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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 $\sum_{i = 1}^{\infty} \frac{h^{i}}{i!} \frac{d^{i - 1}}{d x^{i - 1}} 𝐟 [x, 𝐲 (x)]$ up to $h^{7}$ with the development given by successive insertion into the formula $h . f_{i} (k_{0}, k_{1}, ..., k_{i - 1})$ for $i = 1, 2, ..., 10$ and $k = \sum_{i = 0}^{10} p_{i}, k_{i}$ 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.

A discrete maximum principle [Book]

Tadeusz Styś (1981)

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A discrete maximum principle [Book]

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A Finite Element Merthod Scheme for one Dimensional Elliptic Equations with High Super Convergence at the Nodes.

A Finite Element Method for a Singularly Perturbed Boundary Value Problem.

A finite element method for a two point boundary value problem with a small parameter affecting the highest derivative

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