Semi-Analytical Solution of a Two-Dimensional Time-Fractional Fisher’s Equation via the Homotopy Analysis Method

This paper presents a semi-analytical solution to the two-dimensional time-fractional Fisher’s equation using the Homotopy Analysis Method (HAM). The governing equation models reaction–diffusion processes with memory effects, incorporating the Caputo fractional derivative to account for anomalous temporal behavior. The HAM framework is constructed by first selecting an appropriate initial guess that satisfies both the initial and boundary conditions, followed by the recursive generation of higher-order approximations. The convergence of the series solution is controlled through the auxiliary parameter $\hbar$, whose optimal value is determined at each time level by enforcing consistency with boundary conditions. The analytical results are validated against numerical simulations, demonstrating excellent agreement. This study highlights the flexibility and efficiency of HAM in handling high-dimensional nonlinear fractional partial differential equations and provides a foundation for extending the method to more complex biological or ecological models.