Approximation of the solution of the Cauchy problem for nonlinear ODE systems by means of Fourier series in functions orthogonal in the sense of Sobolev

Systems of functions ${\cal \varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal in the sense of Sobolev with respect to a scalar product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(t)g^{(r)}(x)dx$ generated by a given orthonormal system of functions ${\cal \varphi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series and sums with respect to the system ${\cal \varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).