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）︰2014/04/29
考試時限（分鐘）：15:30~17:20
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.
(1) Solve the following equations.
(a) xu_x + u_y + u = 0, u(x,0) = x.
2
(b) 2u_x + u_y - u_z = x + 2z, u(x,x,0) = x .
(2) Solve the equation: u_tt - 4u_xt - 5u_xx = 0, u(x,0)=Φ(x), u_t(x,0)=Ψ(x)
2
(3) Prove the maximum principle: If u(x,t) is a C function satisfying the
diffusion equation in a rectangle {(x,t) : 0≦x≦l, 0≦t≦T}, then the
maximum value of u(x,t) is assumed either on {(x,0)|0≦x≦l} or on
{(x,t)|x=0 or x=l, 0≦t≦T}. 註：此處的l是英文字母
(4) Let ψ(x) be a bounded continuous function on R and
∞
u(x,t) = ∫ S(x-y,t)ψ(y) dy,
-∞
2
-1 -z /4t
where S(z,t) = (√4πt) e . Show that lim u(x,t) = ψ(x).
t→0＋
(5) Solve the inhomogeneous problem on a half line:
︴u_tt － u_xx = xt, 0 < x < ∞, t > 0,
︴
︴
︴u(x,0) = sinx, u_t(x,0) = 1 ＋ x
︴
︴
︴u(0,t) = t.
(6) Let k > 0. Solve u_t = ku_xx, u(x,0) = 0, u(0,t) = 2 on the domain
0 < x < ∞, t > 0.