The collocation method is one of expansion methods which is used to approximate the linear and non-linear integral equations. This method can be used with basis functions such as standard polynomials {1,x,x^2,x^3,...}, Taylor polynomials, Chebyshev polynomials (first, second, third and fourth kinds) and others. What are the orders of convergence of collocation method with standard polynomials and Chebyshev polynomials ( second, third and fourth kinds)?
Hi, Samad.
I do not think this question can be put across with several words, so I suggest that you refer to Prof. Brunner's work "Collocation Methods for Volterra Integral and Related Functional Equations " or Prof. Shen's book "Spectral Methods: Algorithms, Analysis and Applications"
Dear Samad,
to give a short answer, I give an answer only on general level. Some good references have been given above. In general, the convergence speed depends on
1) appropriate choice of the collocations points,
2) smoothness of the data in appropiately chosen (Sobolev) spaces.
Typically the error estimates are of the form
||u-u_N||_r
May be useful this expression try this Order of Convergence is equal to [log(max(error(N/2))/max(error(N)))]/log(2)
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