### 3-point implicit block method for solving ordinary differential equations.

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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 ${\sum}_{i=1}^{\infty}\frac{{h}^{i}}{i!}\frac{{d}^{i-1}}{d{x}^{i-1}}\mathbf{f}\left[x,\mathbf{y}\left(x\right)\right]$ 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.

This paper deals with modeling the passive behavior of skeletal muscle tissue including certain microvibrations at the cell level. Our approach combines a continuum mechanics model with large deformation and incompressibility at the macroscale with chains of coupled nonlinear oscillators. The model verifies that an externally applied vibration at the appropriate frequency is able to synchronize microvibrations in skeletal muscle cells. From the numerical analysis point of view, one faces...

In this paper, Runge-Kutta methods are discussed for numerical solutions of conservative systems. For the energy of conservative systems being as close to the initial energy as possible, a modified version of explicit Runge-Kutta methods is presented. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in energy-preserving to the standard one. Comparing the modified Runge-Kutta method with the standard Runge-Kutta method, numerical experiments...

Specialized literature concerning studies on Orbital Dynamics usually mentions the Gauss-Jackson or sum squared (∑2) method for the numerical integration of second order differential equations. However, as far as we know, no detailed description of this code is available and there is some confusion about the order of the method and its relation with the Störmer method. In this paper we present a simple way of deriving this algorithm and its corresponding analog for first order equations from the...

In the paper the convergence of a mixed Runge--Kutta method of the first and second orders to a strong solution of the Ito stochastic differential equation is studied under a monotonicity condition.