Aanchal Chandel

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.

Non-linear systems of equations play a fundamental role in various engineering and data science models, where accurate solutions are essential for both theoretical research and practical applications. However, solving such systems is highly challenging due to their inherent non-linearity and computational complexity. This study proposes a novel hybrid iterative method with fourth-order convergence. The foundation of the proposed scheme combines the Walrus Optimization Algorithm and a fourth-order iterative technique. The objective of this hybrid approach is to enhance global search capability, reduce the likelihood of convergence to local optima, accelerate convergence, and improve solution accuracy in solving non-linear problems. The effectiveness of the proposed method is checked on standard benchmark problems and two real-world case studies, hydrocarbon combustion and electronic circuit design, and one non-linear boundary value problem. In addition, a comparative analysis is conducted with several well-established optimization algorithms, based on the optimal solution, average fitness value, and convergence rate. Furthermore, the proposed scheme effectively addresses key limitations of traditional iterative techniques, such as sensitivity to initial point selection, divergence issues, and premature convergence. These findings demonstrate that the proposed hybrid method is a robust and efficient approach for solving non-linear problems.