A Hybrid PSO–Fifth-Order Iterative Technique for Nonlinear Systems with Applications in Biological Models

Nonlinear systems of equations arise across engineering, physics, and biological modeling; however, classical Newton-type methods may fail when the initial approximation lies outside the convergence region of the NJN local solver. This work proposes a two-stage hybrid framework that couples Particle Swarm Optimization (PSO) for global exploration with the fifth-order Newton–Jarratt (NJN) iterative method for local refinement. The fifth-order convergence of the NJN phase, established through a complete Fréchet-derivative Taylor expansion with explicitly computed error constants, guarantees rapid local convergence once PSO delivers a sufficiently close starting point. The framework is validated on four test problems with increasing numbers of dimensions (n=2,5,20,40): a two-dimensional benchmark algebraic system, a five-dimensional metabolic network model for ethanol production in Saccharomyces cerevisiae, and two large-scale Hammerstein integral equation systems. Over 30 independent runs per method and under the tested conditions, PSO-NJN achieves 100% convergence with mean final residuals of order 10−14–10−16, while pure PSO fails completely on the high-dimensional Hammerstein cases (n=20,40) and achieves only 10% success on the metabolic model. These results confirm that combining global metaheuristic search with high-order local refinement yields a robust, scalable solver for complex biological and engineering nonlinear systems, though performance on problems with dense high-dimensional Jacobians may require further adaptation.