Optimized technique and dynamical behaviors of fractional Lax and Caudrey–Dodd–Gibbon models modelized by the Caputo fractional derivative

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Kortweg-de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey-Dodd-Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error's analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approash is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.