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.