General Residual Power Series Method: Explicit Coefficient Derivation and Unified Laplace-like Transform Approach for Fractional PDEs

This work introduces the General Residual Power Series Method (GRPSM) as a unified analytical framework encompassing the conventional Residual Power Series Method (RPSM) and its Laplace-like transform variants. By deriving a universal coefficient formula, the GRPSM clarifies the recursive structure of residual-based series solutions and removes the need for repeated limit evaluations across different transform formulations. It is shown that all Laplace-like RPSM variants yield identical coefficient recursions, indicating that their differences stem only from algebraic reparametrizations of the same underlying mechanism. This analytical invariance reveals that the classical RPSM already represents the simplest and most direct form of the unified approach, providing a clear theoretical basis for transform-based extensions in time-fractional and related differential equations.