NONE 20251217120144957 wseas/wseas content-registration-form+ja@crossref.org content-registration-form WSEAS TRANSACTIONS ON MATHEMATICS 1109-2769 2224-2880 Fares Bekhouche Laboratory of Mathematics and Artificial Intelligence, Department of Mathematics, Abbes Laghrour University, Khenchela 40000, ALGERIA Iqbal M. Batiha Department of Mathematics, Al Zaytoonah University of JORDAN, Amman, 11733, JORDAN Ouidad Boulakour Laboratory of Dynamical Systems and Control, Department of Mathematics and Computer Science, University of Oum El Bouaghi, Oum El Bouaghi 04000, ALGERIA Radwan M. Batyha Department of Computer Science Applied Science Private University Amman 11931, JORDAN Ahmed Bouchenak Department of Mathematics, University Mustapha Stambouli of Mascara, Mascara, 29000, ALGERIA Shaher Momani Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, 346, UAE Due to the limitations of the Laplace transform in solving certain classes of nonlinear partial differential equations, this study introduces a novel numerical technique that combines the Laplace transform with the residual power series approach to effectively handle the nonlinear time-fractional Whitham–Broer–Kaup (WBK) equations. The newly developed method, called the Laplace-Residual Power Series Method (L-RPSM), integrates the strengths of both techniques to overcome the restrictions encountered when each method is applied individually. In this approach, the fractional derivative is defined in the sense of the Caputo derivative. To demonstrate the applicability and efficiency of the proposed method, the nonlinear time-fractional WBK equations are solved step by step, and the obtained approximate solutions are analyzed in detail. The performance of the L-RPSM is validated through a numerical application, where the results are presented in tables, two-dimensional plots, and three-dimensional graphs. The results show that the L-RPSM is not only robust and efficient but also straightforward to implement. It provides rapidly convergent series solutions with very high accuracy, confirming its capability to approximate the exact solutions of the WBK equations. In addition, the flexibility of the method allows its application to various fractional partial differential equations, confirming its role as a reliable and versatile approach for analyzing nonlinear fractional systems. 12 17 2025 12 17 2025 694 69 https://creativecommons.org/licenses/by/4.0/deed.en_US 10.37394/23206.2025.24.69 https://wseas.com/journals/mathematics/2025/b405106-2269.pdf Barbero, G., Evangelista, L. R., Zola, R. S., Lenzi, E. K., & Scarfone, A. M. A brief review of fractional calculus as a tool for applications in physics: Adsorption phenomena and electrical impedance in complex fluids, Fractal and Fractional, vol. 8, no 7, 2024, pp. 369. https://doi.org/10.3390/fractalfract8070369 Elsamani, S. A., Mohmmed, A. A., Kogan, K. M., Salih, S. Y. M., & Ahmed, I. A. Fractional calculus applications in engineering insights into four-dimensional chaotic systems, J. Theory Math. Phys, vol. 3, no 5, 2024, pp. 75-84. Cheow, Y. H., Ng, K. H., Phang, C., & Ng, K. H. The application of fractional calculus in economic growth modelling: An approach based on regression analysis, Heliyon, vol. 10, no 15, 2024. https://doi.org/10.1016/j.heliyon.2024.e35379 Atangana, A., & Gómez‐Aguilar, J. F. Numerical approximation of Riemann‐Liouville definition of fractional derivative: from Riemann‐Liouville to Atangana‐Baleanu. Numerical Methods for Partial Differential Equations, vol. 34, no 5, 2017, pp. 1502-1523. https://doi.org/10.1002/num.22195 Liaqat, M. I., Akgül, A., & Bayram, M. Series and closed form solution of Caputo time-fractional wave and heat problems with the variable coefficients by a novel approach, Optical and Quantum Electronics, vol. 56, no 2, 2024, pp. 203. https://doi.org/10.1007/s11082-023-05751-3 Abro, K. A., & Atangana, A. A comparative analysis of electromechanical model of piezoelectric actuator through Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. Mathematical Methods in the Applied Sciences, vol. 43, no 17, 2020, pp. 9681-9691. https://doi.org/10.1002/mma.6638 El-Sayed, S. M., & Kaya, D. Exact and numerical traveling wave solutions of Whitham–Broer–Kaup equations, Applied Mathematics and Computation, vol. 167, no 2, 2005, pp. 1339-1349. https://doi.org/10.1016/j.amc.2004.08.012 Rafei, M., & Daniali, H. Application of the variational iteration method to the Whitham–Broer–Kaup equations, Computers & Mathematics with Applications, vol. 54, no 7-8, 2007, pp. 1079-1085. https://doi.org/10.1016/j.camwa.2006.12.054 Haq, S., & Ishaq, M. Solution of coupled Whitham–Broer–Kaup equations using optimal homotopy asymptotic method, Ocean engineering, vol. 84, 2014, pp. 81-88. https://doi.org/10.1016/j.oceaneng.2014.03.031 Hadhoud, A. R., & Rageh, A. A. Redefined Quintic B‐Spline Collocation Method to Solve the Time‐Fractional Whitham‐Broer‐Kaup Equations, Computational and Mathematical Methods vol. 2024, no 1, 2024, pp. 7326616. https://doi.org/10.1155/2024/7326616 Nawaz, R., Kumam, P., Farid, S., Shutaywi, M., Shah, Z., & Deebani, W. Application of new iterative method to time fractional Whitham–Broer–Kaup equations, Frontiers in Physics, vol. 8, 2020, pp. 104. https://doi.org/10.3389/fphy.2020.00104 Ismail, G. M., Abdl-Rahim, H. R., Ahmad, H., & Chu, Y. M. Fractional residual power series method for the analytical and approximate studies of fractional physical phenomena, Open Physics, vol. 18, no 1, 2020, pp. 799-805. https://doi.org/10.1515/phys-2020-0190 Eriqat, T., El-Ajou, A., Moa’ath, N. O., Al-Zhour, Z., & Momani, S. A new attractive analytic approach for solutions of linear and nonlinear neutral fractional pantograph equations, Chaos, Solitons & Fractals, vol. 138, 2020, pp. 109957. https://doi.org/10.1016/j.chaos.2020.109957 Aljarrah, H., Alaroud, M., Ishak, A., & Darus, M. Approximate solution of nonlinear time-fractional PDEs by Laplace residual power series method, Mathematics, vol. 10, no 12, 2022, pp. 1980. https://doi.org/10.3390/math10121980 Oqielat, M. A. N., Eriqat, T., Ogilat, O., El-Ajou, A., Alhazmi, S. E., & Al-Omari, S. Laplace-residual power series method for solving time-fractional reaction–diffusion model, Fractal and Acknowledgment Fractional, vol. 7, no 4, 2023, pp. 309. https://doi.org/10.3390/fractalfract7040309 Yadav, S., Vats, R. K., & Rao, A. Constructing the fractional series solutions for time-fractional K-dV equation using Laplace residual power series technique, Optical and Quantum Electronics, vol. 56, no 5, 2024, pp. 721. https://doi.org/10.1007/s11082-024-06412-9 A. S. Hussain, K. D. Pati, A. K. Atiyah, and M. A. Tashtoush. Rate of occurrence estimation in geometric processes with Maxwell distribution: A comparative study between artificial intelligence and classical methods. International Journal of Advances in Soft Computing and its Applications, vol. 17, no 1, 2025, pp. 1-15. DOI: 10.15849/IJASCA.250330.01 M. Berir. Analysis of the effect of white noise on the Halvorsen system of variable-order fractional derivatives using a novel numerical method. International Journal of Advances in Soft Computing and its Applications, vol. 16, no 3, 2024, pp. 294–306. DOI: 10.15849/IJASCA.241130.16 P. Singh, N. Zade, P. Priyadarshi, and A. Gupte. The application of machine learning and deep learning techniques for global energy utilization projection for ecologically responsible energy management. International Journal of Advances in Soft Computing and its Applications, vol. 17, no 1, 2025, pp. 49–66. https://doi.org/10.15849/ijasca.250330.04 H. Al-Taani, M. A. Hammad, O. Alomari, I. Bendib, and A. Ouannas. Finite-time control of the discrete Sel’kov–Schnakenberg model: Synchronization and simulations. AIP Advances, vol. 15, no 2, 2025, pp. 025325. https://doi.org/10.1063/5.0257304 Ali, A., Shah, K., & Khan, R. A. Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations, Alexandria Eng. J, vol. 57, no 3, 2018, pp. 1991-1998. https://doi.org/10.1016/j.aej.2017.04.012 Tenenbaum, M., & Pollard, H. Ordinary differential equations: an elementary textbook for students of mathematics, engineering, and the sciences, Courier Corporation, 1985. El-Ajou, A. Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach, The European Physical Journal Plus, vol. 136, no 2, 2021, pp. 229. https://doi.org/10.1140/epjp/s13360-020-01061-9