課程名稱︰偏微分方程導論
課程性質︰數學系必修課
課程教師︰陳俊全
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/6/20
考試時限（分鐘）：8：10～10：00
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Choose 4 from the following 6 problems. Each problem counts 25 points.
(1) Solve the problem:
︴ 2
︴u_tt = c u_xx, 0 < x < 3, t > 0
︴
︴u(0,t) = u(3,t) = 0
︴
︴u(x,0) = 0, u_t(x,0) = x
︴
(2) Use the method of separating variables to solve u_xx + u_yy = 0 in the
disk {r < a} with the boundary condition
u = 1 + cosθ on r = a.
(3) Solve u_xx + u_yy = 0 in the square {0 < x < 1, 0 < y < 1} with the
boundary conditions u_y(x,0) = 0, u(x,1) = x, u(0,y) = 0, u_x(1,y) = 0
-2
(4) Solve u_xx + u_yy + u_zz = r in the spherical shell {a < r < b} with
the boundary conditions u = 5 on r = a, u = 0 on r = b.
(5) Find the Green's function for Δ on the disk {|(x,y)| < a}.
(6) Prove the strong maximum principle for harmonic functions: Let D be a
connected bounded open set and u be a harmonic function on D which is
_
continuous on D = D ∪ ∂D. Then the maximum value of u is attained on
∂D and nowhere inside unless u is a constant.