課程名稱︰偏微分方程式一
課程性質︰必選
課程教師︰陳逸昆
開課學院：理學院
開課系所︰數學系
考試日期︰2019年01月08日(二)
考試時限：10:20-12:10，共計110分鐘
試題 :
PDE, Fall 2018
Final Exam
DEP.＿＿＿＿＿＿＿＿＿ NAME＿＿＿＿＿＿＿＿＿＿＿＿ ID NUMBER＿＿＿＿＿＿＿＿
1. Write down an explicit formula for a solution of
n
u - △u - u = f, in |R × (0,∞),
t
n
u = g, on |R × {t=0}.
(20%)
2. Let u solve the initial-value problem for the wave equation in one dimension
u_t - u_xx = 0, in |R × (0,∞),
u = g, u_t = h, on |R × {t=0}.
Suppose supports of both g and h are [0,8]. The kinetic energy is
∞ 2 ∞ 2
k(t):=∫u_t(x,t)dx, and the potential energy is p(t):=∫u_x(x,t)dx.
-∞ -∞
(a). Find the support of u. (10%)
(b). Prove k(t)+p(t) is constant in time. (10%)
(c). Show that there exist M>0 such that k(t) = p(t) whenever t>M.
Find the best M. (10%)
3. Solve the following partial differential equation
1 2 2
---(u_x + u_y) = u
2
with Cauch data
u(cosθ,sinθ)=1, 0≦θ≦2π
by characteristic method. (25%)
4. Find an entropy solution to the Cauchy problem:
u_t + uu_x = 0, in |R × (0,∞),
u(x,0) = g(x), on |R × {t=0},
where
1 + x, on [-1,0],
g(x) = 1 - x, on (0,1],
0, otherwise.
Indicate the location of shock waves, if any, and check that the entropy
condition holds. (25%)
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