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

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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/}
}
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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/