Geodesic
A fast algorithm for solving the Cauchy problem for ODE using the Sobolev orthogonal polynomials generated by Chebyshev polynomials of the first kind
Daghestan Electronic Mathematical Reports, Tome 10 (2018), pp. 66-76

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We consider a numerical implementation of iteration process for solving Cauchy problem for ODE using the Sobolev orthogonal polynomials generated by Chebyshev polynomials of the first kind $T_0=1/\sqrt{2}$, $T_k(x)=\cos k\arccos x$ ($k\ge1$). Using the fast DCT, we construct the algorithm for this iteration process and develop the corresponding computer program. A number of numerical experiments show that the Fourier series by Sobolev – Chebyshev polynomials are very convenient for solving Cauchy problem.
Keywords: Chebyshev polynomials, Sobolev orthogonal polynomials, fast Fourier transform, discrete cosine transform, fixed-point iteration. 
@article{DEMR_2018_10_a6,
     author = {M. S. Sultanakhmedov and T. N. Shakh-Emirov},
     title = {A fast algorithm for solving the {Cauchy} problem for {ODE} using the {Sobolev} orthogonal polynomials generated by {Chebyshev} polynomials of the first kind},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {66--76},
     publisher = {mathdoc},
     volume = {10},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2018_10_a6/}
}
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M. S. Sultanakhmedov; T. N. Shakh-Emirov. A fast algorithm for solving the Cauchy problem for ODE using the Sobolev orthogonal polynomials generated by Chebyshev polynomials of the first kind. Daghestan Electronic Mathematical Reports, Tome 10 (2018), pp. 66-76. http://geodesic.mathdoc.fr/item/DEMR_2018_10_a6/