A Newton-Based Tuna Swarm Optimization Algorithm for Solving Nonlinear Problems with Application to Differential Equations

This paper presents two novel hybrid iterative schemes that combine Newton’s method and its variant with the Tuna Swarm Optimization (TSO) algorithm, aimed at solving complex nonlinear equations with enhanced accuracy and efficiency. Newton’s method is renowned for its rapid convergence in root-finding problems, and it is integrated with TSO, a recent swarm intelligence algorithm that surpasses the complex behavior of tuna fish in order to optimize the search for superior solutions. These hybrid methods are reliable and efficient for solving challenging mathematical and applied science problems. Several numerical experiments and applications involving ordinary differential equations have been carried out to demonstrate the superiority of the proposed hybrid methods in terms of convergence rate, accuracy, and robustness compared to traditional optimization and iterative methods. The stability and efficiency of the proposed methods have also been verified. The results indicate that the hybrid approaches outperform traditional methods, making them a promising tool for solving a wide range of mathematical and engineering problems.