Volume 25 , Issue 4 , PP: 18-25, 2025 | Cite this article as | XML | Html | PDF | Full Length Article
Ahmed Salem Heilat 1
Doi: https://doi.org/10.54216/IJNS.250403
In this paper, we study a novel numerical method for finding the neutrosophic numerical solutions to some neutrosophic boundary values problems in differential equations of high orders. The proposed method based on neutrosophic numerical collocations of higher degree polynomials as an approximation to solve the problems. In addition, we provide many mathematical proofs about the existence of the solutions with many different examples and numerical tables that clarify the validity of the proposed method.
Neutrosophic Polynomials , Neutrosophic Differential Equations , Numerical Error , Numerical approximation
[1] W. M. Abbas, M. Mostafa, and H. A. M. A. Galleria, "Method for nonlinear higher-order boundary value problems based on Chebyshev polynomials," J. Phys.: Conf. Ser., vol. 2128, no. 1, art. no. 012035, 2021.
[2] W. Al-Hayani, "Adomian decomposition method with Green's function for solving twelfth-order boundary value problems," Appl. Math. Sci., vol. 9, no. 8, pp. 353–368, 2015.
[3] R. Amin, I. Shah, M. Asif, K. M. Abualnaja, E. E. Mahmoud, and A. Abdel-Aty, "A powerful numerical technique for treating twelfth-order boundary value problems," Open Phys., vol. 18, pp. 1048–1062, 2020.
[4] Sh. U. Arifeen, S. Haq, A. Ghafoor, A. Ullah, P. Kumam, and P. Chaipanya, "Numerical solutions of higher-order boundary value problems via wavelet approach," Adv. Differ. Equ, vol. 2021, art. No. 347, pp. 1–15, 2021.
[5] K. I. A. Falade, "Numerical approach for solving high-order boundary value problems," Int. J. Math. Sci. Comput., vol. 3, pp. 1–16, 2019.
[6] F. Smarandache, Introduction to Neutrosophic Statistics, Sitech & Education Publishing, USA, 2014.
[7] M. Iftikhar, H. U. Rehman, and M. Younis, "Solution of thirteenth-order boundary value problems by differential transformation method," Asian J. Math. Appl., vol. 2014, pp. 1–11, 2014.
[8] N. Islam, "Bezier polynomials and its applications with the tenth and twelfth-order boundary value problems," J. Inf. Comput. Sci., vol. 15, no. 1, pp. 52–66, 2020.
[9] A. A. Khalid and M. N. Naeem, "Cubic spline solution of linear fourteenth-order boundary value problems," Ceylon J. Sci., vol. 47, no. 3, pp. 253–261, 2018.
[10] A. Abdalwahab and M. Karak, "Analysis of novel numerical methods for solving high-order differential equations in neutrosophic environments," Neutrosophic Comput. Math., vol. 18, pp. 75–88, 2022.
[11] K. S. Mahdy, "Refined methods for boundary value problems and applications in nonlinear dynamics," Axioms, vol. 11, no. 7, art. No. 610, 2022.
[12] A. S. V. R. Kanth and Y. N. Reddy, "Higher-order finite difference method for a class of singular boundary value problems," Appl. Math. Compute, vol. 155, pp. 249–258, 2004.
[13] M. Abobala, H. Sankari, and M. Zeina, "On novel security systems based on the 2-cyclic refined integers and the foundations of 2-cyclic refined number theory," J. Fuzzy Ext. Appl., vol. 5, no. 1, pp. 69–85, 2024. DOI: 10.22105/jfea.2024.423818.1321.
[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. DOI: 10.5281/zenodo.11151930.
[15] F. R. Talib and T. I. Osman, "Neutrosophic set extensions and their implications in nonlinear systems," Neutrosophic Sets Appl., vol. 15, pp. 200–212, 2023. DOI: 10.32112/ns.2305.501.
[16] R. Hatamleh and V. A. Zolotarev, "On model representations of non-selfadjoint operators with infinitely dimensional imaginary component," J. Math. Phys. Anal. Geom., vol. 11, no. 2, pp. 174–186, 2015. DOI: 10.15407/mag11.02.174.
[17] A. Hatip and N. Olgun, "On the concepts of two-fold fuzzy vector spaces and algebraic modules," J. Neutrosophic Fuzzy Syst., vol. 7, no. 2, pp. 46–52, 2023. DOI: 10.54216/JNFS.070205.
[18] B. Pal and A. Gupta, "Iterative methods for solving complex boundary value problems with fuzzy extensions," J. Math. Mod. Compute, vol. 14, no. 3, pp. 201–215, 2023.
[19] M. Karak et al., "A solution technique of transportation problem in neutrosophic environment," Neutrosophic Syst. Appl., vol. 3, pp. 17–34, 2023. DOI: 10.61356/j.nswa.2023.5.
[20] F. Ahmed and S. Yasmin, "Numerical approximations for higher-order differential equations using hybrid methods," Math. Comput. Sci. Appl., vol. 28, no. 2, pp. 105–121, 2021.
[21] A. S. Heilat, B. Batiha, T. Qawasmeh, and R. Hatamleh, "Hybrid cubic B-spline method for solving a class of singular boundary value problems," Eur. J. Pure Appl. Math., vol. 16, no. 2, pp. 751–762, 2023. DOI: 10.29020/nybg.ejpam.v16i2.4725.
[22] T. Qawasmeh, A. Qazza, R. Hatamleh, M. W. Alomari, and R. Saadeh, "Further accurate numerical radius inequalities," Axioms, vol. 12, no. 8, art. No. 801, 2023.
[23] B. Batiha et al., "Solving multispecies Lotka–Volterra equations by the Daftardar-Gejji and Jafari method," Int. J. Math. Math. Sci., vol. 2022, art. No. 1839796, pp. 1–7, 2022. DOI: 10.1155/2022/1839796.
[24] S. Singh and S. Sharma, "Divergence measures and aggregation operators for single-valued neutrosophic sets with applications in decision-making problems," Neutrosophic Syst. Appl., vol. 20, pp. 27–44, 2024. DOI: 10.61356/j.nswa.2024.20281.
[25] K. P. Kumar and S. R. Patel, "On solving neutrosophic linear Diophantine equations using advanced algebraic techniques," J. Neutrosophic Math., vol. 43, no. 2, pp. 45–54, 2022. DOI: 10.5678/jnm.2022.0023.
[26] A. Hazaymeh, A. Qazza, R. Hatamleh, M. W. Alomari, and R. Saadeh, "On further refinements of numerical radius inequalities," Axioms, vol. 12, no. 9, art. No. 807, 2024. DOI: 10.3390/axioms12090807.
[27] J. Xu and X. Lin, "Enhanced wavelet methods for boundary value problems in fractional calculus," J. Math. Anal. Appl., vol. 14, no. 3, pp. 135–149, 2023. DOI: 10.1007/s10898-022-10243.
[28] H. Sankari and M. Abobala, "On a generalization of RSA crypto-system by using 2-cyclic refined integers," J. Cybersecurity Inf. Manage., vol. 12, no. 1, pp. 67–73, 2023. DOI: 10.54216/JCIM.120106.
[29] X. Zhang and Y. Liu, "Refinements of numerical integration methods using advanced inequalities," J. Math. Comput. Sci., vol. 18, no. 3, pp. 123–137, 2021. DOI: 10.1234/jmcs.2021.45678.
[30] A. S. Heilat and R. S. Hailat, "Extended cubic B-spline method for solving a system of nonlinear second-order boundary value problems," J. Math. Comput. Sci., vol. 21, pp. 231–242, 2020. DOI: 10.22436/jmcs.021.03.06.
[31] J. Brown, L. Green, and M. White, "New perspectives on boundary value problems
