Abstract: 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.

Keywords: Caputo fractional derivative, Laplace transform, Laplace-residual power series method, the nonlinear time-fractional Whitham–Broer–Kaup equations

Pages: 694-706