
x
where g and h are given, with g'(0) = f'(0) = 0.
4. The Kirchhoff's formula reads
1 3
u(x,t) = -------- ∫ tf(y) + g(y) +▽g(y)．(y-x) dS(y), (x∈|R , t>0),
4πt^2 ∂B(x,t)
3 3 2 3
where g∈C (|R ) and f∈C (|R ). Prove
lim u(x,t) = g(x0); lim u(x,t) = f(x0).
t>0 t>0 t
(x,t)→(x0,0) (x,t)→(x0,0)
--
※ 發信站: 批踢踢實業坊(ptt.cc), 來自: 39.10.13.27
※ 文章網址: https://webptt.com/m.aspx?n=bbs/NTU-Exam/M.1537087404.A.D76.html
※ 編輯: t0444564 (39.10.13.27), 09/16/2018 16:47:58