Green's function method

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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

Here the specific position is and the general coordinate position is in 3D. ==

A typical physical sciences problem may be written as




Here is the source, is the particular solution called the field. The quantity is the Dirac delta function.

The problem is set up in a volume bounded by a surface There may be boundary conditions, that is to say, prescribed values of or its derivatives on the boundary . The 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

where the integrand of the volume integral is an exact divergence

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

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

Self adjoint case

The situation where the operator is called the self-adjoint case. We leave the , but note that there is an explicit form for the right hand side

Here is the normal derivative which is the directional derivative in the direction normal to the surface

Integral equation for the field, self adjoint operator

Substituting from the equalities above

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

This result is an integral equation for the field observed at the specific position , written in the form of an integral equation. The boundary may have specific values of and/or or may be sent to infinity, if an unbounded medium is desired. In the latter case, the boundary conditions are replaced with radiation conditions.

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).

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Green's function method
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