Adaptive Spectral Homotopy Perturbation Method for Long-Time Integration of Fractional Partial Differential Equations: Rigorous Treatment of the Memory Effect (مجلة الفاروق للعلوم, 2026)
Fractional partial differential equations (FPDEs) are widely used to model systems with memory effects and anomalous diffusion, yet their analytical solutions are rarely available. This paper presents an Adaptive Spectral Homotopy Perturbation Method (A-SHPM) for the accurate and stable long-time solution of FPDEs. The proposed method extends the Spectral Homotopy Perturbation Method (SHPM) through systematic time-domain decomposition while rigorously preserving the memory effect of the Caputo fractional derivative across successive sub-intervals. The history term is evaluated numerically using Gauss–Lobatto quadrature based on approximate solutions from previous sub-intervals, eliminating the need for the exact solution. A convergence analysis based on the Banach Fixed Point Theorem proves that the history term preserves the contraction property of the iterative scheme, and explicit criteria for selecting sub-interval sizes are derived. Numerical results for time-fractional diffusion and reaction–diffusion equations demonstrate that A-SHPM maintains high accuracy and numerical stability for long-time integration up to (t=5.0), significantly extending the applicability of SHPM beyond its theoretical stability limit. At an error tolerance of (10^{-4}), the proposed method achieves accuracy comparable to finite difference methods with substantially lower computational cost.