I am a graduate student of Applied Mathematics. (Numerical Analysis)
I like to do an interdisciplinary work for my dissertation. (The application of radial basis functions in electromagnetics)
I would be grateful that you tell me more about choosing a good topic or introduce me some good articles and references.
Thank you for your favor!
Dear Hassan you can find out more about this topic in applied math. and numerical analysis book published in 1995 by Princeton publications.
Look at this book. Good luck
http://bookzz.org/s/?q=numerical+analysis+Princeton+&yearFrom=&yearTo=&language=&extension=&t=0
This is an excellent question with many possible answers. Apart from the very helpful answers already given, there is a bit more to consider.
In the run up to answers to this question, you may find the work by Bengt Fornberg helpful:
https://amath.colorado.edu/faculty/fornberg/
For an interesting illustration, see Figure 3 on page 31.
More to the point, for a Ph.D. thesis on radial point interpolation in electromagnetics, see
http://e-collection.library.ethz.ch/eserv/eth:2880/eth-2880-02.pdf
An introduction to radial basis functions with electomagnetics in mind is given in Section 3, starting on page 5.
Thank you for your attention to this matter.
My problem was solved.
Thank you again for everything you’ve done!
I believe this paper
Narcowich, Francis J., and Joseph D. Ward. "Generalized Hermite interpolation via matrix-valued conditionally positive definite functions." Mathematics of Computation 63.208 (1994): 661-687.
could be useful for the applications you have in mind. It contains a generalization of the standard scalar RBF concept to matrix valued RBFs, that can be employed for example to obtain divergence free vector fields, see e.g.
Wendland, Holger. "Divergence-free kernel methods for approximating the Stokes problem." SIAM Journal on Numerical Analysis 47.4 (2009): 3158-3179.
McNally, Colin P. "Divergence-free interpolation of vector fields from point values—exact∇⋅ B= 0 in numerical simulations." Monthly Notices of the Royal Astronomical Society: Letters 413.1 (2011): L76-L80.
The Narcowich-Ward paper is quite theoretical and may be difficult to read. An application to fluid dynamics problems with a simpler case of matrix RBF written out in more elementary fashion is contained in the following paper
Bonaventura, L., A. Iske, and E. Miglio. "Kernel‐based vector field reconstruction in computational fluid dynamic models." International Journal for Numerical Methods in Fluids 66.6 (2011): 714-729.
I hope it helps...
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