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︰2018年01月
考試時限：10:20-12:10，共計110分鐘
試題 :
Partial Differential Equations, fall 2017
Final Exam
DEP. ＿＿＿＿＿＿＿＿ NAME. ＿＿＿＿＿＿＿＿＿＿＿ ID NUMBER. ＿＿＿＿＿＿＿＿
2
1. Let F(X,Y,Z,P,Q) be a C function. We consider a system of ODEs with
parameter s
dX/ds = F_P　　　　　　dY/ds = F_Q
dZ/ds = P * F_P + Q * F_Q
dP/ds = -F_X - P * F_Z, 　dQ/ds = -F_Y - Q * F_Z,
with initial
X(s,0) = f(s), Y(s,0) = g(s), Z(s,0) = h(s), P(s,0) = ψ(s), Q(s,0) = φ(s),
such that
h'(s) = ψ(s)f'(s) + φ(s)g'(s), F(f(s),g(s),h(s),ψ(s),φ(s)) = 0.
Suppose the above system has a solution in a neighborhood of (s0,0), D.
(a) Show that F(X(s,t),Y(s,t),Z(s,t),P(s,t),Q(s,t)) = 0 in D. (10%)
(b) Show that the strip conditions
　　　　　　　Z_s = P * X_s + Q * Y_s, Z_t = P * X_t + Q * Y_t
hold in D. (15%)
2. Solve the boundary value problem in [0,∞) × |R:
u u = u,　　x > 0
x y
u(x,0) = y^2.
3. Find an entropy solution to the following Cauchy problem.
u + uu = 0 in |R × (0,∞)
t x
u(x,0) = g(x),
where
/ 1 - |x| if -1≦x≦1,
g(x) = |
\ 0 otherwise.
Check the entropy condition for shock waves if any occur.
n
4. Let U be a bounded smooth open set in |R . We consider
u - △u = f in U × (0,T],
tt
u = g on (∂U×[0,T])∪(U×{t=0}),
u = h on U × {t=0}.
t
2 _
Show that there exists at most one function u∈C (U ×[0,T]) solving the
above problem.