Analytical solutions of time-fractional non-linear model Clannish RandomWalker’s Parabolic equation and its sensitivity

In this article, the exact travelling wave solutions for the nonlinear time-fractional Clannish Random Walker’s Parabolic equation are discussed. This study establishes the new extended direct algebraic method in which periodic, bright, multiple U-shaped bright and kink-type wave solitons are obtained with exact solutions offered by the mixed hyperbolic and trigonometry solutions, mixed periodic and periodic solutions, plane solutions, shock solutions, mixed trigonometric solutions, mixed singular solutions, mixed shock single solutions, complex solitary singular solutions, shock solutions and shock wave solutions. The obtained solutions of the non-linear time-fractional Clannish Random Walker’s Parabolic equation model are graphically presented for different values of the involved parameters using Wolfram Mathematica software. The propagating behaviours are visualised through 3D, contour, and 2D surface plots to illustrate the influence of key parameters on the solution profiles. The time-fractional derivative introduces a memory effect into the model, making it more suitable for describing real-world physical processes that involve hereditary and nonlocal behaviour. The presence of novel soliton structures, such as multiple U-shaped solitons and bright-shaped solitons, further highlights the novelty and complexity of the model’s dynamics. In this study, a planar dynamical system is constructed from the proposed model to carry out a sensitivity analysis of initial conditions. This transformation enables the investigation of how small variations in the initial values influence the system’s long-term behaviour. The proposed method proves to be efficient, reliable, and broadly applicable for generating new analytical solutions to both integer and non-integer-order differential equations arising in mathematical physics and engineering.