SINTA

0.0

Impact

Scholar

9

H-Index

Interval: Indonesian Journal of Mathematical Education

an Open Access Journal

Modification of the Fourth Order Runge Kutta Method Based on the Contra Harmonic Average

Peshawa Mohammed Khudhur
University of Salahaddin
Iraq
peshawa.khu71@gmail.com
Dara Maghdid
Soran University ROR
Iraq
Share

  • Purpose of the Study
    This study aims to modify the fourth-order Runge-Kutta method based on the contra harmonic mean. The research discusses the theoretical modification of the fourth-order Runge-Kutta method.

    Methodology
    The research was conducted using a literature study method. The study begins by introducing the general form of the Runge-Kutta method up to the nth order. This general form is then specialized to the fourth order. Additionally, the concept of the contra harmonic mean is introduced. After obtaining the general form of the fourth-order Runge-Kutta method and the contra harmonic mean, these two general forms are modified to derive a new formula.

    Main Findings
    Based on the results, the modified fourth-order Runge-Kutta method has the following equation form:
    yi+1 = yi + (h/4) * [(k1^2 + k2^2) / (k1 + k2) + 2 * (k2^2 + k3^2) / (k2 + k3) + (k3^2 + k4^2) / (k3 + k4)],
    with an error of order O(h^5). Numerical simulations demonstrate that the modified method provides better results compared to the original fourth-order Runge-Kutta method.

    Novelty/Originality of this Study
    The numerical simulations using the RKKCM method show improved accuracy compared to the unmodified fourth-order Runge-Kutta method, highlighting the innovation and contribution of this study.

  • How to cite

    Modification of the Fourth Order Runge Kutta Method Based on the Contra Harmonic Average. (2025). Interval: Indonesian Journal of Mathematical Education, 3(1), 69-81. https://doi.org/10.37251/ijome.v3i1.1588

    More Citation Formats

  • 535
    Abstract views
    247
    Downloads

    Metrics — Badges

    1. K. Velten, D. M. Schmidt, and K. Kahlen, Mathematical Modeling and Simulation: Introduction for Scientists and Engineers. Hoboken, NJ: John Wiley & Sons, 2024. DOI: https://doi.org/10.1002/9783527849604
    2. E. Lozada, C. Guerrero-Ortiz, A. Coronel, and R. Medina, “Classroom methodologies for teaching and learning ordinary differential equations: A systemic literature review and bibliometric analysis,” Mathematics, vol. 9, no. 7, p. 745, 2021, doi: 10.3390/math9070745. DOI: https://doi.org/10.3390/math9070745
    3. S. Elango, L. Govindarao, J. Mohapatra, R. Vadivel, and N. T. Hu, “Numerical analysis for second order differential equation of reaction-diffusion problems in viscoelasticity,” Alexandria Engineering Journal, vol. 92, pp. 92–101, 2024, doi: 10.1016/j.aej.2024.02.046. DOI: https://doi.org/10.1016/j.aej.2024.02.046
    4. A. A. Khan, M. Ahsan, I. Ahmad, and M. Alwuthaynani, “Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach,” The European Physical Journal Special Topics, pp. 1–14, 2024, doi: 10.1140/epjs/s11734-024-01254-8. DOI: https://doi.org/10.1140/epjs/s11734-024-01254-8
    5. R. Sharma and O. P. Yadav, “The Evolution of Finite Element Approaches in Reaction-Diffusion Modeling,” Archives of Computational Methods in Engineering, pp. 1–22, 2025, doi: 10.1007/s11831-025-10222-x. DOI: https://doi.org/10.1007/s11831-025-10222-x
    6. M. Genčev and D. Šalounová, “First-and second-order linear difference equations with constant coefficients: suggestions for making the theory more accessible,” International Journal of Mathematical Education in Science and Technology, vol. 54, no. 7, pp. 1349–1372, 2023, doi: 10.1080/0020739X.2022.2140081. DOI: https://doi.org/10.1080/0020739X.2022.2140081
    7. G. Policastro, V. Luongo, L. Frunzo, and M. Fabbricino, “A comprehensive review of mathematical models of photo fermentation,” Critical Reviews in Biotechnology, vol. 41, no. 4, pp. 628–648, 2021, doi: 10.1080/07388551.2021.1873241. DOI: https://doi.org/10.1080/07388551.2021.1873241
    8. G. Shi et al., “A new passive non-ideal floating memristor emulator circuit,” AEU - International Journal of Electronics and Communications, vol. 170, p. 154823, 2023, doi: 10.1016/j.aeue.2023.154823. DOI: https://doi.org/10.1016/j.aeue.2023.154823
    9. H. Wang, M. Zhu, R. Fan, and Y. Li, “Parametric model‐based deinterleaving of radar signals with non‐ideal observations via maximum likelihood solution,” IET Radar, Sonar & Navigation, vol. 16, no. 8, pp. 1253–1268, 2022, doi: 10.1049/rsn2.12258. DOI: https://doi.org/10.1049/rsn2.12258
    10. S. Banerjee, Mathematical Modeling: Models, Analysis and Applications. Boca Raton, FL: Chapman and Hall/CRC, 2021, doi: 10.1201/9781351022941. DOI: https://doi.org/10.1201/9781351022941
    11. C. Fu, J. J. Sinou, W. Zhu, K. Lu, and Y. Yang, “A state-of-the-art review on uncertainty analysis of rotor systems,” Mechanical Systems and Signal Processing, vol. 183, p. 109619, 2023, doi: 10.1016/j.ymssp.2022.109619. DOI: https://doi.org/10.1016/j.ymssp.2022.109619
    12. G. Noorsumar, S. Rogovchenko, K. G. Robbersmyr, and D. Vysochinskiy, “Mathematical models for assessment of vehicle crashworthiness: a review,” International Journal of Crashworthiness, vol. 27, no. 5, pp. 1545–1559, 2022, doi: 10.1080/13588265.2021.1929760. DOI: https://doi.org/10.1080/13588265.2021.1929760
    13. X. Qin, Z. Jiang, J. Yu, L. Huang, and C. Yan, “Strong stability-preserving three-derivative Runge–Kutta methods,” Computational and Applied Mathematics, vol. 42, no. 4, p. 171, 2023, doi: 10.1007/s40314-023-02285-y. DOI: https://doi.org/10.1007/s40314-023-02285-y
    14. M. Saqib, M. Akram, S. Bashir, and T. Allahviranloo, “A Runge–Kutta numerical method to approximate the solution of bipolar fuzzy initial value problems,” Computational and Applied Mathematics, vol. 40, no. 4, p. 151, 2021, doi: 10.1007/s40314-021-01535-1. DOI: https://doi.org/10.1007/s40314-021-01535-1
    15. F. Yılmaz, H. Öz Bakan, and G. W. Weber, “Weak second-order conditions of Runge–Kutta method for stochastic optimal control problems,” Journal of Optimization Theory and Applications, vol. 202, no. 1, pp. 497–517, 2024, doi: 10.1007/s10957-023-02324-y. DOI: https://doi.org/10.1007/s10957-023-02324-y
    16. B. B. Sanugi and D. J. Evans, "A new fourth order Runge-Kutta formula based on the harmonic mean," International Journal of Computer Mathematics, vol. 50, no. 1–2, pp. 113–118, 1994, doi: 10.1080/00207169408804246 DOI: https://doi.org/10.1080/00207169408804246
    17. D. J. Evans, "A new 4th order Runge-Kutta method for initial value problems with error control," International Journal of Computer Mathematics, vol. 39, no. 3–4, pp. 217–227, 1991, doi: 10.1080/00207169108803994. DOI: https://doi.org/10.1080/00207169108803994
    18. O. Y. Ababneh and R. Rozita, "New third order Runge-Kutta based on contraharmonic mean for stiff problems," Applied Mathematical Sciences, vol. 3, no. 8, pp. 365–376, 2009.
    19. G. Arora, V. Joshi, and I. S. Garki, "Developments in Runge–Kutta method to solve ordinary differential equations," in Recent Advances in Mathematics for Engineering, CRC Press, 2020, pp. 193–202. DOI: https://doi.org/10.1201/9780429200304-9
    20. Y. Workineh, H. Mekonnen, and B. Belew, "Numerical methods for solving second-order initial value problems of ordinary differential equations with Euler and Runge-Kutta fourth-order methods," Frontiers in Applied Mathematics and Statistics, vol. 10, 2024, doi: 10.3389/fams.2024.1360628. DOI: https://doi.org/10.3389/fams.2024.1360628
    21. Z. Kalogiratou, T. Monovasilis, G. Psihoyios, and T. E. Simos, "Runge–Kutta type methods with special properties for the numerical integration of ordinary differential equations," Physics Reports, vol. 536, no. 3, pp. 75–146, 2014, doi: 10.1016/j.physrep.2013.11.003. DOI: https://doi.org/10.1016/j.physrep.2013.11.003
    22. I. Godswill and G. Isobeye, "Comparative convergence analysis of Runge-Kutta fourth order and Runge-Kutta-Fehlberg methods implementation in Matlab and Python applied to a series RLC circuit," International Journal of Innovative Science and Research Technology, vol. 10, no. 3, pp. 2957–2975, 2025, doi: 10.38124/ijisrt/25mar599. DOI: https://doi.org/10.38124/ijisrt/25mar599
    23. T. Lai, T. H. Yi, H. N. Li, and X. Fu, "An explicit fourth-order Runge–Kutta method for dynamic force identification," International Journal of Structural Stability and Dynamics, vol. 17, no. 10, p. 1750120, 2017, doi: 10.1142/S0219455417501206. DOI: https://doi.org/10.1142/S0219455417501206
    24. W. Zhai, D. Tao, and Y. Bao, "Parameter estimation and modeling of nonlinear dynamical systems based on Runge–Kutta physics-informed neural network," Nonlinear Dynamics, vol. 111, no. 22, pp. 21117–21130, 2023, doi: 10.1007/s11071-023-08933-6. DOI: https://doi.org/10.1007/s11071-023-08933-6
    25. R. C. Rockne et al., "The 2019 mathematical oncology roadmap," Physical Biology, vol. 16, no. 4, p. 041005, 2019, doi: 10.1088/1478-3975/ab1a09. DOI: https://doi.org/10.1088/1478-3975/ab1a09
    26. Y. Liu, R. Wu, and A. Yang, "Research on medical problems based on mathematical models," Mathematics, vol. 11, no. 13, p. 2842, 2023, doi: 10.3390/math11132842. DOI: https://doi.org/10.3390/math11132842
    27. P. Luo et al., "Urban flood numerical simulation: Research, methods and future perspectives," Environmental Modelling & Software, vol. 156, p. 105478, 2022, doi: 10.1016/j.envsoft.2022.105478. DOI: https://doi.org/10.1016/j.envsoft.2022.105478
    28. E. Elpianora, M. Berou, X. Kong, K. Hun, and E. Azadegan, “Fourth order Runge-Kutta and gill methods in numerical analysis of Predator-Prey models,” Interval: Indonesian Journal of Mathematical Education, vol. 2, no. 2, pp. 164–177, 2024, doi: 10.37251/ijome.v2i2.1366. DOI: https://doi.org/10.37251/ijome.v2i2.1366
    29. H. Ranocha, L. Dalcin, M. Parsani, and D. I. Ketcheson, “Optimized Runge-Kutta methods with automatic step size control for compressible computational fluid dynamics,” Communications on Applied Mathematics and Computation, vol. 4, no. 4, pp. 1191–1228, 2022, doi: 10.1007/s42967-021-00159-w. DOI: https://doi.org/10.1007/s42967-021-00159-w
    30. M. Zhang, H. Chen, A. A. Heidari, Z. Cai, N. O. Aljehane, and R. F. Mansour, “OCRUN: An oppositional Runge Kutta optimizer with cuckoo search for global optimization and feature selection,” Applied Soft Computing, vol. 146, p. 110664, 2023, doi: 10.1016/j.asoc.2023.110664. DOI: https://doi.org/10.1016/j.asoc.2023.110664
    31. V. Chauhan and P. K. Srivastava, “Tetra geometric mean Runge-Kutta analysis of bioeconomic fisheries model,” Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science, pp. 169–180, 2023, doi: 10.31926/but.mif.2023.3.65.2.15. DOI: https://doi.org/10.31926/but.mif.2023.3.65.2.15
    32. A. Soomro, S. Qureshi, and A. A. Shaikh, “A new nonlinear hybrid technique with fixed and adaptive step-size approaches,” Sigma Journal of Engineering and Natural Sciences, vol. 40, no. 1, pp. 162–178, 2022, doi: 10.14744/sigma.2022.00013. DOI: https://doi.org/10.14744/sigma.2022.00013
    33. N. S. Zulkifli et al., “Centroidal-polygon: a new modified Euler to improve speed of resistor-inductor circuit equation,” Indonesian Journal of Electrical Engineering and Computer Science, vol. 24, no. 3, pp. 1399–1404, 2021, doi: 10.11591/ijeecs.v24.i3.pp1399-1404. DOI: https://doi.org/10.11591/ijeecs.v24.i3.pp1399-1404
    34. E. Setiawan and M. Imran, "A weighted fourth order Runge-Kutta method based on contra-harmonic mean," Journal of Mathematics and Computer Science, vol. 6, no. 6, pp. 989–1001, 2016.
    35. G. E. Masri, A. Ali, W. H. Abuwatfa, M. Mortula, and G. A. Husseini, "A comparative analysis of numerical methods for solving the leaky integrate and fire neuron model," Mathematics, vol. 11, no. 3, p. 714, 2023, doi: 10.3390/math11030714. DOI: https://doi.org/10.3390/math11030714
    36. S. Suvitha and R. G. Sharmila, "Solving larger order uncertain differential equations by newly attained fourth order RK method," Indian Journal of Science and Technology, vol. 17, no. 33, pp. 3447–3455, 2024, doi: 10.17485/IJST/v17i33.1498. DOI: https://doi.org/10.17485/IJST/v17i33.1498
    37. C. Chicone, Ordinary Differential Equations with Applications. Springer, 2006, doi: 10.1007/0-387-35794-7. DOI: https://doi.org/10.1007/0-387-35794-7
    38. A. D. Polyanin, V. G. Sorokin, and A. I. Zhurov, Delay Ordinary and Partial Differential Equations. Chapman and Hall/CRC, 2023, doi: 10.1201/9781003042310. DOI: https://doi.org/10.1201/9781003042310
    39. B. Moseley, A. Markham, and T. Nissen-Meyer, "Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations," Advances in Computational Mathematics, vol. 49, no. 4, p. 62, 2023, doi: 10.1007/s10444-023-10065-9. DOI: https://doi.org/10.1007/s10444-023-10065-9
    40. P. Hehenberger, M. Habib, and D. Bradley, EcoMechatronics. Springer Nature, 2023, doi: 10.1007/978-3-031-07555-1. DOI: https://doi.org/10.1007/978-3-031-07555-1
    41. A. Alongi, A. Angelotti, and L. Mazzarella, "A numerical model to simulate the dynamic performance of breathing walls," Journal of Building Performance Simulation, vol. 14, no. 2, pp. 155–180, 2021, doi: 10.1080/19401493.2020.1868578 DOI: https://doi.org/10.1080/19401493.2020.1868578