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Keywords:
Runge-Kutta formulas; rational coefficients; systems; 7th order formulas
Runge-Kutta formulas; rational coefficients; systems; 7th order formulas
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.
References:
[1] W. Kutta: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. und Phys. 46 (1901), 435-453.
[2] E. J. Nyström: Über die numerische Integration von Differentialgleichungen. Acta Soc. Sci. Fennicae 50 (1926), 13.
[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
[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
[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.)
[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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