In this paper, we establish a novel fractional numerical modification of the 5-point classical central formula; called the modified 5-point fractional formula for approximating the first fractional-order derivative in the sense of the Caputo operator. Accordingly, we then introduce a new methodology for Richardson extrapolation depending on the fractional central formula in order to obtain a high accuracy for the gained approximations. We compare the efficiency of the proposed methods by using tables and figures to show their reliability.

A study of a class of nonlinear differential equations involving the $\phi$-Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results.