Find the adjoint of the following operators L dened on linear spaces S,
and indicate whether they are self-adjoint. All quantities are real
a) L=d/dt, S = { x屬於C1([0,1];IR) : x(0) = 3x(1)}.
這題我是令y也屬於C1[0,1],然後積分內積，得到∫dx*y/dt=xy|-∫dy*x/dt，
請問之後是直接代入x(0) = 3x(1)嗎？ 因為沒看過上下界不為值的題目....
b) The Laplace operator▽= d/dx^2+d/dy^2.
S={u屬於C2([0,1]×[0,1],IR):u(0,y) = u(x,0) = u(x,1) = u(1,y) = 0}
這題完全不會，請問集合裡[0,1]×[0,1]是甚麼意思呢?
If L = p(t)d^2/dt^2 + q(t)d/dt + r(t); with usual inner product
<x,y>=∫x(t)y(t)dt on [a,b], and L and determine conditions on p;q;r
such that L is self-adjoint and repeat,using <x,y>=∫w(t)x(t)y(t)dt;
p.s.w(t) is weighting function on [a,b]
請問一下在有weighting function下，L*要怎麼用積分證出來呢？
Find the eigenvalues and eigenvectors of the following matrices: |1 1|
|0 1|
這題我算出來的λ=1,eigenvector為(0,0)，請問eigenvector可能為0嗎?
Expand vector x = [ 2 0 1 ]T in terms of the normalized eigenvec-
tors of A= 0 0 1
0 0 1
1 1 1
請問這題是把A的所有特徵向量，個別乘以x嗎?
solve the following eigenvalue problem:
a) y"+λy=0 y'(0)=0 , y(π/2)=0
b) Expand function f(x)={0 0<=π/4
{100 π/4 <x<=π/2
這題我算出 eigenfunction y=c1*cos2nx ,但是b)我完全沒頭緒.....
不好意思，題目有點多，謝謝各位...
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