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

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.