Volume 25 • Issue 4 • PP: 10-17 • 2025
The Runge-Kutta Numerical Method of Rank Seven for the Solutions of Some Refined Neutrosophic Differential Problems
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© 2025 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
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In this paper, we present a numerical approach to the seventh rank refined neutrosophic Runge-Kutta numerical method, where we provide the theoretical basis of this formula to be applicable on refined neutrosophic differential equations. In addition, we provide numerical tables to compare the validity of this new method with other methods, as well as a clear computation of absolute errors in terms of refined neutrosophic numbers.
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References
[1] J. C. Butcher and M. T. Diamantakis, "DESIRE: Diagonally extended singly implicit Runge-Kutta effective order methods," Numeric. Algorithm, vol. 17, pp. 121–145, 1998.
[2] J. C. Butcher, "Numerical methods for differential equations and applications," Arabian J. Sci. Eng., vol. 22, no. 2, pp. 17–29, 1997.
[3] J. R. Cash, "Block Runge-Kutta methods for numerical integration of initial value problems in ordinary differential equations Part II: The stiff case," Math. Comput, vol. 40, no. 161, pp. 193–206, 1983.
[4] J. Wang, Y. Luo, and X. Li, "A class of efficient implicit Runge-Kutta methods for stiff ordinary differential equations," J. Comput. Appl. Math., vol. 401, art. No. 113794, 2022. DOI: 10.1016/j.cam.2021.113794.
[5] J. R. Cash, "Runge-Kutta methods for the solution of stiff two-point boundary value problems," Appl. Numer. Math., vol. 22, pp. 165–177, 1996.
[6] F. Smarandache, Introduction to Neutrosophic Statistics, Sitech & Education Publishing, USA, 2014.
[7] D. A. Voss and M. J. Casper, "Efficient split linear multistep methods for stiff ordinary differential equations," SIAM J. Sci. Stat. Comput., vol. 19, no. 5, pp. 990–999, 1989.
[8] D. A. Voss, "Factored two-step Runge-Kutta methods," Appl. Math. Compute, vol. 31, pp. 361–368, 1989.
[9] D. A. Voss, "Fifth-order exponentially fitted formula," SIAM J. Numer. Anal., vol. 25, no. 3, pp. 670–678, 1988.
[10] A. A. Abubaker, M. Abualhomos, K. Matarneh, and A. Al-Husban, "A numerical approach for the algebra of two-fold," Neutrosophic Sets Syst., vol. 75, pp. 181–195, 2025.
[11] M. A. Khoshnam, S. A. Kheirandish, and H. R. Naderi, "Numerical Methods for Neutrosophic Boundary Value Problems: Applications and Solutions," International Journal of Neutrosophic Science, vol. 24, no. 3, pp. 150–162, 2023. DOI: 10.54216/IJNS.240301.
[12] S. M. and A. N. Mera, "Fuzzy logic used to solve ODEs of second order under neutrosophic initial conditions," Int. J. Neutrosophic Sci., vol. 23, no. 1, pp. 51–58, 2024. DOI: 10.54216/IJNS.230104.
[13] S. Topal, F. Tas, S. Broumi, and O. Ayhan, "Applications of neutrosophic logic of smart agriculture via Internet of Things," Int. J. Neutrosophic Sci., vol. 12, no. 2, pp. 105–115, 2020. DOI: 10.54216/IJNS.120205.
[14] A. Shihadeh, K. A. M. Matarneh, R. Hatamleh, R. B. Y. Hijazeen, M. O. Al-Qadri, and A. Al-Husban, "An example of two-fold fuzzy algebras based on neutrosophic real numbers," Neutrosophic Sets Syst., vol. 67, pp. 169–178, 2024.
[15] A. F. Salamah and R. M. Dallah, "A study of neutrosophic Bernoulli and Riccati equations using the one-dimensional geometric AH-Isometry," J. Neutrosophic Fuzzy Syst., vol. 5, no. 1, pp. 30–40, 2023. DOI: 10.54216/JNFS.050104.
[16] T. Hamadneh, A. Abbes, I. A. Falahah, Y. A. Al-Khassawneh, A. S. Heilat, A. Al-Husban, and A. Ouannas, "Complexity and chaos analysis for two-dimensional discrete-time predator-prey Leslie-Gower model with fractional orders," Axioms, vol. 12, no. 6, art. no. 561, 2023.
[17] M. Sahin and N. Olgun, “On the Refined AH-Isometry and Its Applications in Refined Neutrosophic Surfaces,” Galoitica: Journal of Mathematical Structures and Applications, vol. 2, no. 1, pp. 21–28, 2022.
[18] F. Al-Sharqi, “Exploring the Algebraic Structures of Q-Complex Neutrosophic Soft Fields,” Journal of Neutrosophic and Fuzzy Systems, vol. 5, no. 2, pp. 45–58, 2023.
[19] A. M. Al-Odhari, “Some Algebraic Structure of Neutrosophic Matrices,” Journal of Algebraic Structures and Their Applications, vol. 10, no. 3, pp. 158–167, 2023.
[20] B. Batiha, "New solution of the Sine-Gordon equation by the Daftardar-Gejji and Jafari Method," Symmetry, vol. 14, no. 1, art. No. 57, 2022.
[21] B. Batiha, G. F. Alayed, O. Hatamleh, A. S. Heilat, H. Zureigat, and O. Bazighifan, "Solving multispecies Lotka-Volterra equations by the Daftardar-Gejji and Jafari Method," Int. J. Math. Math. Sci., vol. 2022, art. No. 1839796.
[22] O. Ala’yed, B. Batiha, R. Abdelrahim, and A. Jawarneh, "On the numerical solution of the nonlinear Bratu type equation via quintic B-spline method," J. Interdiscip. Math., vol. 22, no. 4, pp. 405–413, 2019.
[23] M. Abualhomos, W. M. M. Salameh, M. Bataineh, M. O. Al-Qadri, A. Alahmade, and A. Al-Husban, "An effective algorithm for solving weak fuzzy complex Diophantine equations in two variables," Int. J. Neutrosophic Sci., vol. 23, no. 4, pp. 386–394, 2024.
[24] A. Al-Husban, R. C. Karoun, A. S. Heilat, M. Al Horani, A. A. Khennaoui, G. Grassi, and A. Ouannas, "Chaos in a two-dimensional fractional discrete Hopfield neural network and its control," Alexandria Eng. J., vol. 75, pp. 627–638, 2023.
[25] A. S. Heilat, R. C. Karoun, A. Al-Husban, A. Abbes, M. Al Horani, G. Grassi, and A. Ouannas, "The new fractional discrete neural network model under electromagnetic radiation: Chaos, control and synchronization," Alexandria Eng. J., vol. 76, pp. 391–409, 2023.
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