A conformable fractional iterative fixed-point method for solving one-dimensional nonlinear equations

In this paper, we propose a new fixed-point iterative method for the approximate solution of one-dimensional nonlinear equations. Motivated by the limitations of the classical Newton-Raphson method, particularly its sensitivity to vanishing or near-zero derivatives, we employ the conformable derivative operator introduced by Anderson and Ulness (Adv. Dyn. Syst. Appl 10(2), 109–137, 2015), which provides a combination of the function and its classical derivative. Unlike fractional derivatives of nonlocal nature, this local conformable operator preserves differentiability while extending the class of functions it can handle. The proposed method defines a fixed-point iteration function that incorporates the conformable derivative and offers a robust alternative when classical or fractional Newton-type methods fail. We present the theoretical foundations of the method, its convergence analysis, experimental verification of the order of convergence, and a visual stability analysis, compared to the classical Newton-Raphson method. Applications to five benchmark problems demonstrate the effectiveness of the proposed method in overcoming classical difficulties, such as division by a near-zero value, division by zero, divergence at inflection points, iterations falling outside the domain of the function, and oscillations near a local minimum.