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SOLVING NON-LINEAR EQUATIONS BY FIXED POINT ITERATION METHOD AND ITS ACCELERATING APPROACH

ABSTRACT
Due to the inherent complexity of non-linear equations, finding their analytical solutions is often challenging, leading to a reliance on numerical solutions. This article examines key methods for solving these solutions, beginning with an introduction to the fixed point iteration method. Through examples, we explore the nuances of this technique in solving non-linear equations. We then examine Aitken's accelerated method and Steffensen's iteration approach, and discuss their convergence properties. By applying these methods to a consistent example, we compare their effectiveness in achieving similar error accuracy. The analysis shows that both the Aitken method and the Steffensen method require significant-ly fewer iterations than the fixed point iteration approach for solving non-linear equations, with the Steffensen method showing superior performance over the Aitken method. Finally, we have applied the Steffensen method to the flash evaporation problem, and the computational results are highly efficient.
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PAPER SUBMITTED: 2023-12-16
PAPER REVISED: 2024-07-07
PAPER ACCEPTED: 2024-07-07
PUBLISHED ONLINE: 2025-07-06
DOI REFERENCE: https://doi.org/10.2298/TSCI2503031C
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2025, VOLUME 29, ISSUE Issue 3, PAGES [2031 - 2039]
REFERENCES
  1. He, J.-H., Linearization and Correction Method for Nonlinear Problems, Applied Mathematics and Mechanics, 23 (2002), 3, pp. 241-248
  2. He, J.-H., Application of Homotopy Perturbation Method to Nonlinear Wave Equations, Chaos, Solitons & Fractals, 26 (2005), 3, pp. 695-700
  3. He, J.-H., et al., Good Initial Guess for Approximating Nonlinear Oscillators by the Homotopy Perturbation Method, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 1, pp. 21-29
  4. He, J.-H., Wu, X. H., Construction of Solitary Solution and Compacton-Like Solution by Variational Iteration Method, Chaos, Solitons & Fractals, 29 (2006), 1, pp. 108-113
  5. Anjum, N., et al., Free Vibration of a Tapered Beam by the Aboodh Transform-Based Variational Iteration Method, Journal of Computational Applied Mechanics, 55 (2024), 3, 440-450
  6. He, J.-H., Variational Iteration Method - Some Recent Results and New Interpretations, Journal of Computational Applied Mathematics and Mechanics, 207 (2007), 1, pp. 3-17
  7. Elsayed, E. M., et al., The Analysis of the Fractional-Order Navier-Stokes Equations by a Novel Approach, Journal of Function Spaces, 2022 (2022), 8979447
  8. Boykov, I., et al., New Iterative Method for Solving Linear and Nonlinear Hypersingular Integral Equations, Applied Numerical Mathematics, 127 (2018), May, pp. 280-305
  9. Chun, C., A New Iterative Method for Solving Nonlinear Equations, Applied Mathematics Computation, 178 (2006), 2, pp. 415-422
  10. Darvishi, M. T., Barati, A., A Third-Order Newton-Type Method to Solve Systems of Nonlinear Equations, Applied Mathematics Computation, 187 (2007), 2, pp. 630-635
  11. Lotfi, T., Tavakoli, E., On a New Efficient Steffensen-Like Iterative Class by Applying a Suitable Self-Accelerator Parameter, The Scientific World Journal, 2014 (2014), 4, pp. 769-758
  12. Altaee, A. A. H., et al., A New Approach to Find Roots of Nonlinear Equations by Hybrid Algorithm to Bisection and Newton-Raphson Algorithms, Iraq. J. Inform. Technol, 7 (2015), May, pp. 75-82
  13. Sihwail, R., et al., A Hybrid Approach for Solving Systems of Nonlinear Equations Using Harris Hawks Optimization and Newton's Method, IEEE Access, 9 (2021), July, pp. 95791-95807
  14. Li, Q. Y., et al., Numerical Analysis, Tsinghua University Press, Beijing, 2011
  15. Fatoorehchi, H., et al., A New Parametric Algorithm for Isothermal Flash Calculations by the Adomian Decomposition of Michaelis-Menten Type Nonlinearities, Fluid Phase Equilibria, 395 (2015), June, pp. 44-50
  16. Liu, Y. Q., He, J.-H., On Relationship between Two Ancient Chinese Algorithms and Their Application to Flash Evaporation, Results in Physics, 7 (2017), 12, pp. 320-322
  17. He, J.-H., et al., Fast Identification of the Pull-In Voltage of a Nano/Micro-Electromechanical System, Journal of Low Frequency Noise Vibration and Active Control, 41 (2022), 2, pp.566-571
  18. He,J.-H., et al., Piezoelectric Biosensor Based on Ultrasensitive MEMS system, Sensors and Actuators A: Physical, 376 (2024), 115664

2025 Society of Thermal Engineers of Serbia. Published by the VinĨa Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence