Geodesic
A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials
Daghestan Electronic Mathematical Reports, Tome 12 (2019), pp. 13-24.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider a numerical realization of an iterative method for solving the Cauchy problem for ordinary differential equations, based on representing the solution in the form of a Fourier series by the system of polynomials $\{L_{1,n}(x;b)\}_{n=0}^\infty$, orthonormal with respect to the Sobolev-type inner product $$ \langle f,g\rangle=f(0)g(0)+\int_{0}^\infty f'(x)g'(x)\rho(x;b)dx $$ and generated by the system of modified Laguerre polynomials $\{L_{n}(x;b)\}_{n=0}^\infty$, where $b>0$. In the approximate calculation of the Fourier coefficients of the desired solution, the Gauss–Laguerre quadrature formula is used.
Keywords: Laguerre polynomials, ordinary differential equation (ODE), Cauchy problem, inner product of Sobolev-type, Gauss–Laguerre quadrature formula. 
@article{DEMR_2019_12_a1,
     author = {G. G. Akniev and R. M. Gadzhimirzaev},
     title = {A numerical method for solving the {Cauchy} problem for {ODEs} using a system of polynomials generated by a system of modified {Laguerre} polynomials},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {13--24},
     publisher = {mathdoc},
     volume = {12},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2019_12_a1/}
}
TY  - JOUR
AU  - G. G. Akniev
AU  - R. M. Gadzhimirzaev
TI  - A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials
JO  - Daghestan Electronic Mathematical Reports
PY  - 2019
SP  - 13
EP  - 24
VL  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DEMR_2019_12_a1/
LA  - ru
ID  - DEMR_2019_12_a1
ER  - 
%0 Journal Article
%A G. G. Akniev
%A R. M. Gadzhimirzaev
%T A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials
%J Daghestan Electronic Mathematical Reports
%D 2019
%P 13-24
%V 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DEMR_2019_12_a1/
%G ru
%F DEMR_2019_12_a1
G. G. Akniev; R. M. Gadzhimirzaev. A numerical method for solving the Cauchy problem for ODEs using a system of polynomials generated by a system of modified Laguerre polynomials. Daghestan Electronic Mathematical Reports, Tome 12 (2019), pp. 13-24. http://geodesic.mathdoc.fr/item/DEMR_2019_12_a1/

[1] Sharapudinov I.I., Sharapudinov T.I., “Ortogonalnye po Sobolevu polinomy, porozhdennye modifitsirovannymi polinomami Lagerra, i zadacha Koshi dlya sistem ODU”, Dagestanskie elektronnye matematicheskie izvestiya, 2018, no. 10, 23–40

[2] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integraly i ryady, V 3 t., v. 2, Spetsialnye funktsii, 2-e izd., isprav., FIZMATLIT, M., 2003, 664 pp.

[3] Krylov V.I., Priblizhennoe vychislenie integralov, 2-e izd., dopolnennoe, Nauka, M., 1967, 500 pp.