# The Green's function method  

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

A typical physical sciences problem may be written as

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

The problem is set up in a volume ${\mathcal {D}}$ bounded by a surface $D.$ There may be boundary conditions, that is to say, prescribed values of $u$ or its derivatives on the boundary $\partial D$ . The $\star$ 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

$\int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\right\}\;dV=\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS,$ where the integrand of the volume integral is an exact divergence

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

$\int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\left[-f(\mathbf {x} )\right]-u(\mathbf {x} )\left[-\delta (\mathbf {x} -\mathbf {x_{0}} )\right]\right\}\;dV=\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS,$ applying the sifting property of the delta function, and rearranging terms, we have

$u(\mathbf {x_{0}} )=\int _{D}f(\mathbf {x} )g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\;dV+\int _{\partial D}\left\{\mathbf {\hat {n}} \cdot \mathbf {Q} \right\}\;dS$ The situation where the operator ${\mathcal {L}}^{\star }={\mathcal {L}}$ is called the self-adjoint case. We leave the $\star$ , but note that there is an explicit form for the right hand side

$\int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\mathcal {L}}u(\mathbf {x} )-u(\mathbf {x} ){\mathcal {L}}^{\star }g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\right\}\;dV=\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS,$ Here $\partial /\partial n$ is the normal derivative which is the directional derivative in the direction normal to the surface $\partial /\partial n\equiv \mathbf {\hat {n}} \cdot \nabla .$ ### Integral equation for the field, self adjoint operator

Substituting from the equalities above

$\int _{D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\left[-f(\mathbf {x} )\right]-u(\mathbf {x} )\left[-\delta (\mathbf {x} -\mathbf {x_{0}} )\right]\right\}\;dV=\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS,$ applying the sifting property of the delta function, and rearranging terms, we have the integral equation of the field

$u(\mathbf {x_{0}} )=\int _{D}f(\mathbf {x} )g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )\;dV+\int _{\partial D}\left\{g^{\star }(\mathbf {x} ,\mathbf {x_{0}} ){\frac {\partial u(\mathbf {x} )}{\partial n}}-u(\mathbf {x} ){\frac {\partial g^{\star }(\mathbf {x} ,\mathbf {x_{0}} )}{\partial n}}\right\}\;dS.$ This result is an integral equation for the field observed at the specific position $\mathbf {x_{0}}$ , written in the form of an integral equation. The boundary $\partial D$ may have specific values of $u(\mathbf {x} )$ and/or $\partial u/\partial n,$ 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) uses the term representation theorem for such a result.