Analytical Dynamics of Phase Separation with Memory: Solving the Fractional Allen–Cahn Equation via Laplace-Residual Series

This paper adapts a semi-analytical framework the Laplace-Residual Power Series Method (LRPSM) to solve the time-fractional Allen–Cahn equation under the Caputo derivative. While the classical Allen–Cahn model successfully describes phase separation, its fractional counterpart is essential for capturing sub-diffusive memory effects in complex heterogeneous materials. However, the interplay between the non-local fractional temporal operator and the cubic nonlinearity of the bistable double-well potential creates significant computational bottlenecks for conventional time-domain series solvers. The proposed approach projects the governing fractional partial differential equation into the Laplace domain, systematically replacing the computation of iterative fractional derivatives with the algebraic evaluation of asymptotic limits at infinity. Furthermore, the nonlinear cubic interactions are managed through Laplace-space convolution theorems. The structural convergence of this approach is evaluated against multi-scenario one-dimensional phase transitions. Graphical analyses, featuring 2D profile trajectories and 3D spatiotemporal surface mappings, visually illustrate the retarded interfacial propagation driven by fractional memory. Ultimately, this study presents the LRPSM as an applicable, continuous mathematical tool for approximating anomalous diffusion in the specific phase-field dynamics evaluated herein.