# The Green's function method [1] [2]

The Green's function may be used in conjunction with Green's theorem to construct solutions for problems that are governed by ordinary or partial differential equations.

## Integral equation for the field at ${\displaystyle u(\mathbf {x_{0}} )}$

Here the specific position is ${\displaystyle \mathbf {x_{0}} \equiv (x_{01},x_{02},x_{03})}$ and the general coordinate position is ${\displaystyle \mathbf {x} \equiv (x_{1},x_{2},x_{3}).}$ in 3D. ==

A typical physical sciences problem may be written as

${\displaystyle {\mathcal {L}}u(\mathbf {x} )=-f(\mathbf {x} )}$

${\displaystyle {\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )=-\delta (\mathbf {x} -\mathbf {x_{0}} ).}$

Here ${\displaystyle f(\mathbf {x} )}$ is the source, ${\displaystyle u(\mathbf {x} )}$ is the particular solution called the field. The quantity ${\displaystyle \delta (\mathbf {x} -\mathbf {x_{0}} )}$ is the Dirac delta function.

The problem is set up in a volume ${\displaystyle {\mathcal {D}}}$ bounded by a surface ${\displaystyle D.}$ There may be boundary conditions, that is to say, prescribed values of ${\displaystyle u}$ or its derivatives on the boundary ${\displaystyle \partial D}$. The ${\displaystyle \star }$ indicates adjoint meaning that this is the operator and its respective Green's function allow the expression to be written as an exact divergence.

### General form

We write the more general form of Green's theorem making use of these definitions as

${\displaystyle \int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\right\}\;dV=\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS,}$

where the integrand of the volume integral is an exact divergence

${\displaystyle \left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\right\}=\nabla \cdot \mathbf {Q} }$

and the left hand side follows from an application of the divergence theorem.

### Integral equation for the field, non-self adjoint operator

Substituting from the equalities above

${\displaystyle \int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\left[-f(\mathbf {x} )\right]-u(\mathbf {x} )\left[-\delta (\mathbf {x} -\mathbf {x_{0}} )\right]\right\}\;dV=\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS,}$

applying the sifting property of the delta function, and rearranging terms, we have

${\displaystyle u(\mathbf {x_{0}} )=\int _{D}f(\mathbf {x} )g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\;dV+\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS}$

The situation where the operator ${\displaystyle {\mathcal {L}}^{\star }={\mathcal {L}}}$ is called the self-adjoint case. We leave the ${\displaystyle \star }$, but note that there is an explicit form for the right hand side

${\displaystyle \int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\right\}\;dV=\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS,}$

Here ${\displaystyle \partial /\partial n}$ is the normal derivative which is the directional derivative in the direction normal to the surface ${\displaystyle \partial /\partial n\equiv \mathbf {\hat {n}} \cdot \nabla .}$

### Integral equation for the field, self adjoint operator

Substituting from the equalities above

${\displaystyle \int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\left[-f(\mathbf {x} )\right]-u(\mathbf {x} )\left[-\delta (\mathbf {x} -\mathbf {x_{0}} )\right]\right\}\;dV=\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS,}$

applying the sifting property of the delta function, and rearranging terms, we have the integral equation of the field

${\displaystyle u(\mathbf {x_{0}} )=\int _{D}f(\mathbf {x} )g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\;dV+\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS.}$

This result is an integral equation for the field observed at the specific position ${\displaystyle \mathbf {x_{0}} }$, written in the form of an integral equation. The boundary ${\displaystyle \partial D}$ may have specific values of ${\displaystyle u(\mathbf {x} )}$ and/or ${\displaystyle \partial u/\partial n,}$ or may be sent to infinity, if an unbounded medium is desired. In the latter case, the boundary conditions are replaced with a radiation condition.

Aki and Richards (2002)[3] uses the term representation theorem for such a result.

## References

1. Bleistein, N. (1984). Mathematical methods for wave phenomena. Academic Press.
2. Bleistein, N., J. K. Cohen, & J. W. Stockwell Jr., (2001). Mathematics of multidimensional seismic imaging, migration, and inversion. Springer Verlag.
3. Aki, K., & Richards, P. G. (2002). Quantitative seismology (Vol. 1).