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Title: A contribution to Runge-Kutta formulas of the 7th order with rational coefficients for systems of differential equations of the first order (English)
Author: Huťa, Anton
Author: Penjak, Vladimír
Language: English
Journal: Aplikace matematiky
ISSN: 0373-6725
Volume: 29
Issue: 6
Year: 1984
Pages: 411-422
Summary lang: English
Summary lang: Slovak
Summary lang: Russian
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Category: math
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Summary: The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development $\sum^\infty_{i=1} \frac {h^i} {i!} \frac {d^{i-1}} {dx^{i-1}} \bold f\left[x,\bold y(x)\right]$ up to $h^7$ with the development given by successive insertion into the formula $h.f_i(k_0,k_1,\ldots, k_{i-1})$ for $i=1,2,\ldots, 10$ and $k=\sum^{10}_{i=0} p_i, k_i$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers. (English)
Keyword: Runge-Kutta formulas
Keyword: rational coefficients
Keyword: systems
Keyword: 7th order formulas
MSC: 34A34
MSC: 65L05
idZBL: Zbl 0558.65047
idMR: MR0767494
DOI: 10.21136/AM.1984.104115
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Date available: 2008-05-20T18:26:11Z
Last updated: 2020-07-28
Stable URL: http://hdl.handle.net/10338.dmlcz/104115
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Reference: [1] W. Kutta: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen.Z. Math. und Phys. 46 (1901), 435-453.
Reference: [2] E. J. Nyström: Über die numerische Integration von Differentialgleichungen.Acta Soc. Sci. Fennicae 50 (1926), 13.
Reference: [3] A. Huťa: Une amélioration de la méthode de Runge-Kutta-Nyström pour la résolution numérique des équations différentielles du premier ordre.Acta Fac. Rer. Nat. Univ. Comen. 1 (1956), IV-VI, 201-224. MR 0097156
Reference: [4] A. Huťa: Contribution à la formule de sixième ordre dans la méthode de Runge-Kutta-Nyström.Acta Fac. Rer. Nat. Univ. Comen 2 (1957), I-II, 21 - 24. MR 0097157
Reference: [5] V. Penjak: Condition equations of the 7th order Runge-Kutta methods.Štrbské Pleso 1982. Proceedings from the symposium on the numerical methods and graph theory, 105-108. (Slovak.)
Reference: [6] A. Huťa: The Algorithm for Computation of the n-order Formula for Numerical Solution of Initial Value Problem of Differential Equations.5th Symposium on Algorithms - Algorithms' 79. Proceedings of Lectures (1979), 53 - 61.
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