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課程名稱︰偏微分方程導論
課程性質︰數學系大三必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試日期︰2013年05月31日(五)，08:10-09:10
考試時限：60分鐘
是否需發放獎勵金：是
試題 :
Math 2206 (Intorduction to PDE) Quiz No.7 (5/31/2013)
Solve the following problems. Please write down your computational or proof
steps clearly on the answer sheets.
A. (20 points) Solve u_t = u_xx for x > 0, t > 0 satisfying the initial
condition u(x,0) = g(x) for x≧0, and the boundary condition u_x(0,t) = 0
for t≧0, where g(x) is a bounded continuous function defined in x≧0.
B. (25 points) Let G⊂R^n be a bounded domain so that ∂G is a C^2 boundary.
Consider the initial-boundary value problem
u_t = △u for x∈G, t > 0
u(x,0) = f(x) for x∈bar(G)
u(x,t) = g(x) fpr x∈∂G, t > 0
where f∈C^2(bar(G)), and g∈C(∂G). Prove that lim u(x,t) = v(x) so that
t→∞
△v(x) = 0 for x∈G, and v(x) = g(x) for x∂G.
C. Consider the Porous media equation u_t = △(u^γ), where γ>1 is a constant.
(a) (5 points) Find necessary conditions on a,b,c∈|R such that the
a b c
equation has a solution u(x,t) satisfying u(x,t) = λ u(λ x,λ t) for
all λ > 0.
(b) (20 points) Find an example of such solution in (a) which is also
rotationally invariant.
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