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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 1, Pages 144–154 (Mi vuu316)

COMPUTER SCIENCE

Exact formulas for coefficients and residual of optimal approximate spline of simplest wave equation

N. V. Rodionova

Udmurt State University, Izhevsk, Russia

Abstract: We define the parameter family of finite-dimensional spaces of special quadratic splines of Lagrange's type. In each space, the optimal spline which gives the smallest residual being a square of the norm in the space $\mathrm L_2$, is proposed as a solution to the initial-boundary problem for the simplest wave equation. The exact formulas for the coefficients of the spline and its residual are obtained. The formula for the coefficients of this spline is a linear form of finite differences of the discretely given initial and boundary conditions of the original problem. The formula for the residual $J$ is a positive definite quadratic form of these quantities. The coefficients of both forms are computable via Chebyshev's polynomials of the second kind. The explicit form of the formula for the residual allows to solve the inequality $J<\varepsilon^2$ for a given computing accuracy $\varepsilon>0$ and to receive a priori sufficient number of nodes of a difference scheme.
The investigations were carried out for one time layer, which has two sublayers. We obtained difference formulas of the initial condition for the partial derivative with respect to time. They allow to create a difference scheme for the new layer, which in turn allows to continue the iterative computational process in time as far as desired.

Keywords: interpolation, approximate spline, residual, Chebyshev's polynomials.

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UDC: 519.651+517.518.823
MSC: 41A15

Citation: N. V. Rodionova, “Exact formulas for coefficients and residual of optimal approximate spline of simplest wave equation”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 1, 144–154

Citation in format AMSBIB
\Bibitem{Rod12} \by N.~V.~Rodionova \paper Exact formulas for coefficients and residual of optimal approximate spline of simplest wave equation \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2012 \issue 1 \pages 144--154 \mathnet{http://mi.mathnet.ru/vuu316} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. N. V. Rodionova, “Tochnoe reshenie odnoi zadachi optimizatsii, porozhdennoi prosteishim volnovym uravneniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 1, 141–152
2. V. I. Rodionov, “O reshenii odnoi zadachi optimizatsii, porozhdennoi prosteishim uravneniem teploprovodnosti”, Izv. IMI UdGU, 2014, no. 1(43), 49–67
3. V. I. Rodionov, “O lineinom algoritme chislennogo resheniya kraevoi zadachi dlya prosteishego volnovogo uravneniya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:1 (2015), 126–144
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