Numerical approach for approximating the Caputo fractional-order derivative operator

This work aims to propose a new simple robust power series formula with its truncation error to approximate the Caputo fractional-order operator D₀^y (t) of order m-1 m, where m. The proposed formula, which are derived with the help of the weighted mean value theorem, is expressed ultimately in terms of a fractional-order series and its reminder term. This formula is used successfully to provide approximate solutions of linear and nonlinear fractional-order differential equations in the form of series solution. It can be used to determine the analytic solutions of such equations in some cases. Some illustrative numerical examples, including some linear and nonlinear problems, are provided to validate the established formula.