- #1
DrJekyll
- 6
- 0
I am trying to find the solution to a problem defined as follows:
[tex]
(\partial_x^2-K^2)^2 G(\vec{x},\vec{x}')=\delta(\vec{x}'-\vec{x})
[/tex]
where K is simply a constant and x is three dimensional.
[tex]
A \left[ e^{-K(\vec{x}-\vec{x}')}H(\vec{x}-\vec{x}') + e^{K(\vec{x}-\vec{x}')}H(\vec{x}'-\vec{x}) \right]
+ B \left[ (\vec{x}-\vec{x}') e^{-K(\vec{x}-\vec{x}')}H(\vec{x}-\vec{x}') + (\vec{x}'-\vec{x}) e^{K(\vec{x}-\vec{x}')}H(\vec{x}'-\vec{x}) \right]
[/tex]
Here, H is the step function.
and I am trying to find A and B such that the first equation is satisfied. Does anyone have any advice on how to proceed?
My initial guess was to apply [tex] (\partial_x^2-K^2)^2 [/tex] to my proposed solution and then solve for A and B. This gets a bit tricky since once I start taking derivatives I get derivatives of the delta function.
[tex]
(\partial_x^2-K^2)^2 G(\vec{x},\vec{x}')=\delta(\vec{x}'-\vec{x})
[/tex]
where K is simply a constant and x is three dimensional.
[tex]
A \left[ e^{-K(\vec{x}-\vec{x}')}H(\vec{x}-\vec{x}') + e^{K(\vec{x}-\vec{x}')}H(\vec{x}'-\vec{x}) \right]
+ B \left[ (\vec{x}-\vec{x}') e^{-K(\vec{x}-\vec{x}')}H(\vec{x}-\vec{x}') + (\vec{x}'-\vec{x}) e^{K(\vec{x}-\vec{x}')}H(\vec{x}'-\vec{x}) \right]
[/tex]
Here, H is the step function.
and I am trying to find A and B such that the first equation is satisfied. Does anyone have any advice on how to proceed?
My initial guess was to apply [tex] (\partial_x^2-K^2)^2 [/tex] to my proposed solution and then solve for A and B. This gets a bit tricky since once I start taking derivatives I get derivatives of the delta function.