Green's function method

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

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

Self adjoint case

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.

External links

find literature about Green's function method