In this work, we derived a novel fractional numerical differentiation formula, named the new Katugampola Caputo \(GL_{1-2}\) formula for approximating the Katugampola fractional derivative of order \((0 < \alpha < 1)\) . The formula is constructed using quadratic interpolation of three points \(\left(t_{j-2}^{p},f\left(t_{j-2}^{p}\right)\right),\left(t_{j-1}^{p},f\left(t_{j-1}^{p}\right)\right)\) , and \(\left(t_{j}^{p},f(t_{j}^{p})\right)\) . In the small internal \(\left[t_{j-1}^p, t_{j}^{p}\right]\) ( \(j\geq2\) ) a linear interpolation method is employed. This approach can be viewed as an enhancement of the classical \(L_{1}\) formula, which relies on piecewise linear interpolation for \(f(t)\) . The new Katugampola Caputo \(GL_{1-2}\) formula improves computational efficiency and numerical accuracy over the classical \(L_{1}\) formula. This paper provides a theoretical proof of proposed truncation errors of the new Katugampola \(GL_{1}\) and new Katugampola Caputo \(GL_{1-2}\) formula. Finally, the new Katugampola Caputo \(GL_{1-2}\) formula is applied in various neural network systems and analyzes its dynamic behaviors.
