A fast algorithm for solving the Cauchy problem for ODE using the Sobolev orthogonal polynomials generated by Chebyshev polynomials of the first kind

M. S. Sultanakhmedov, T. N. Shakh-Emirov

Daghestan Scientific Centre of Russian Academy of Sciences, Makhachkala

Abstract: 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.

UDC: 517.538

Received: 15.10.2018
Revised: 28.11.2018
Accepted: 29.11.2018

DOI: 10.31029/demr.10.7