Abstract
In this paper, we employ the Laplace-residual power series technique to present an analytical approximation of the solutions of the time-fractional nonlinear water wave partial differential equation via Caputo fractional derivative with different initial value conditions. The importance of this study lies in providing solutions identical to the previous results of the mentioned equation, which confirms the efficiency of the old and new solutions. In addition, this method avoids using the fractional derivative during solution construction due to its disappearance in the Laplace space. To show the effectiveness and simplicity of our technique, numerical and graphical results are introduced and compared with the exact and the approximate Laplace-Homotopy solutions. The results suggest that the sub-figures are almost identical and confirm the vigorous agreement between the exact and the approximate Laplace-residual power series solutions. Finally, the behavior of the solutions to the problem is studied at different values of alpha.