A class of high-order fractional parallel iterative methods for nonlinear engineering problems: convergence, stability, and neural network-based acceleration

Abstract

Conventional analytical techniques often fail to yield efficient or closed-form solutions for nonlinear fractional problems due to their inherent nonlocality and complexity. This study introduces a new class of high-order parallel iterative methods for solving nonlinear equations, with a focus on fractional-order formulations. We first develop a sixth-order single-root finding scheme, which is then extended to a fractional-order method with convergence order 5𝜎 + 1, and further generalized into a parallel scheme achieving order 20𝜎 + 8. To improve computational performance, we propose a hybrid neural network-based parallel scheme, in which optimal parameter values are identified through dynamical systems analysis. The resulting methods exhibit strong stability, accuracy, and efficiency, and are robust with respect to both accurate and perturbed initial approximations. Comparative experiments on real-world engineering problems demonstrate that the proposed fractional parallel schemes consistently outperform existing methods in terms of residual error, convergence rate, and computational cost.