Iterative Methods: Theory, Dynamics, Algorithms, and Fractal Structures

Dear Colleagues,

Iterative methods constitute a fundamental pillar in modern applied mathematics and computational science. They play a central role in solving nonlinear equations, optimization problems, and large-scale systems arising in science, engineering, and data-driven applications. Over the past decades, the development of iterative schemes has evolved significantly, encompassing not only convergence theory and efficiency analysis but also dynamical behavior and geometric structures associated with these methods.

This Topic aims to bring together recent advances in the theory, dynamics, and applications of iterative methods, with particular emphasis on their algorithmic development and the emergence of fractal structures in complex dynamical systems. The interplay between numerical analysis, dynamical systems, and fractal geometry has opened new perspectives for understanding convergence properties, stability regions, and global behavior of iterative processes.

This Topic also explores the integration of fractional-order operators and non-local derivatives within iterative frameworks to better account for memory effects and hereditary properties in complex systems. By examining how these advanced mathematical tools influence the topology of basins of attraction, researchers can develop more robust algorithms for modeling nonlinear phenomena in fields such as epidemiology and pharmacokinetic dynamics. The synergy between high-order iterative schemes and fractal–fractional calculus provides a sophisticated lens for evaluating the global stability and long-term behavior of multi-scale models.

Submissions addressing both theoretical contributions and practical implementations are welcome. Topics of interest include, but are not limited to:

• Design and analysis of iterative methods for nonlinear equations and systems;
• High-order and optimal iterative schemes;
• Convergence analysis and computational efficiency;
• Dynamical behavior of iterative processes;
• Basins of attraction and stability regions;
• Fractal structures associated with iterative methods;
• Iterative algorithms for optimization and inverse problems;
• Applications in engineering, physics, data science, and artificial intelligence;
• Numerical experiments and visualization techniques.

Prof. Dr. Alicia Cordero
Prof. Dr. Juan Ramón Torregrosa Sánchez
Prof. Dr. Younghee Geum
Prof. Dr. Xiaofeng Wang
Topic Editors