A Class of Higher-Order Iterative Methods for Simultaneous Solutions of Nonlinear Equations with Applications to Biological Problems

A new class of high-order simultaneous iterative methods is constructed for the simultaneous computation of all simple roots of a nonlinear equation. The objective is to develop efficient iterative schemes with improved convergence characteristics and computational performance. The schemes are constructed by applying suitable weight functions and attain a convergence order of 2p+1, where p denotes the order of the predictor iterative method. Their performance is investigated through comprehensive numerical experiments on several nonlinear biological models, including the Population Model with Allee Effect and Density–Dependent Migration, the Epidemic Disease Model, and the Michaelis–Menten Enzyme Kinetics Model, to illustrate the accuracy, efficiency, and convergence behavior of the proposed schemes. Additionally, error graphs are plotted to compare the proposed methods with existing methods. The numerical results show that some of the proposed schemes achieve faster convergence and lower residual errors than existing schemes and have better computational performance. The results indicate that the proposed methods are competitive and reliable for solving nonlinear equations arising in biological and applied mathematical models.