Fractal analysis of fractional numerical scheme for resolving nonlinear engineering problems

Abstract

Fractional calculus has evolved as an effective mathematical tool for simulating complex dynamical systems in science and engineering. This study develops an enhanced fractional iterative technique for solving nonlinear equations that incorporates fractional calculus to improve accuracy, stability and computational efficiency. Traditional approaches frequently struggle with high computing costs and slow convergence, but the suggested method effectively balances accuracy and efficiency by adding Caputo fractional-order derivatives. Analysis of convergence reveals that the order of convergence of the suggested family of approaches is 2ß + 1. Fractal analysis of the suggested numerical methods for solving nonlinear equations reveals that they outperform existing classical approaches in terms of convergence and stability. A comparison with existing methods shows that the newly developed schemes perform better in terms of residual error reduction, CPU time and convergence rate. Since increasing the fractional parameter ß from 0 to 1 greatly increases the method’s efficiency — near-optimal performance is seen around ß ≈ 1 — it plays a critical role. The novel strategy generates fractals with greater elapsed time and consistency than existing strategies, according to numerical studies on engineering applications.