Geodesic
An approximate solution of the Cauchy problem for an ODE system by means of system $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$
Daghestan Electronic Mathematical Reports, Tome 9 (2018), pp. 33-51

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We consider a system of functions $\xi_0(x)=1,\, \{\xi_n(x)=\sqrt{2}\cos(\pi nx)\}_{n=1}^\infty$ and the system $$ \xi_{1,0}(x)=1,\, \xi_{1,1}(x)=x,\, \xi_{1,n+1}(x)=\int_0^x \xi_{n}(t)dt=\frac{\sqrt{2}}{\pi n}\sin(\pi nx),\, n=1,2,\ldots, $$ generated by it, which is Sobolev orthonormal with respect to a scalar product of the form $$. It is shown that the Fourier series and sums with respect to the system $\{\xi_{1,n}(x)\}_{n=0}^\infty$ are a convenient and very effective tool for the approximate solution of the Cauchy problem for systems of nonlinear ordinary differential equations (ODEs).
Keywords: Cauchy problem, ODE, Fourier series, Fourier sums, approximate solution. 
@article{DEMR_2018_9_a4,
     author = {I. I. Sharapudinov},
     title = {An approximate solution of the {Cauchy} problem for an {ODE} system by means of system $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {33--51},
     publisher = {mathdoc},
     volume = {9},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2018_9_a4/}
}
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I. I. Sharapudinov. An approximate solution of the Cauchy problem for an ODE system by means of system $1,\, x,\, \{\frac{\sqrt{2}}{\pi n}\sin(\pi nx)\}_{n=1}^\infty$. Daghestan Electronic Mathematical Reports, Tome 9 (2018), pp. 33-51. http://geodesic.mathdoc.fr/item/DEMR_2018_9_a4/