Volume 25 • Issue 3 • PP: 25-36 • 2025
On The Numerical Solutions of the Neutrosophic One-Dimensional Sine-Gordon System
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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).
Abstract
This paper uses finite difference methods to study the numerical solution for neutrosophic Sine-Gordon system in one dimension. We use the explicit method and Crank-Nicholson method. Also, an effective comparison between the results of the two methods has been made, where we obtain the result that Crank-Nicholson method is more accurate than the explicit method, but the explicit method is easier. We also study the stability analysis for each method by using Fourier (Von-Neumann) method and get that Crank-Nicholson method is unconditionally stable while the Explicit method is stable under the condition 𝑟2≤1𝑐2 and 𝑟2≤1.
Keywords
References
[1] Pelloni and D. A. Pinotsis. (2010).The elliptic sine-Gordon equation in a half plane, Nonlinearity, 23, 77–88.
[2] M. Jaworski and D. Kaup, Direct and inverse scattering problem associated with the elliptic sinh-Gordon equation, Inverse Problems 6 (1990) 543–556.
[3] ARODZ, H. and KLIMAS, P., (2005),"Chain of impacting pendulums as non-analytically perturbed
Sine-Gordon system", Acta Phys polonica B, No.3, Vol.36.
[4] Gordon D. Smith (1965);"Numerical Solution of Partial Differential Equations: Finite Difference Methods", second edition, Oxford University press.
[5] Griffiths, S.D., R.H.J. Grimshaw, K.R. Khusnutdinova, D.E. Pelinovsky. (2003).Energy exchange in coupled Sine-Gordon equations and the influence of modulational instability, Geophysical Research Abstracts, Vol.5, 00471.
[6] Griffiths,S.D., R.H.J. Grimshaw, K.R. Khusnutdinova.(2003).The influence of modulational instability on energy exchange in coupled Sine-Gordon equations,Theor. Math. Phys. 137, 1446-1456.
[7] A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF regional conference series in applied mathematics, SIAM 2008
[8] Mathewes, J.H. and Fink K.D. (2004); “Numerical Method using Matlab", Prentice-Hall, Inc.
[9] A. S. Fokas. (2000). on the integrability of certain linear and nonlinear partial differential equations, J. Math.Phys. 41,4188–4237.
[10] Scott, A. C. (2003). Nonlinear Science: "Emergence and dynamics of coherent structures",Second Edition, Oxford and New York :Oxford University Press.
[11] Shanthakumar , M. (1989); "Computer Based Numerical Analysis", Khanna Publishers.
[12] Heilat, A.S., Batiha,B ,T.Qawasmeh ,Hatamleh R.(2023). Hybrid Cubic B-spline Method for Solving A
Class of Singular Boundary Value Problems. European Journal of Pure and Applied Mathematics, 16(2),
751-762. (DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4725).
[13] Ayman Hazaymeh, Ahmad Qazza, Raed Hatamleh, Mohammad W Alomari, Rania Saadeh.(2023).On
Further Refinements of Numerical Radius Inequalities, Axiom-MDPI, 12(9), 807.
[14] T.Qawasmeh, A. Qazza, R. Hatamleh, M.W. Alomari, R. Saadeh. (2023). Further accurate numerical radius inequalities, Axiom-MDPI, 12 (8), 801.
[15] Ayman Hazaymeh, Rania Saadeh, Raed Hatamleh, Mohammad W. Alomari, Ahmad Qazza. (2023). A
Perturbed Milne’s Quadrature Rule for n-Times Differentiable Functions with Lp-Error Estimates, Axioms- MDPI, 12(9), and 803.
[16] T.Qawasmeh,R.Hatamleh,(2023).A new contraction based on H-simulation functions in the frame of Extended b-metric spaces and application, International Journal of Electrical and Computer Engineering,13 (4),4212-4221.
[17] R.Hatamleh. (2024). On the Compactness and Continuity of Uryson's Operator in Orlicz Space, International Journal of Neutrosophic Science, 24 (3), 233-239.
[18] Hatamleh, R. (2003). On the Form of Correlation Function for a Class of Nonstationary Field with a Zero Spectrum, Rocky Mountain Journal of Mathematics, 33(1)159-173. (DOI:https://doi.org/10.1216/rmjm/1181069991).
[19] Heilat, A.S., Batiha,B ,T.Qawasmeh ,Hatamleh R.(2023). Hybrid Cubic B-spline Method for Solving a Class of Singular Boundary Value Problems, European Journal of Pure and Applied Mathematics, 16(2) 751-762. (DOI: https://doi.org/10.29020/nybg.ejpam.v16i2.4725).
[20] Abdallah Shihadeh, Khaled Ahmad Mohammad Matarneh, Raed Hatamleh, Randa Bashir Yousef Hijazeen, Mowafaq Omar Al-Qadri, Abdallah Al-Husban.(2024). An Example of Two-Fold Fuzzy Algebras Based On Neutrosophic Real Numbers, Neutrosophic Sets and Systems,vol.67, pp. 169-178 . (DOI: 10.5281/zenodo.11151930)
[21] Batiha, B., Ghanim, F., Alayed, O., Hatamleh, R., Heilat, A. S., Zureigat, H., & Bazighifan, O. (2022).
Solving Multispecies Lotka–Volterra Equations by the Daftardar-Gejji and Jafari Method. International
Journal of Mathematics and Mathematical Sciences, 2022, 1–7. (DOI:https://doi.org/10.1155/2022/1839796).
[22] F. Y. Kamsu, C. K. Issa, and T. Z. Ngo, "Numerical solutions for singular boundary value problems using Chebyshev collocation method," Mathematics and Computers in Simulation, vol. 181, pp. 49–60, 2021, doi: 10.1016/j.matcom.2021.02.012.
[23] Hamadneh, T., Abbes, A., Falahah, I. A., Al-Khassawneh, Y. A., Heilat, A. S., Al-Husban, A., & Ouannas, A. (2023). Complexity and chaos analysis for two-dimensional discrete-time predator–prey Leslie–Gowermodel with fractional orders. Axioms, 12(6), 561.
[24] Heilat, A. S., Karoun, R. C., Al-Husban, A., Abbes, A., Al Horani, M., Grassi, G., & Ouannas, A. (2023). The new fractional discrete neural network model under electromagnetic radiation: Chaos, control and synchronization. Alexandria Engineering Journal, 76, 391-409.
[25] Abu Falahah, I., Hioual, A., Al-Qadri, M. O., Al-Khassawneh, Y. A., Al-Husban, A., Hamadneh, T., &
Ouannas, A. (2023). Synchronization of Fractional Partial Difference Equations via Linear Methods. Axioms, 12(8), 728.
[26] Alsayyed, O., Hioual, A., Gharib, G. M., Abualhomos, M., Al-Tarawneh, H., Alsauodi, M. S., & Ouannas, A. (2023). On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model. Fractal and Fractional, 7(10), 729.
[27] Hamadneh, T., Hioual, A., Alsayyed, O., AL-Khassawneh, Y. A., Al-Husban, A., & Ouannas, A. (2023). Local stability, global stability, and simulations in a fractional discrete glycolysis reaction–diffusion model. Fractal and Fractional, 7(8), 587.
[28] MB Zeina, M Abobala, On The Refined Neutrosophic Real Analysis Based on Refined Neutrosophic Algebraic AH-Isometry, Neutrosophic Sets and Systems, 2023.
[29] Bal, M. D., K. Ali, R. (2022). A Review Study on Neutrosophic AH-Algebraic Structures. Journal of Neutrosophic and Fuzzy Systems, (), 40-60. DOI: https://doi.org/10.54216/JNFS.020105
[30] Salama, A. Dalla, R. Al, M. Ali, R. (2022). On Some Results About The Second Order Neutrosophic Differential Equations By Using Neutrosophic Thick Function. Journal of Neutrosophic and Fuzzy Systems, (), 30-40. DOI: https://doi.org/10.54216/JNFS.040104
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