Approximate analytical solutions of fractional coupled Whitham-Broer-Kaup equations via novel transform

Abstract

In this study, the approximated analytical solution for the time-fractional coupled Whitham-Broer-Kaup (WBK) equations describing the propagation of shallow water waves are obtained with the aid of an efficient computational technique called, homotopy analysis Shehu transform methodm(briefly, HASTM). The Caputo operator is utilized to describe fractional-order derivatives. Our proposed approach combines the Shehu transformation with the homotopy analysis method, employing homotopy polynomials to handle nonlinear terms. To validate the correctness of our method, we offer a comparison of obtained and exact solutions with different fractional order values. Given its novelty and straightforward implementation, our method is considered a reliable and efficient analytical technique for solving both linear and non-linear fractional partial differential equations.

Keywords

Coupled WBK equations / Homotopy analysis method / Shehu transform / Caputo fractional derivative

Funding

None.

Declaration of competing interest

The authors declare no conflict of interest.

References

[1] Ali, A., Shah, K., & Khan, R. A. (2017). Numerical treatment for travelling wave solutions of fractional Whitham-Broer-Kaup equations. Alexandria Engineering Journal, 57(3), 1991-1998. https://doi.org/10.1016/j.aej.2017.04.012

[2] Chen, Z., Manafian, J., Raheel, M., Zafar, A., Alsaikhan, F., & Abotaleb, M. (2022). Extracting the exact solitons of time-fractional three coupled nonlinear Maccari’s system with complex form via four different methods. Results in Physics, 36, 105400. https://doi.org/10.1016/j.rinp.2022.105400

[3] Dehghan, M., & Manafian, J. (2018). Analytical treatment of nonlinear conformable time-fractional Boussinesq equations by three integration methods. Optical and Quantum Electronics, 50(4), 1-31. https://doi.org/10.1007/s11082-017-1268-0

[4] Haq, S., & Ishaq, M. (2014). Solution of coupled Whitham-Broer-Kaup equations using optimal homotopy asymptotic method. Ocean Engineering, 84, 81-88. https://doi.org/10.1016/j.oceaneng.2014.03.031

[5] Manafian, J., & Dehghan, M. (2015). Optical solitons with Biswas-Milovic equation for Kerr law nonlinearity. European Physical Journal Plus, 130, 1112. https://doi.org/10.1140/epjp/i2015-15061-1

[6] Ray, S. S. (2015). A novel method for traveling wave solutions of fractional Whitham-Broer-Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water. Mathematical Methods in the Applied Sciences, 38, 1352-1368. https://doi.org/10.1002/mma.3151

[7] Wang, L., & Chen, X. (2015). Approximate analytical solutions of time fractional Whitham-Broer-Kaup equations by a residual power series method. Entropy, 17, 6519-6533. https://doi.org/10.3390/e17096519

[8] Xie, F., Yan, Z., & Zhang, H. (2001). Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations. Physics Letters A, 285, 76-80. https://doi.org/10.1016/S0375-9601(01)00333-4

[9] Dubey, R. S., Belgacem, F. B. M., & Goswami, P. (2016). Homotopy perturbation approximate solutions for Bergman’s minimal blood glucose-insulin model. Fractal Geometry and Nonlinear Analysis in Medicine and Biology, 2(3). https://doi.org/10.15761/FGNAMB.1000140

[10] Gour, M. M., Yadav, L. K., Purohit, S. D., & Suthar, D. L. (2023). Homotopy decomposition method to analysis fractional hepatitis B virus infection model. Applied Mathematics in Science and Engineering, 31(1). https://doi.org/10.1080/27690911.2023.2260075

[11] Veeresha, P., & Prakasha, D. G. (2020). An efficient technique for two-dimensional fractional order biological population model. International Journal of Modeling, Simulation, and Scientific Computing, 11, 2050005. https://doi.org/10.1142/S1793962320500051

[12] Sheergojri, A., Iqbal, P., Agarwal, P., & Ozdemir, N. (2022). Uncertainty-based Gompertz growth model for tumor population and its numerical analysis. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12, 137-150. https://doi.org/10.11121/ijocta.2022.1208

[13] Nasrolahpour, H. (2013). A note on fractional electrodynamics. Communications in Nonlinear Science and Numerical Simulation, 18, 2589-2593. https://doi.org/10.1016/j.cnsns.2013.01.005

[14] Drapaca, C. S., & Sivaloganathan, S. (2012). A fractional model of continuum mechanics. Journal of Elasticity, 107, 105-123. https://doi.org/10.1007/s10659-011-9346-1

[15] Kumar, D., Seadwy, A. R., & Joarder, A. K. (2018). Modified kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chinese Journal of Physics, 56(1), 75-85. https://doi.org/10.1016/j.cjph.2017.11.020

[16] Baleanu, D., Wu, G. C., & Zeng, S. D. (2017). Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos, Solitons & Fractals, 102, 99-105. https://doi.org/10.1016/j.chaos.2017.02.007

[17] Demirtas, M., & Ahmad, F. (2023). Fractional fuzzy PI controller using particle swarm optimization to improve power factor by boost converter, An International Journal of Optimization and Control:Theories & Applications (IJOCTA), 13(2), 205-213. https://doi.org/10.11121/ijocta.2023.1260

[18] Tajadodi, H., Jafari, H., & Ncube, M. N. (2022). Genocchi polynomials as a tool for solving a class of fractional optimal control problems, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(2), 160-168. https://doi.org/10.11121/ijocta.2022.1263

[19] Uzun, P. Y., Uzun, K., & Koca, I. (2023). The effect of fractional order mathematical modelling for examination of academic achievement in schools with stochastic behaviors, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 13(2), 244-258. https://doi.org/10.11121/ijocta.2023.1366

[20] Sene, N. (2022). Theory and applications of new fractional-order chaotic system under Caputo operator. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 12(1), 20-38. https://doi.org/10.11121/ijocta.2022.1108

[21] Kazem, S. (2013). Exact solution of some linear fractional differential equations by Laplace transform. International Journal of Nonlinear Sciences, 16(1), 3-11.

[22] Kumar, S. (2014). A new analytical modelling for fractional telegraph equation via Laplace transform. Applied Mathematical Modelling, 38, 3154-3163. https://doi.org/10.1016/j.apm.2013.11.035

[23] Agarwal, G., Yadav, L.K., Albalawi, W., Abdel-Aty, A.H., Nisar, K.S., & Shefeeq, T. (2022). Two analytical approaches for space-and time-fractional coupled burger’s equations via Elzaki transform. Progress in Fractional Differentiation and Applications, 8(1), 177-190. https://doi.org/10.18576/pfda/080111

[24] Neamaty, A., Agheli, B., & Darzi, R. (2016). Applications of homotopy perturbation method and Elzaki transform for solving nonlinear partial differential equations of fractional order. Journal of Nonlinear Evolution Equations and Applications, 6, 91-104.

[25] Yang, X. J. (2016). A new integral transform method for solving steady heat-transfer problem. Thermal Science, 20, 639-642. https://doi.org/10.2298/TSCI16S3639Y

[26] Aboodh, K.S. (2013). The new integral transform ”Aboodh Transform”. Global Journal of Pure and Applied Mathematics, 9(1), 35-43.

[27] Kiliçman, A., & Gadain, H. E. (2010). On the applications of Laplace and Sumudu transforms. Journal of the Franklin Institute, 347, 848-862. https://doi.org/10.1016/j.jfranklin.2010.03.008

[28] Guo, S. M., Mei, L. Q., Li, Y., & Sun, Y. F. (2012). The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics. Physics Letters A, 376, 407-411. https://doi.org/10.1016/j.physleta.2011.10.056

[29] Whitham, G. B. (1967). Variational methods and applications to water waves. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 299, 6-25. https://doi.org/10.1098/rspa.1967.0119

[30] Broer, L. J. F. (1975). Approximate equations for long water waves. Applied Scientific Research, 31, 377-395. https://doi.org/10.1007/BF00418048

[31] Kaup, D. J. (1975). A higher-order water-wave equation and the method for solving it. Progress of Theoretical Physics, 54, 396-408. https://doi.org/10.1143/PTP.54.396

[32] Liao, S. J. (1992). The proposed homotopy analysis technique for the solution of nonlinear problems (PhD thesis). Shanghai Jiao Tong University, Shanghai.

[33] Liao, S. J. (1992). An approximate solution technique not depending on small parameters: A special example. International Journal of Non-Linear Mechanics, 30(3), 371-380. https://doi.org/10.1016/0020-7462(94)00054-E

[34] El-Sayed, S. M., & Kaya, D. (2005). Exact and numerical travelling wave solutions of Whitham-Broer-Kaup equations. Applied Mathematics and Computation, 167, 1339-1349. https://doi.org/10.1016/j.amc.2004.08.012

[35] Rafei, M., & Daniali, H. (2007). Application of the variational iteration method to the Whitham-Broer-Kaup equations. Computers & Mathematics with Applications, 54, 1079-1085. https://doi.org/10.1016/j.camwa.2006.12.054

[36] Nawaz, R., Kumam, P., Farid, S., Shutaywi, M., Shah, Z., & Deebani, W. (2020). Application of new iterative method to time-fractional Whitham-Broer-Kaup equations. Frontiers in Physics, 8, 1-10. https://doi.org/10.3389/fphy.2020.00104

[37] Veeresha, P., Prakasha, D. G., Qurashi, M. A., & Baleanu, D. (2019). A reliable technique for fractional modified Boussinesq and approximate long wave equations. Advances in Difference Equations, 2019, 253-323. https://doi.org/10.1186/s13662-019-2185-2

[38] Yasmin, H. (2022). Numerical analysis of time-fractional Whitham-Broer-Kaup equations with exponential-decay kernel. Fractal and Fractional, 6(3), 142. https://doi.org/10.3390/fractalfraction6030142

[39] Yadav, L. K., Agarwal, G., Gour, M. M., & Kumari, M. (2023). Analytical approach to study weakly nonlocal fractional Schrödinger equation via novel transform. International Journal of Dynamics and Control, 12, 271-282. https://doi.org/10.1007/s40435-023-01246-x

[40] Maitama, S., & Zhao, W. (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. International Journal of Analysis and Applications, 17(2), 167-190. https://doi.org/10.28924/2291-8639-17-2019-167