Heat Kernel Likewise
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The first integration is 1 Z x Z ∞ e−²t ²(x − η) √ u(x, t) = {cos √ }d²g(η)dη 2π 0 2a ² − a ²−a where a is a positive constant and t ∈ (0, ∞). When I use Laplace inverse transform in my calculation, I met such a integration. Basically what I want to show is that when g(x),the initial data is small, for instance g ∈ Lp , I want to have u(x, t) ∈ Lp ,and ku(., t)kLp ∼ 1 1 O(1/(t + 1) 2 − 2p ) or around. So I’ve got to analyze the kernel
K(x, η, t) =
Z ∞ 2a
²(x − η) e−²t √ {cos √ }d² ²−a ²−a
The kernel acts as a dirac function when t approaches zero, and the numerical analysis verify it.I think it’s interesting to do analysis on it.But I don’t know whether other people already did this before, or whether it’s worth while. Another one is
v(x, t) =
Z πZ x 0
0
√ θ θ eat(cos(θ)−1) cos{ + atsin(θ) − 2 a(x − η)sin( )}g(η)dηdθ 2 2
Hopefully someone can tell me there WERE existing results about those two integral kernel before I spend time to re-find it.
1
K(x, η, t) =
Z ∞ 2a
²(x − η) e−²t √ {cos √ }d² ²−a ²−a
The kernel acts as a dirac function when t approaches zero, and the numerical analysis verify it.I think it’s interesting to do analysis on it.But I don’t know whether other people already did this before, or whether it’s worth while. Another one is
v(x, t) =
Z πZ x 0
0
√ θ θ eat(cos(θ)−1) cos{ + atsin(θ) − 2 a(x − η)sin( )}g(η)dηdθ 2 2
Hopefully someone can tell me there WERE existing results about those two integral kernel before I spend time to re-find it.
1
