- #1
nkinar
- 76
- 0
In my research, I'm using a modified version of the wave equation:
[tex]
\[
c^2 \left( {\frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }}} \right) = - \tau c^2 \left( {\frac{{\partial ^3 p}}{{\partial t\partial x^2 }} + \frac{{\partial ^3 p}}{{\partial t\partial y^2 }}} \right) + \frac{{\partial ^2 p}}{{\partial t^2 }}
\]
[/tex]
I would like to take this PDE, and split the equation into a system of PDEs or ODEs. There is a PDF document on the internet which deals with this type of splitting on page 4, but I do not understand what is being mentioned when the author writes about an "auxiliary field."
Here is a link to the PDF:
http://math.mit.edu/~stevenj/18.369/pml.pdf
In this PDF, the author gives the source-free scalar wave equation:
[tex]
\[
\nabla \cdot \left( {a\nabla u} \right) = \frac{1}{b}\frac{{\partial ^2 u}}{{\partial t^2 }} = \frac{{\ddot u}}{b}
\]
[/tex]
The author then introduces an "auxiliary field", and re-writes the source-free scalar wave equation as the system of two coupled PDEs:
[tex]
\[
\frac{{\partial u}}{{\partial t}} = b\nabla \cdot {\bf{v}}
\]
[/tex]
[tex]
\[
\frac{{\partial {\bf{v}}}}{{\partial t}} = a\nabla u
\]
[/tex]
I would like to do the same for my modified version of the wave equation, but I am uncertain as how to deal with the mixed partial derivatives.
[tex]
\[
c^2 \left( {\frac{{\partial ^2 p}}{{\partial x^2 }} + \frac{{\partial ^2 p}}{{\partial y^2 }}} \right) = - \tau c^2 \left( {\frac{{\partial ^3 p}}{{\partial t\partial x^2 }} + \frac{{\partial ^3 p}}{{\partial t\partial y^2 }}} \right) + \frac{{\partial ^2 p}}{{\partial t^2 }}
\]
[/tex]
I would like to take this PDE, and split the equation into a system of PDEs or ODEs. There is a PDF document on the internet which deals with this type of splitting on page 4, but I do not understand what is being mentioned when the author writes about an "auxiliary field."
Here is a link to the PDF:
http://math.mit.edu/~stevenj/18.369/pml.pdf
In this PDF, the author gives the source-free scalar wave equation:
[tex]
\[
\nabla \cdot \left( {a\nabla u} \right) = \frac{1}{b}\frac{{\partial ^2 u}}{{\partial t^2 }} = \frac{{\ddot u}}{b}
\]
[/tex]
The author then introduces an "auxiliary field", and re-writes the source-free scalar wave equation as the system of two coupled PDEs:
[tex]
\[
\frac{{\partial u}}{{\partial t}} = b\nabla \cdot {\bf{v}}
\]
[/tex]
[tex]
\[
\frac{{\partial {\bf{v}}}}{{\partial t}} = a\nabla u
\]
[/tex]
I would like to do the same for my modified version of the wave equation, but I am uncertain as how to deal with the mixed partial derivatives.