How to deal with the singularity of partial derivative for 3-D translating-pulsating

Recently I am stuty on the Numeric Calculation of 3-D translating-pulsating source(Haskind source) Green's function in Havelock form.LOBATTO rule could be used to eliminate the singulatity of of the Green's function expressed in single integration form and the Hess &Smith Polynomial Method could be adopted to calculated the complex exponentila integral.But the complex exponential integral for derivatives of Green's function oscillates with high frequency when theta=pi due to its tending to intinity.And how to deal with this singularity of partial derivative for 3-D translating-pulsating source Green's function????????
If you need a more particular knowledge of this problem ,you could read the attachment named the singularity of partial derivative.pdf
If anyone willing to tell me any method to deal with singularity of partial derivative for 3-D translating-pulsating source Green's function in Havelock form,or recommend me any material about this problem,it will be much appreciated.
Your response will be highly appreciated.
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Thanking you in advance

 

There is a chapter in a book by Cheung Hun Kim that might be helpful:

https://books.google.com.au/books?i...an green's function haskind pulsating&f=false

J.N. Newman published several papers in the 1980s and 1990s dealing with
fast algorithms to calculate Green's functions in the time domain and the
frequency domain. Search for his name and the Journal of Engineering Mathematics.

Also see: J.N. Newman,
"Double-precision evaluation of the oscillatory source potential",
J. Ship Research. 28 (1984).

 

There is the "finite part integral operator" that was developed to handle some of these problems with indefinite kernels in partial differential equations.

See if the examples in this doc look promising.

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930092099.pdf

 

thanks for the kind help of Leo Lazauskas and philSweet.I will pick up the file and read it as soon as possible.

 

Sorry for taking so long to get back to this post.

There is a paper by Matiur Rahman that shows how to calculate the
3D Green's function for ship motions at forward speed, and that
handles the singularities that are probably giving you trouble.

It is available on the net at:
http://www.emis.de/journals/HOA/IJMMS/Volume13_3/347046.pdf

I would also have a look at the method given in:
Wu, G.X., and Eatock-Taylor, R.,
Green's function form for ship motions at forward speed,
Int. Shipbuilding Progress, Vol. 34, 1987, pp 189-196.

Someone at Shanghai University has some work that might be
interesting but the paper on the internet is not complete:
http://www.isope.org/publications/proceedings/ISOPE/ISOPE 1995/Abstract Pages/I95v3p226.pdf

Good luck!