In this study, a detailed analytical-numerical study of the complex modified Korteweg–De Vries (mKdV) model with truncated M-fractional derivative is carried out to investigate the effects of the fractional order on nonlinear wave propagation. The fractional partial differential equation is solved by an appropriate fractional traveling wave transformation, which transforms it into a nonlinear ordinary differential equation. Two very powerful analytical methods are then used: the modified sub-equation method and the Kumar–Malik method, which give the exact closed-form solutions. The obtained semi-analytical numerical approximations are then obtained from the Differential Transformation Method (DTM). Bright and dark solitons, kink-type waves, periodic and rational solutions, exponential solutions, and Jacobi elliptic functions are found for a variety of parametric regimes. Explicit compatibility conditions and parametric constraints, which control the amplitude, width, and propagation, are derived. The DTM approximations are found to converge to the exact solutions with good accuracy, and the absolute errors are almost negligible, which validates the accuracy of the approximations and reliability of the solution. The three-dimensional visualizations of surface plots, two-dimensional profiles, and contour visualization further illustrate the dispersive dynamics and stability properties. Significance: This study shows that the truncated M-fractional derivative is a good operator to model memory-dependent nonlinear wave propagation. A new precise solution and reliable validation methods have been obtained for high-dimensional fractional nonlinear evolution equations in the hybrid analytical-numerical framework, which can be useful in plasma physics, nonlinear optics, and complex media. The present study contains restrictions for constant coefficients, a specific parametric regime, one fractional derivative definition, and experimental validation is not included. Future directions are limitations on constant coefficients, specific parametric regimes, one fractional derivative definition, and experimental validation is not included. The approach is to be extended in the future to variable coefficients, other fractional operators (Caputo, Riemann–Liouville), and to higher-order nonlinearities, and then to be experimentally tested in optical or plasma systems.
Aljethi et al. (Wed,) studied this question.