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) =