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 Daghestan Electronic Mathematical Reports, 2017, Issue 7, Pages 77–85 (Mi demr40)

Cauchy problem for the difference equation and Sobolev orthogonal functions on the finite grid, generated by discrete orthogonal functions

M. S. Sultanakhmedov

Daghestan scientific center of RAS

Abstract: We consider the system of functions ${\psi}_{1,n}(x, N)$ ($n=0,1,\ldots,$ $N$), orthonormal in Sobolev sense and generated by a given orthonormal on finite grid $\Omega_N=\{ 0,1,\ldots,N-1 \}$ system of functions ${\psi}_{n}(x,N)$ $( n=0,1,\ldots,N-1)$. These new functions are orthonormal with respect to the inner product of the following type: $\langle f,g\rangle = f(0)g(0)+ \sum_{j=0}^{N-1}\Delta f(j)\Delta g(j)\rho(j)$. It is shown that the finite Fourier series by the functions ${\psi}_{1,n}(x)$ and their partial sums are convenient and a very effective tool for the approximate solution of the Cauchy problem for nonlinear difference equations.

Keywords: Sobolev orthogonal functions; functions orthogonal on the finite grid; finite grid; uniform grid; approximation of discrete functions; mixed series by the functions orthogonal on a uniform grid; iterative process for the approximate solution of difference equations

DOI: https://doi.org/10.31029/demr.7.9

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UDC: 517.912
Revised: 17.04.2017
Accepted:18.04.2017
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Citation: M. S. Sultanakhmedov, “Cauchy problem for the difference equation and Sobolev orthogonal functions on the finite grid, generated by discrete orthogonal functions”, Daghestan Electronic Mathematical Reports, 2017, no. 7, 77–85

Citation in format AMSBIB
\Bibitem{Sul17} \by M.~S.~Sultanakhmedov \paper Cauchy problem for the difference equation and Sobolev orthogonal functions on the finite grid, generated by discrete orthogonal functions \jour Daghestan Electronic Mathematical Reports \yr 2017 \issue 7 \pages 77--85 \mathnet{http://mi.mathnet.ru/demr40} \crossref{https://doi.org/10.31029/demr.7.9} 

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This publication is cited in the following articles:
1. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Sobolev orthogonal functions on the grid, generated by discrete orthogonal functions and the Cauchy problem for the difference equation”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 29–39
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