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06月16日，09:00-12:00
考試時限：180分鐘
是否需發放獎勵金：是
試題 :
Math 2206 (Introduction to PDE) Final (6/16/2013)
There are problems A to F with a total of 150 points. Please write down your
computational or proof steps clearly on the answer sheets. Problems (A) to (E)
each has 22 points, and every sub-problem in (F) has 10 points.
A. Solve the heat equation in the semi-infinite rod x≧0:
u_t = u_xx for x > 0, t > 0;
u(x,0) = f(x) for x > 0;
(u_x)(0,t) = g(t) for t > 0
where f(x) and g(t) are sufficient smooth functions. Is f'(0) = g(0) a
necessary condition in order for the solution to exist?
B. Consider the exterior Dirichlet problem for the Laplace equation:
△u = 0 for x^2 + y^2 > 1;
u(x,y) = f(x,y) for x^2 + y^2 = 1;
u(x,y) → γ as x^2 + y^2 →∞
where f(x,y) is a given C^1 function defined on x^2 + y^2 = 1, and γ∈|R
is a constant. Find a solution whenever it is solvable. Is the solution you
get the unique solution?
C. Assume that u(x,t) solves the heat equation u_t = △u in x∈|R^n, t > 0 so
that u∈C(|R^n ×[0,∞)), and u is bounded. If ∫ |u(x,0)|dx < ∞, prove
|R^n
that lim u(x,t) = 0 uniformly in x∈|R^n. Do you conclude that
t→∞
lim ∫ u(x,t)dx = 0?
t→∞ |R^n
D. Consider the open domain D = {x=(x1,x2,x3)∈R^3 | |x| < R, x3 > 0}(R > 0 is
the radius). Write down the explicit form of the Dirichlet Green's function
G(x,y), x,y∈D, x≠y for D. Then compute the Dirichlet Poisson kernel for D
E. Consider the inviscid Burgers equation u_t + u(u_x) = 0 in t≧0. Find
necessary conditions on a,b,c∈|R such that the equation has a solution
u(x,t) satisfying u(x,t) = (λ^a) u((λ^b)x,(λ^c)t) for all λ > 0. Then
find two distinct non-trival, non-constant such solution.
F. Determine which of the following statements is true. Prove your answer.
(a) Let G = {(x,y)∈|R^2 | x∈|R, y > 0}. If u∈C^2(G)∩C(bar(G)) is
harmonic in G, then |u(x,y)| ≦ sup |u(x,0)| for all (x,y)∈G.
x∈|R
(b) Let u∈C((a,b)), where (a,b)⊂|R is an open interval. If u((x+y)/2) =
[u(x) + u(y)]/2 for all x,y∈(a,b). Then u∈C^∞((a,b)), and u'' = 0.
(c) Consider the initial boundary value problem
u_t = u_xx for 0 < x <L, t > 0,
u(x,0) = 0 for 0≦ x≦L,
(u_x)(0,t) = α for t≧0,
(u_x)(L,t) = β for t≧0,
where α,β are constants.
Then lim u(x,t) = U(x) exists, and U_xx = 0.
t→∞
(d) Let G⊂|R^n be a bounded open domain, and U = G ×(0,T)(where T > 0 is
a constant). If u∈C(bar(U)) is smooth in U, and satisfies
u_t = △u + cu, where c∈|R is another constant.
Then max |u(x,t)| = max |u(x,t)|, where ∂'U = (G ×{0})∪(∂G ×[0,T])
(x,t)∈bar(U) (x,t)∈∂'U
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.252.31