Page 52 - IJOCTA-15-1
P. 52
L.K. Yadav et.al. / IJOCTA, Vol.15, No.1, pp.35-49 (2025)
obtained solutions for Example 1 with δ ≤ 1. Vikash Kumar Meena
Fig. 4 represent the ℏ-curves of the HASTM so- Writing – review & editing: Ebenezer Bonyah,
lution for Example 1. Sunil Dutt Purohit
Availability of data
Moreover, Figs. 5-6 represents the nature of
HASTM solution in comparison with exact so-
lution of fractional approximate long wave equa- Not applicable.
tion (45). Fig. 7 explore the obtained solutions
for example 2 with δ ≤ 1. Fig. 8 represent the References
ℏ-curves of the HASTM solution for example 2,
which helps to adjust the convergence region. [1] Ali, A., Shah, K., & Khan, R. A. (2017). Numeri-
cal treatment for travelling wave solutions of frac-
tional Whitham-Broer-Kaup equations. Alexan-
8. Conclusions
dria Engineering Journal, 57(3), 1991-1998. http
s://doi.org/10.1016/j.aej.2017.04.012
In this study, we successfully implemented the [2] Chen, Z., Manafian, J., Raheel, M., Zafar, A.,
HASTM to examine coupled WBK equations with Alsaikhan, F., & Abotaleb, M. (2022). Extracting
the fractional Caputo derivative operator. We ob- the exact solitons of time-fractional three coupled
nonlinear Maccari’s system with complex form
tained both analytical and numerical solutions for
via four different methods. Results in Physics, 36,
two applications of fractional WBK equations to
105400. https://doi.org/10.1016/j.rinp.202
demonstrate the effectiveness and accuracy of the
2.105400
proposed scheme. Notably, the proposed method
[3] Dehghan, M., & Manafian, J. (2018). Analyt-
yielded convergent series solutions with easily de-
ical treatment of nonlinear conformable time-
terminable components, without the need for per- fractional Boussinesq equations by three integra-
turbation, linearization, or limiting assumptions. tion methods. Optical and Quantum Electronics,
The numerical and graphical solutions obtained 50(4), 1-31. https://doi.org/10.1007/s11082
through the proposed method demonstrated that -017-1268-0
the technique is computationally accurate, pro- [4] Haq, S., & Ishaq, M. (2014). Solution of cou-
viding an alternative approach for examining so- pled Whitham-Broer-Kaup equations using opti-
lutions of time fractional differential equations. mal homotopy asymptotic method. Ocean Engi-
neering, 84, 81-88. https://doi.org/10.1016/
j.oceaneng.2014.03.031
Acknowledgments [5] Manafian, J., & Dehghan, M. (2015). Optical
solitons with Biswas-Milovic equation for Kerr
law nonlinearity. European Physical Journal Plus,
None.
130, 1112. https://doi.org/10.1140/epjp/i
2015-15061-1
Funding [6] Ray, S. S. (2015). A novel method for travel-
ling wave solutions of fractional Whitham-Broer-
Kaup, fractional modified Boussinesq and frac-
None. tional approximate long wave equations in shal-
low water. Mathematical Methods in the Applied
Conflict of interest Sciences, 38, 1352-1368. https://doi.org/10.1
002/mma.3151
[7] Wang, L., & Chen, X. (2015). Approximate an-
The authors declare no conflict of interest. alytical solutions of time fractional Whitham-
Broer-Kaup equations by a residual power series
method. Entropy, 17, 6519-6533. https://doi.
Author contributions org/10.3390/e17096519
[8] Xie, F., Yan, Z., & Zhang, H. (2001). Explicit and
exact traveling wave solutions of Whitham-Broer-
Conceptualization: Ebenezer Bonyah, Sunil Dutt
Kaup shallow water equations. Physics Letters A,
Purohit
285, 76-80. https://doi.org/10.1016/S0375-9
Formal analysis: Vikash Kumar Meena, Murli
601(01)00333-4
Manohar Gour [9] Dubey, R. S., Belgacem, F. B. M., & Goswami,
Methodology: Lokesh Kumar Yadav, Murli P. (2016). Homotopy perturbation approximate
Manohar Gour solutions for Bergman’s minimal blood glucose-
Writing – original draft: Lokesh Kumar Yadav, insulin model. Fractal Geometry and Nonlinear
46
