An efficient technique to study of time fractional Whitham-Broer-Kaup equations

In this study, we derive the approximate analytical solution for the fractional coupled Whitham-Broer-Kaup (WBK) equations, a significant mathematical model for representing wave propagation in shallow water. The solution is obtained through the utilization of the q-homotopy analysis rho-Laplace transform method (q-HAL(rho)TM), a hybrid approach combining rho-Laplace transformation and the homotopy analysis method. Homotopy polynomials are employed to address nonlinear terms, and the introduced algorithm incorporates the auxiliary parameter h to regulate and fine-tune the convergence region of the resulting series solution. Comparative numerical analyses are conducted with outcomes from the Adomian decomposition method (ADM), variational iteration method (VIM), and optimal homotopy asymptotic method (OHAM), demonstrating the superior accuracy of the proposed method. The method's novelty and straightforward implementation establish it as a reliable and efficient analytical technique for solving both linear and nonlinear fractional partial differential equations.