Fast algorithm for finding approximate solutions to the Cauchy problem for ODE

The present article considers the quick algorithm for finding an approximate solution for the Cauchy problem for ODE by calculating the coefficients of expansion of this solution in terms of the system $\{\varphi_{1,n}(x)\}_{n=0}^{\infty}$, where $\varphi_{1,0}(x)=1$, $\varphi_{1,1}(x)=x$, $\varphi_{1,n+1}(x)=\frac{\sqrt{2}}{\pi n}\sin(\pi nx),$ $n=1,2,\ldots$. This system is orthonormal with respect to the Sobolev scalar product $\langle f, g\rangle=f(0)g(0)+\int_0^1f'(x)g'(x)dx$ and generated by cosines $\varphi_0(x)=1$, $ \{\varphi_n(x)=\sqrt{2}\cos(\pi nx)\}_{n=1}^\infty$. The calculation of these coefficients is performed by an iterative process based on the fast Fourier transform.