  In this paper, an efficient technique is employed to study the modified
Boussinesq and approximate long wave equations of the Caputo fractional time
derivative, namely q-homotopy analysis transform method. These equations are
playing a vital rule in describing the properties of shallow water waves
through distinct dispersion relation. The convergence analysis and error
analysis has been presented in the present investigation for the future scheme.
We illustrate two examples to demonstrate the leverage and effectiveness of the
proposed scheme, and the error analysis has been discussed to verify the
accuracy. The numerical simulation has been conducted to ensure the exactness
of the future technique. The obtained numerical and graphical results are
divulge, the proposed scheme is computationally very accurate and
straightforward to study and find the solution for fractional coupled nonlinear
complex phenomena arised in science and technology.
