Construction and applications of iterative methods for finding approximate solutions of nonlinear equations having unknown zeros of multiplicity with fractal geometry and dynamical behavior.

In this study, several new iterative schemes are developed to compute the multiple zeros of nonlinear equations. The construction of these methods is based on the variational iteration approach which provides a systematic framework for formulating efficient and flexible algorithms. The proposed schemes generalize and encompass well-known classical methods such as Newton's and Halley's methods, along with their modified versions as special cases. This generality enhances their adaptability to a broader class of nonlinear problems involving both known and unknown multiplicities. To assess the effectiveness of the proposed iterative schemes, extensive numerical experiments are conducted, comparing their convergence speed and accuracy with existing methods. The results demonstrate that the newly developed methods exhibit superior performance in terms of stability, precision, and computational efficiency. Furthermore, to visualize and analyse the global convergence behaviour fractal basin plots are presented. These fractals illustrate the basins of attraction in the complex plane, providing deeper insight into the dynamical behaviour, convergence regions and boundary structures associated with each iterative process.•Developing efficient and flexible iterative methods using the variational iteration method.•Generalizing classical methods to tackle nonlinear problems with known and unknown multiplicity.•Validating performance of various methods through detailed numerical experiments and fractal basin plots.