Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs
Izvestiya. Mathematics , Tome 83 (2019) no. 2, pp. 391-412
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We consider systems of functions ${\varphi}_{r,n}(x)$ ($r=1,2,\dots$,
$n=0,1,\dots$) that are Sobolev-orthonormal with respect to a scalar
product of the form $\langle f,g\rangle=
\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+
\int_{a}^{b}f^{(r)}(x)g^{(r)}(x)\rho(x)\,dx$
and are generated by a given orthonormal system of functions
$\varphi_{n}(x)$ ($n=0,1,\dots$). The Fourier series and sums with respect
to the system $\varphi_{r,n}(x)$ ($r=1,2,\dots$, $n=0,1,\dots$) are shown
to be a convenient and efficient tool for the approximate solution of the
Cauchy problem for ordinary differential equations (ODEs).
Keywords:
Sobolev-orthogonal systems, Cauchy problem for ODEs,
systems generated by Haar functions, cosines or Chebyshev polynomials.
@article{IM2_2019_83_2_a10,
author = {I. I. Sharapudinov},
title = {Sobolev-orthogonal systems of functions and the {Cauchy} problem for {ODEs}},
journal = {Izvestiya. Mathematics },
pages = {391--412},
publisher = {mathdoc},
volume = {83},
number = {2},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_2_a10/}
}
I. I. Sharapudinov. Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs. Izvestiya. Mathematics , Tome 83 (2019) no. 2, pp. 391-412. http://geodesic.mathdoc.fr/item/IM2_2019_83_2_a10/