Fractional-order multi-parametric methods for nonlinear problems

In this research, a two-step bi- and tri-parametric iterative method is developed for solving nonlinear equations employing the Caputo fractional derivative. The theoretical convergence analysis demonstrates that the proposed schemes achieve an order of convergence of $ 2\tau+1 $, where $ \tau $ denotes the order of the Caputo fractional operator. In addition, fractal analysis is employed to identify effective initial values that enhance numerical performance. To compare the suggested scheme's efficacy and stability to existing approaches, several nonlinear engineering problems are examined. The numerical results demonstrate that, compared with existing methods, the proposed schemes achieve lower residual errors, higher convergence rates, reduced memory consumption, improved error profiles, and superior computational orders of convergence, thereby making them a more efficient and reliable alternative for solving problems in science and engineering.