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I am going to post here a problem that I've been working on for quite some time, and I haven't managed to obtain a good answer.

To approximate free-field conditions in the numerical solution of wave equation PDEs, the following coordinate transformation is often applied to implement a Perfectly Matched Layer (PML):

[tex]

\[

\frac{\partial }{{\partial x}} \to \frac{1}{{1 + \frac{{i\sigma (x)}}{\omega }}}\frac{\partial }{{\partial x}}

\]

[/tex]

There is further information given in this application note:

http://www-math.mit.edu/~stevenj/18.369/pml.pdf" [Broken]

Normally this coordinate-stretching is applied to [tex]\partial /\partial x[/tex]

But is it possible to apply the coordinate-stretching in the following fashion?

[tex]

\[

\frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{u}\frac{{\partial ^2 }}{{\partial x^2 }}

\]

[/tex]

That is, the coordinate stretching is applied without recourse to this:

[tex]

\[

\frac{{\partial ^2 }}{{\partial x^2 }} \to \frac{1}{s}\frac{\partial }{{\partial x}}\left( {\frac{1}{s}\frac{\partial }{{\partial x}}} \right)

\]

[/tex]

where

[tex]

\[

s = 1 + \frac{{i\sigma (x)}}{\omega }

\]

[/tex]

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# Application of perfectly matched layer transformation to second order derivative

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