Numerical Simulation of Nonlinear Equations by Modified Bisection and Regula Falsi Method - Inderjeet; Rashmi Bhardwaj (Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences, 2025)
The study of nonlinear equations and their effective numerical solutions is crucial to mathematical research because nonlinear models are prevalent in nature and require thorough analysis and solution. Many methodologies have been developed to obtain the roots of nonlinear equations, which have significant applications in several areas, especially engineering. However, all of these methods have certain challenges. The development of efficient and effective iterative methods is, therefore, very important and can positively impact the task of finding numerical solutions to many real-world problems. This paper presents a thorough analysis of a numerical approach for solving nonlinear equations using a recently proposed technique, which is a modification of the Regula-Falsi and Bisection numerical methods. The purpose of this work is to provide a novel and effective approach to solving nonlinear equations. The iterative technique for solving nonlinear equations, which has been examined in many scientific and technical domains, is based on the conventional Bisection and Regula-Falsi methods. The proposed approach for finding roots of nonlinear equations achieves second-order convergence. The performance of the newly developed technique was compared with conventional Bisection, Regula-Falsi, Steffensen, and Newton-Raphson methods, and its convergence was validated using several benchmark problems with different iterations. The results showed that, in terms of iterations, the newly developed method performed better than the traditional Bisection, Regula-Falsi, Steffensen, and Newton-Raphson approaches. This supports the credibility of the recently developed method and offers promise for future studies aimed at further refinement. Excel and MATLAB software were used for obtaining results and graphical representations. Besides this, the newly developed technique also has certain limitations. For instance, it cannot cover all possible types of nonlinear equations. Further testing on a broader range of functions, particularly those arising from specific scientific and engineering applications, would be valuable. Additionally, our current study focuses on one-dimensional root finding. Extending the approach to systems of nonlinear equations is an important direction for future research.