Home / Current Issue / Paper 1722139
Numerical Methods for Solving Nonlinear Equations in Engineering Problems
Subject area: Science,Engineering and Technology · Area of research: Numerical Methods in Engineering
Abstract
Nonlinear equations are fundamental in engineering and applied physics, appearing in structural mechanics, thermodynamics, electrical systems, fluid dynamics, and control theory. Due to the inherent nonlinearity of governing equations, analytical solutions are rarely obtainable, necessitating numerical approximation techniques. This paper presents a rigorous mathematical framework for analyzing iterative methods used to solve nonlinear equations of the form. We study five classical numerical methods: Bisection Method, Newton–Raphson Method, Secant Method, Fixed Point Iteration, and Regula Falsi Method. The focus is on convergence theory, stability conditions, and error propagation behavior. Using tools from real analysis, including the Intermediate Value Theorem, Taylor series expansion, and Banach contraction mapping theorem, we derive sufficient conditions for convergence and explicit error bounds. The study further investigates convergence rates, showing that Newton’s method achieves quadratic convergence while secant method achieves superlinear convergence. The results provide a mathematically rigorous foundation for selecting numerical methods in computational engineering problems where accuracy, efficiency, and stability are critical.
References
[1] Burden, R. L., & Faires, J. D. (2015). Numerical analysis. Cengage Learning.
[2] Atkinson, K. (2018). An introduction to numerical analysis. Wiley.
[3] Stoer, J., & Bulirsch, R. (2016). Introduction to numerical analysis. Springer.
[4] Chapra, S. C. (2017). Applied numerical methods with MATLAB. McGraw-Hill.
[5] Kelley, C. T. (2018). Iterative methods for nonlinear equations. SIAM.
[6] Ortega, J. M., & Rheinboldt, W. C. (2019). Iterative solution of nonlinear equations. SIAM.
[7] Saad, Y. (2016). Iterative methods for sparse linear systems. SIAM.
[8] Kincaid, D., & Cheney, W. (2017). Numerical analysis: Mathematics of scientific computing. AMS.
[9] Isaacson, E., & Keller, H. B. (2016). Analysis of numerical methods. Dover.
[10] Traub, J. F. (2015). Iterative methods for the solution of equations. Chelsea.
[11] Press, W. H. (2017). Numerical recipes in C. Cambridge University Press.
[12] Quarteroni, A. (2018). Numerical mathematics. Springer.
[13] Trefethen, L. N. (2019). Approximation theory and numerical analysis. Oxford University Press.
[14] Demmel, J. (2017). Applied numerical linear algebra. SIAM.
[15] Higham, N. J. (2016). Accuracy and stability of numerical algorithms. SIAM.
[16] Greenspan, D. (2015). Numerical analysis for applied sciences. CRC Press.
[17] Jain, M. K. (2018). Numerical methods for scientific and engineering computation. New Age International.
[18] Conte, S. D. (2016). Elementary numerical analysis. McGraw-Hill.
[19] Ralston, A. (2017). A first course in numerical analysis. Dover.
[20] Smith, G. D. (2018). Numerical solution of partial differential equations. Oxford.
How to cite this paper
@article{1722139,
author = {Lavkush Pandey},
title = {Numerical Methods for Solving Nonlinear Equations in Engineering Problems},
journal = {Iconic Research And Engineering Journals},
year = {2026},
volume = {10},
number = {1},
pages = {3964-3970},
issn = {2456-8880},
url = {https://www.irejournals.com/formatedpaper/1722139.pdf},
abstract = {Nonlinear equations are fundamental in engineering and applied physics, appearing in structural mechanics, thermodynamics, electrical systems, fluid dynamics, and control theory. Due to the inherent nonlinearity of governing equations, analytical solutions are rarely obtainable, necessitating numerical approximation techniques. This paper presents a rigorous mathematical framework for analyzing iterative methods used to solve nonlinear equations of the form. We study five classical numerical methods: Bisection Method, Newton–Raphson Method, Secant Method, Fixed Point Iteration, and Regula Falsi Method. The focus is on convergence theory, stability conditions, and error propagation behavior. Using tools from real analysis, including the Intermediate Value Theorem, Taylor series expansion, and Banach contraction mapping theorem, we derive sufficient conditions for convergence and explicit error bounds. The study further investigates convergence rates, showing that Newton’s method achieves quadratic convergence while secant method achieves superlinear convergence. The results provide a mathematically rigorous foundation for selecting numerical methods in computational engineering problems where accuracy, efficiency, and stability are critical.},
month = {July},
}