Geodesic
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
Daghestan Electronic Mathematical Reports, Tome 7 (2017), pp. 66-76

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Consider the systems of functions ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ orthonormal with respect to a Sobolev-type inner product of the form $\langle f,g\rangle= \sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b}f^{(r)}(x)g^{(r)}\rho(x)(x)dx$ generated by a given orthonormal system of functions ${\varphi}_{n}(x)$ $( n=0,1,\ldots)$. It is shown that the Fourier series in the system ${\varphi}_{r,n}(x)$ $(r=1,2,\ldots, n=0,1,\ldots)$ and their partial sums are a convenient and very effective tool for the approximate solution of the Cauchy problem for ordinary differential equations (ODEs).
Keywords: the Cauchy problem, Fourier series, Sobolev orthogonal functions. 
@article{DEMR_2017_7_a7,
     author = {I. I. Sharapudinov},
     title = {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}},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {66--76},
     publisher = {mathdoc},
     volume = {7},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2017_7_a7/}
}
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I. I. Sharapudinov. 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. Daghestan Electronic Mathematical Reports, Tome 7 (2017), pp. 66-76. http://geodesic.mathdoc.fr/item/DEMR_2017_7_a7/