In this study, a reliable implicit finite difference method based on the modified trapezoidal quadrature rule, backward Euler differences, nonstandard central approximations, and the Hadamard finite-part integral is being considered to solve a viscous asymptotical model named as fractional Kakutani–Matsuuchi water wave model. The fractional derivative is used in the Riemann–Liouville sense. Based on the properties of Brouwer's fixed-point theorem, the existence, uniqueness, convergence, and stability of the proposed method are proved. Furthermore, we show that the global convergence order of the method in maximum norm is O ( 휏 , h min { 훽 ,3 − 훼 } ), where 0 <훼 ≤ 1and 훽> 0 are the order of fractional derivative and the Lipschitz constant. Also, 휏 and h are the time step and space step, respectively. Finally, several examples are used to illustrate the accuracy and performance of the method.
