Using a finite difference scheme together with neural networks analysis for propagation of the long waves equation

Thabet Abdeljawad; Kamal Shah; Israr Ahmad; Manel Hleili; Maryam Salem Alatawi; Thabet Abdeljawad; Kamal Shah; Israr Ahmad; Manel Hleili; Maryam Salem Alatawi

doi:10.3934/math.2026036

AIMS Mathematics

2026, Volume 11, Issue 1: 825-856. doi: 10.3934/math.2026036

Previous Article Next Article

Research article Special Issues

Using a finite difference scheme together with neural networks analysis for propagation of the long waves equation

1.
Department of Fundamental Sciences, Faculty of Engineering and Architecture, Istanbul Gelisim University, 34310 Avcilar, Istanbul, Turkey
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics, Government Post Graduate Jahanzeb College, Swat, KP, Pakistan
5.
Department of Mathematics, Faculty of Science, University of Tabuk, 71491 Tabuk, Saudi Arabia

Received: 13 October 2025 Revised: 17 December 2025 Accepted: 27 December 2025 Published: 12 January 2026
MSC : 26A33, 34A08

In this paper, we explored the Benjamin-Bona-Mahony-Burger (BBMB) equation in the context of the conformable fractional derivative. The major results are the existence, the uniqueness, and inaddition, the Ulam-Hyers (UH) stability of the solutions that are obtained, which guarantee the mathematical soundness of the proposed model. For the numerical study, the standard finite difference method (SFDM) and the non-standard finite difference method (NSFDM) were developed, and their performance was assessed through a comparison with the exact solution. The conclusions showed that NSFDM is more accurate and stable compared to SFDM. Additionally, a neural network (NN) scheme was used as a further validation tool, which was complemented by regression analysis and error distribution measures. The change of fractional order significantly affects the solution profiles, as shown by two or three-dimensional plots in numerical simulations. The fractional dynamics, therefore, play a crucial role in modifying wave propagation in more dimensions. The unique feature of this research is the joint use of conformable fractional calculus with NSFDM and neural computing for the BBMB equation, providing a new way for the treatment of nonlinear dispersive wave models.
- conformable derivative,
- BBMB equation,
- qualitative analysis,
- SFDM,
- NSFDM,
- neural network
Citation: Thabet Abdeljawad, Kamal Shah, Israr Ahmad, Manel Hleili, Maryam Salem Alatawi. Using a finite difference scheme together with neural networks analysis for propagation of the long waves equation[J]. AIMS Mathematics, 2026, 11(1): 825-856. doi: 10.3934/math.2026036

Related Papers:

Abstract

In this paper, we explored the Benjamin-Bona-Mahony-Burger (BBMB) equation in the context of the conformable fractional derivative. The major results are the existence, the uniqueness, and inaddition, the Ulam-Hyers (UH) stability of the solutions that are obtained, which guarantee the mathematical soundness of the proposed model. For the numerical study, the standard finite difference method (SFDM) and the non-standard finite difference method (NSFDM) were developed, and their performance was assessed through a comparison with the exact solution. The conclusions showed that NSFDM is more accurate and stable compared to SFDM. Additionally, a neural network (NN) scheme was used as a further validation tool, which was complemented by regression analysis and error distribution measures. The change of fractional order significantly affects the solution profiles, as shown by two or three-dimensional plots in numerical simulations. The fractional dynamics, therefore, play a crucial role in modifying wave propagation in more dimensions. The unique feature of this research is the joint use of conformable fractional calculus with NSFDM and neural computing for the BBMB equation, providing a new way for the treatment of nonlinear dispersive wave models.

References

[1]	I. Ahmad, Z. Ali, B. Khan, K. Shah, T. Abdeljawad, Exploring the dynamics of Gumboro-Salmonella co-infection with fractal fractional analysis, Alex. Eng. J., 117 (2025), 472–489, http://doi.org/10.1016/j.aej.2024.12.119 doi: 10.1016/j.aej.2024.12.119
[2]	G. Ali, I. Ahmad, K. Shah, T. Abdeljawad, Iterative analysis of nonlinear BBM equations under nonsingular fractional order derivative, Advances in Mathematical Physics, 2020 (2020), 3131856. http://doi.org/10.1155/2020/3131856 doi: 10.1155/2020/3131856
[3]	I. Podlubny, Fractional Differential Equations, San Diego: Academic Press, 1999.
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.
[5]	R. Hilfer, Applications of Fractional Calculus in Physics, Singapore: World Scientific, 2000.
[6]	R. L. Magin, Fractional Calculus in Bioengineering, New York: Begell House, 2006.
[7]	H. Alrabaiah, I. Ahmad, K. Shah, I. Mahariq, G. U. Rahman, Analytical solution of non-linear fractional order Swift-Hohenberg equations, Ain Shams Eng. J., 12 (2021), 3099–3107. http://doi.org/10.1016/j.asej.2020.11.019 doi: 10.1016/j.asej.2020.11.019
[8]	V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Berlin: Springer, 2011.
[9]	F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Soliton. Fract., 7 (1996), 1461–1477. http://doi.org/10.1016/0960-0779(95)00125-5 doi: 10.1016/0960-0779(95)00125-5
[10]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, Singapore: World Scientific, 2012.
[11]	A. Bekir, Ö. Güner, Ö. Ünsal, The first integral method for exact solutions of nonlinear fractional differential equations, J. Comput. Nonlinear Dyn. 10 (2015), 021020–5. http://doi.org/10.1115/1.4028065 doi: 10.1115/1.4028065
[12]	E. A. Az-Zo'bi, Q. M. M. Alomari, K. Afef, M. Inc, Dynamics of generalized time-fractional viscous-capillarity compressible fluid model, Opt. Quant. Electron., 56 (2024), 629. http://doi.org/10.1007/s11082-023-06233-2 doi: 10.1007/s11082-023-06233-2
[13]	Y. Zhou, Basic theory of fractional differential equations, Singapore: World scientific, 2023.
[14]	E. Hussain, Z. Li, S. A. A Shah, E. A. Az-Zo'bi, M. Hussien, Dynamics study of stability analysis, sensitivity insights and precise soliton solutions of the nonlinear (STO)-Burger equation, Opt. Quant. Electron., 55 (2023), 1274. http://doi.org/10.1007/s11082-023-05588-w doi: 10.1007/s11082-023-05588-w
[15]	M. A. Al Zubi, K. Afef, E. A. Az-Zo'bi, Assorted spatial optical dynamics of a generalized fractional quadruple nematic liquid crystal system in non-local media, Symmetry, 16 (2024), 778. http://doi.org/10.3390/sym16060778 doi: 10.3390/sym16060778
[16]	J. H. He, Homotopy perturbation technique, Comput. Math. Appl., 57 (1999), 410–412. http://doi.org/10.1016/S0045-7825(99)00018-3 doi: 10.1016/S0045-7825(99)00018-3
[17]	R. Khalil, M. Al Horani, A. Yousef, M. A. Sababheh, New definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. http://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[18]	H. U.Rehman, I. Iqbal, S. S. Aiadi, N. Mlaiki, M. S. Saleem, Soliton solutions of Klein-Fock-Gordon equation using Sardar subequation method, Mathematics, 10 (2022), 3377.
[19]	H. U. Rehman, I. Iqbal, M. Mirzazadeh, S. Haque, N. Mlaiki, W. Shatanawi, Dynamical behavior of perturbed Gerdjikov-Ivanov equation through different techniques, Bound. Value Prob., 2023 (2023), 105. http://doi.org/10.1186/s13661-023-01792-5 doi: 10.1186/s13661-023-01792-5
[20]	C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett., 19 (1967), 1095. http://doi.org/10.1103/PhysRevLett.19.1095 doi: 10.1103/PhysRevLett.19.1095
[21]	T. B. Benjamin, J. L. Bona, J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. T. R. Soc. A, 272 (1972), 47–78. http://doi.org/10.1098/rsta.1972.0032 doi: 10.1098/rsta.1972.0032
[22]	C. Li, Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term, Appl. Math. Comput., 311 (2017), 240–250. http://doi.org/10.1016/j.amc.2017.05.022 doi: 10.1016/j.amc.2017.05.022
[23]	A. Saadatmandi, M. A. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326–1336. http://doi.org/10.1016/j.camwa.2009.07.006 doi: 10.1016/j.camwa.2009.07.006
[24]	E. Hussain, S. A. A. Shah, A. Bariq, Z. Li, M. R. Ahmad, A. E. Ragab, et al., Solitonic solutions and stability analysis of Benjamin Bona Mahony Burger equation using two versatile techniques, Sci. Rep., 14 (2024), 13520. http://doi.org/10.1038/s41598-024-60732-0 doi: 10.1038/s41598-024-60732-0
[25]	M. Kamran, M. Abbas, A. Majeed, H. Emadifar, T. Nazir, Numerical simulation of time fractional BBM-Burger equation using cubic B-spline functions, J. Function Spaces, 2022 (2022), 2119416. http://doi.org/10.1155/2022/2119416 doi: 10.1155/2022/2119416
[26]	A. M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102 (1999), 77–86. http://doi.org/10.1016/S0096-3003(98)10024-3 doi: 10.1016/S0096-3003(98)10024-3
[27]	Z. Yan, J. Li, S. Barak, S. Haque, N. Mlaiki, Delving into quasi-periodic type optical solitons in fully nonlinear complex structured perturbed Gerdjikov-Ivanov equation, Sci. Rep., 15 (2025), 8818. http://doi.org/10.1038/s41598-025-91978-x doi: 10.1038/s41598-025-91978-x
[28]	S. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, New York: Chapman & Hall/CRC, 2003.
[29]	A. R. Appadu, A. S. Kelil, N. W. Nyingong, Solving a fractional diffusion PDE using some standard and nonstandard finite difference methods with conformable and Caputo operators, Front. Appl. Math. Stat., 10 (2024), 1358485. http://doi.org/10.3389/fams.2024.1358485 doi: 10.3389/fams.2024.1358485
[30]	Y. Zou, Y. Cui, Uniqueness criteria for initial value problem of conformable fractional differential equation, Electron. Res. Arch., 31 (2023), 4077–4087. http://doi.org/10.3934/era.2023207 doi: 10.3934/era.2023207
[31]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. http://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[32]	H. Khan, J. Alzabut, D. Baleanu, G. Alobaidi, M. U. Rehman, Existence of solutions and a numerical scheme for a generalized hybrid class of n-coupled modified ABC-fractional differential equations with an application, AIMS Mathematics, 8 (2023), 6609–6625. http://doi.org/10.3934/math.2023334 doi: 10.3934/math.2023334
[33]	E. A. Az-Zo'bi, R. Shah, H. A. Alyousef, C. G. L. Tiofack, S. A. El-Tantawy, On the feed-forward neural network for analyzing pantograph equations, AIP Adv., 14 (2024), 025042. http://doi.org/10.1063/5.0195270 doi: 10.1063/5.0195270
[34]	A. R. Appadu, G. N. de Waal, C. J. Pretorius, Nonstandard finite difference methods for a convective predator-prey pursuit and evasion model, J. Differ. Equ. Appl., 30 (2024), 1808–1841. http://doi.org/10.1080/10236198.2024.2361115 doi: 10.1080/10236198.2024.2361115
[35]	R. E. Mickens, Applications of Nonstandard Finite Difference Schemes, Singapore: World Scientific, 2000.
[36]	M. Raissi, P. Perdikaris, G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys., 378 (2019), 686–707. http://doi.org/10.1016/j.jcp.2018.10.045 doi: 10.1016/j.jcp.2018.10.045
[37]	L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators, Nat. Mach. Intell., 3 (2021), 218–229. http://doi.org/10.1038/s42256-021-00302-5 doi: 10.1038/s42256-021-00302-5
[38]	H. S. Alruhaili, A. S. Hussain, A. Ajlouni, F. Türk, E. A. Az-Zo'bi, M. Tashtoush, Solving Time-Fractional Nonlinear Variable-Order Delay PDEs Using Feedforward Neural Networks, Iraqi J. Comput. Sci. Math., 6 (2025), 12. http://doi.org/10.52866/2788-7421.1284 doi: 10.52866/2788-7421.1284

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)