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International Journal of Neutrosophic Science

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International Journal of Neutrosophic Science

Volume 25 , Issue 3 , PP: 76-91, 2025 | Cite this article as | XML | Html | PDF | Full Length Article

Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem

Iqbal M. Batiha 1 * , Mohammad W. Alomari 2 , Nidal Anakira 3 , Saad Meqdad 4 , Iqbal H. Jebril 5 * , Shaher Momani 6

  • 1 Department of Mathematics, Al Zaytoonah University, Amman 11733, Jordan - (i.batiha@zuj.edu.jo)
  • 2 Department of Mathematics, Jadara University, P.O. Box 733, Irbid, P.C. 21110, Jordan - (mwomath@gmail.com)
  • 3 Faculty of Education and Arts, Sohar University, Sohar 3111, Oman; Jadara Research Center, Jadara University, Irbid 21110, Jordan - (nanakira@su.edu.om)
  • 4 Applied science private university, Amman 11937, Jordan - (s meqdad@asu.edu.jo)
  • 5 Department of Mathematics, Al Zaytoonah University, Amman 11733, Jordan - (i.jebril@zuj.edu.jo)
  • 6 Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman 346, UAE; Department of Mathematics, The University of Jordan, Amman, Jordan - (s.momani@ju.edu.jo)

    • Doi: https://doi.org/10.54216/IJNS.250308

    Received: February 13, 2024 Revised: May 14, 2024 Accepted: September 18, 2024
    Abstract

    This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by the Euler-Maclaurin formula. This results in a higher-order implicit corrected method that outperforms the Runge-Kutta method in terms of accuracy. We derive an error bound for the Euler-Maclaurin higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Runge-Kutta method. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficiency of our proposed method over the traditional well-known methods. In conclusion, the proposed method consistently outperforms the Runge-Kutta method experimentally in all practical problems.

    Keywords :

    Euler-Maclaurin formula , Runge-Kutta method , ODE , Darboux&rsquo , s formula , Approximations

    References

    [1] M. Abualhomos, Numerical ways for solving fuzzy differential equations, Int. J. Appl. Engin. Res., vol. 13, pp. 4610-4613, 2018.

    [2] S. Al-Ahmad, N. R. Anakira, M. Mamat, I. M. Suliman, R. AlAhmad, Modified differential transformation scheme for solving classes of non-linear differential equations, TWMS J. Appl. Eng. Math., vol. 12, no. 1, pp. 107-119, 2022.

    [3] A. Burqan, M. Khandaqji, Z. Al-Zhour, A. El-Ajou, T. Alrahamneh, Analytical approximate solutions of Caputo fractional KdV-Burgers equations using Laplace residual power series technique, Journal of Applied Mathematics, vol. 2024, no. 1, 7835548, 2024.

    [4] F. Awawdeh, M. Khandaqji, Z. Mustafa, M. Younis, A new approach for the solution of the electrostatic potential differential equations, Mathematical Problems in Engineering, vol. 2009, 980987, 2009.

    [5] A. Jameel, N. R. Anakira, A. K. Alomari, N. H. Man, Solution and analysis of the fuzzy Volterra integral equations via homotopy analysis method, Computer Modeling in Engineering & Sciences, vol. 127, no. 3, pp. 875-99, 2021.

    [6] A. F. Jameel, N. R. Anakira, A. K. Alomari, D. M. Alsharo, A. Saaban, New semi-analytical method for solving two point nth order fuzzy boundary value problem, International Journal of Mathematical Modelling and Numerical Optimisation, vol. 9, no. 1, pp. 12-31, 2019.

    [7] S. A. Altaie, N. Anakira, A. Jameel, O. Ababneh, A. Qazza, A. K. Alomari, Homotopy analysis method analytical scheme for developing a solution to partial differential equations in fuzzy environment, Fractal Fract., vol. 6, 419, 2022.

    [8] I. M. Batiha, A. Bataihah, A. Al-Nana, S. Alshorm, I. H. Jebril, A. Zraiqat, A numerical scheme for dealing with fractional initial value problem, International Journal of Innovative Computing, Information and Control, vol. 19, no. 3, pp. 763-774, 2023.

    [9] T. Hamadneh, A. Abbes, I. Abu Falahah, Y. A. AL-Khassawneh, A. S. Heilat, A. Al-Husban, A. Ouannas, Complexity and chaos analysis for two-dimensional discrete-time predator–prey Leslie–Gower model with fractional orders, Axioms, vol. 12, no. 6, 561, 2023.

    [10] I. M. Batiha, A. A. Abubaker, I. H. Jebril, S. B. Al-Shaikh, K. Matarneh, New algorithms for dealing with fractional initial value problems, Axioms, vol. 12, no. 5, 488, 2023.

    [11] A. A. Al-Nana, I. M. Batiha, S. Momani, A numerical approach for dealing with fractional boundary value problems, Mathematics, vol. 11, no. 19, 4082, 2023.

    [12] I. M. Batiha, I. H. Jebril, A. Abdelnebi, Z. Dahmani, S. Alkhazaleh, N. Anakira, A new fractional representation of the higher order Taylor scheme, Computational and Mathematical Methods, vol. 2024, 2849717, 2024.

    [13] M. W. Alomari, I. M. Batiha, S. Momani, New higher-order implict method for approximating solutions of the initial value problems, Journal of Applied Mathematics and Computing, vol. 70, pp. 3369–3393, 2024.

    [14] A. Zraiqat, I. M. Batiha, S. Alshorm, Numerical comparisons between some recent modifications of fractional Euler methods, WSEAS Transactions on Mathematics, vol. 23, no. 1, pp. 529-535, 2024.

    [15] A. Borri, F. Carravetta, P. Palumbo, Quadratized Taylor series methods for ODE numerical integration, Applied Mathematics and Computation, vol. 458, 128237, 2023.

    [16] A. Baeza, S. Boscarino, P. Mulet, G. Russo, D. Zor´ıo, Reprint of: approximate Taylor methods for ODEs, Computers & Fluids, vol. 169, pp. 87-97, 2018.

    [17] H. Carrillo, E. Macca, C. Par´es, G. Russo, D. Zor´ıo, An order-adaptive compact approximation Taylor method for systems of conservation laws, Journal of Computational Physics, vol. 438, no. 1, 110358, 2021.

    [18] K. L. Narayanan, J. Kavitha, R. U. Rani, M. E. Lyons, L. Rajendran, Mathematical modelling of Amperometric Glucose biosensor based on immobilized Enzymes: new approach of Taylors series method, International Journal of Electrochemical Science, vol. 17, no. 10, 221064, 2022.

    [19] K. Wang, Q. Wang, Taylor collocation method and convergence analysis for the Volterra–Fredholm integral equations, Journal of Computational and Applied Mathematics, vol. 260, pp. 294-300, 2014.

    [20] L. Euler, De summis serierum reciprocarum, Translated to English On the sums of series of reciprocals, Commentarii academiae scientiarum Petropolitanae, vol. 7, pp. 123-134, 1744.

    [21] C. Maclaurin, A Treatise of Fluxionsin Two Books, Ruddimans, Edinburgh, 1742.

    [22] T. M. Apostol, An elementary view of Euler’s summation formula, American Mathematical Monthly, vol. 106, no. 5, pp. 409-418, 1999.

    [23] E. T. Whittaker, G. N. Watson, A Course of Modern Analysis, Cambridge University Press, London, 1940.

    [24] Z. X.Wang, D. R. Guo, Special functions, Translated from the Chinese by Guo and X. J. Xia.World Scientfic Publishing Co., Inc., Teaneck, NJ, 1989; Available online at https://doi.org/10.1142/ 0653.

    [25] R. L. Burden, J. D. Faires, Numerical Analysis, Brooks/Cole, Cengage Learning, Boston, USA, 2011.

    [26] F. W. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, NIST Handbook of Mathematical Functions, Cambridge University Press, London, 2010.

    Cite This Article As :
    M., Iqbal. , W., Mohammad. , Anakira, Nidal. , Meqdad, Saad. , H., Iqbal. , Momani, Shaher. Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem. International Journal of Neutrosophic Science, vol. , no. , 2025, pp. 76-91. DOI: https://doi.org/10.54216/IJNS.250308
    M., I. W., M. Anakira, N. Meqdad, S. H., I. Momani, S. (2025). Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem. International Journal of Neutrosophic Science, (), 76-91. DOI: https://doi.org/10.54216/IJNS.250308
    M., Iqbal. W., Mohammad. Anakira, Nidal. Meqdad, Saad. H., Iqbal. Momani, Shaher. Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem. International Journal of Neutrosophic Science , no. (2025): 76-91. DOI: https://doi.org/10.54216/IJNS.250308
    M., I. , W., M. , Anakira, N. , Meqdad, S. , H., I. , Momani, S. (2025) . Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem. International Journal of Neutrosophic Science , () , 76-91 . DOI: https://doi.org/10.54216/IJNS.250308
    M. I. , W. M. , Anakira N. , Meqdad S. , H. I. , Momani S. [2025]. Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem. International Journal of Neutrosophic Science. (): 76-91. DOI: https://doi.org/10.54216/IJNS.250308
    M., I. W., M. Anakira, N. Meqdad, S. H., I. Momani, S. "Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem," International Journal of Neutrosophic Science, vol. , no. , pp. 76-91, 2025. DOI: https://doi.org/10.54216/IJNS.250308

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