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Application of minimal interpolation splines to solve the Cauchy problem. (English. Russian original) Zbl 1152.65080

Vestn. St. Petersbg. Univ., Math. 40, No. 4, 302-305 (2007); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2007, No. 4, 114-117 (2007).
The classical Adams method for the numerical solution of an initial value problem for an ordinary differential equation consists of rewriting the problem as an equivalent Volterra equation, then replacing the integrand by an interpolating polynomial and finally integrating this polynomial exactly to compute the required approximate solution. The authors of the paper under review suggest to replace the interpolating polynomial in this construction by a spline function that satisfies an Hermite interpolation condition. The other parts of the algorithm remain unchanged. A numerical example is given but there is no theoretical investigation of the modified scheme with respect to properties like convergence, consistency or stability.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
Full Text: DOI

References:

[1] I. P. Mysovskikh, Lectures on Computational Methods (St. Petersburg, 1998) [in Russian].
[2] I. G. Burova and Yu. K. Dem’yanovich, The Theory of Minimal Splines (St. Petersburg, 2000) [in Russian].
[3] I. G. Burova, Vestn. S.-Peterburg. Univ., Mat. Mekh. Astron., No. 1, 9–13 (2005).
[4] I. G. Burova and V. A. Timofeev, Metody Vychisl., Vol. 21, 31–39 (2005).
[5] B. I. Kvasov, Methods for Spline Isogeometric Approximation (Moscow, 2006) [in Russian].
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