On memory-based higher-order iterative schemes for nonlinear equations: complex dynamics and applications

Purpose The limitations of traditional methods such as – slow convergence, smaller regions of convergence and the requirement of derivative computations near the desired root – have motivated the researchers to derive better alternate techniques. Design/methodology/approach This article introduces new memory-based two-step and three-step derivative-free iterative schemes for solving nonlinear equations. These methods utilize the approximations of self-accelerating parameters with the help of Hermite interpolating polynomials, enhancing the convergence order compared to existing methods without memory. Specifically, the convergence order of fifth and ninth order methods without memory is increased up to 5.70 and 11.68, respectively, by employing the proposed memory-based approaches. To demonstrate the efficacy of these methods, numerical experiments are carried out across various nonlinear equations, including engineering problems, confirming the theoretical advancements. Additionally, basins of attraction are presented to investigate the convergence behavior of the proposed methods compared to existing techniques in the complex plane. Findings The numerical examples illustrate that our iteration methods are superior. The theoretical results are also validated by the numerical and dynamical results. The proposed iteration schemes create portraits of basins of attraction faster with wider regions of convergence outperforming existing well-known methods. Research limitations/implications The proposed methods depend upon an initial guess chosen in the neighborhood of the desired root. Practical implications Our methods relax the constraint of f′(v) ≠ 0 near the root, overcoming a key limitation of Newton-type methods, thereby broadening their applicability and robustness in practical scenarios. The application based problems such as Neural Activation in Response to Complex Stimuli, Population Growth in Ecosystems with Periodic Resources, Perfectly Mixed Reactor, Planck's Radiation Law and Charge between two parallel plates are presented in the paper. Originality/value It is hereby certified that this work is original, is not currently submitted to other journals and that it will not be submitted to other journals during the reviewing process.