- #1

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## Main Question or Discussion Point

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.