Lie Symmetries, Conservation Laws and Solutions for (4+1)-Dimensional Time Fractional KP Equation with Variable Coefficients in Fluid Mechanics

In recent years, high-dimensional fractional equations have gained prominence as a pivotal focus of interdisciplinary research spanning mathematical physics, fluid mechanics, and related fields. In this paper, we investigate a (4+1)-dimensional time-fractional Kadomtsev-Petviashvili (KP) equation with variable coefficients. We first derive the (4+1)-dimensional time-fractional KP equation with variable coefficients in the sense of the Riemann-Liouville fractional derivative using the semi-inverse and variational methods. The symmetries and conservation laws of this equation are analyzed through Lie symmetry analysis and a new conservation theorem, respectively. Finally, both exact and numerical solutions of the fractional-order equation are obtained using the Hirota bilinear method and the pseudo-spectral method. The effectiveness and reliability of the proposed approach are demonstrated by comparing the numerical solutions of the derived models with exact solutions in cases where such solutions are known.