multistep

Existing variable order step size numerical techniques for solving a system of higher-order ordinary differential equations (ODEs)  requires direct calculating the integration coefficients at each step change.  In this study, a variable order step size is presented for direct solving higher-order orbital equations.  The proposed algorithm calculates the integration coefficients only once at the beginning and, if necessary, once at the end.  The accuracy of the numerical approximation is validated with well-known orbital differential equations.  To reduce computational costs, we obtain the r

The current study provides a numerical method that is derived in a backward difference formulation for ordinary differential equations.  The proposed method employs a constant step size algorithm of order 12.  The backward difference formulation serves as a competitive algorithm for solving ordinary differential equations.  In the current study, the backward difference method is used to analyze the dynamics of capital stocks in terms of depreciation rate for the capital–labor ratio.  Results provided in this study will validate the accuracy of the backward difference algorithm hence proving