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October 7, 2012, 07:00 

#21  
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October 7, 2012, 17:06 

#22  
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Nikola Mirkov
Join Date: Mar 2009
Location: Belgrade, Serbia
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The path that led me to these is a curious one. I first used Bernstein polynomials to approximate a shape of an "flying saucer" shaped aircraft in an aerodynamic study of Coanda effect. Here is the paper from ICAS 2010 in Nice, France where it was presented. In presented scheme they seemed powerful in representation of various complicated shapes. Then I was thinking  because solutions of ODE's and PDE's are also some shapes (curves and surfaces), that Bernstein polynomials can approximate them as well. There were some mathematical results that backed that (e.g. StoneWeierstrass theorem). I'm still discovering some of their properties. I even managed to generalize some (formula for nonrecursive evaluation of kth order derivative for polynomials defined on a generalized interval [a,b] is still not published  I will do it when I find time...my PhD thesis is suffering because of my infidelity  I hope it won't ask for a divorce ) I found a couple of papers from 2008, but they are using Galerkin approach (I use collocation), and they solve linear system, where they should only evaluate function... I really hope I will find time to put everything together and write a paper!
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October 7, 2012, 17:15 

#23  
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Filippo Maria Denaro
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did you analize the spectral properties of such polynomials? What about the equivalent modified wavenumber for the numerical derivative? 

October 7, 2012, 17:35 

#24 
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Nikola Mirkov
Join Date: Mar 2009
Location: Belgrade, Serbia
Posts: 18
Rep Power: 9 
@ FM Denaro
Not yet! Now I'm reading John Boyd's book on Chebyshev and Fourier spectral methods (awesome and cheap book btw) and I collected couple of ideas what tests to do and what problems to try to tackle. P.S. I'm unfortunately mostly working on my own, without guidance, and I really like this kind of interaction. People around me at work are more into commercial software...
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