### Approximation theory

### Daghestan Electronic Mathematical Reports: Issue 10 (2018)

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

### UDK: 517.538

### Pages: 66 - 77

### DOI: 10.31029/demr.10.7

In this work, 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 carried out with the help of this program, show that the Fourier series by generated polynomials are very convenient for solving differential equations.

**Keywords: **
Chebyshev polynomials, Sobolev orthogonal polynomials, fast Fourier transform, discrete cosine transform, iterative process.