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

Pisamai Kittipoom () and Jessada Tanthanuch
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Pisamai Kittipoom: Division of Computational Science, Faculty of Science, Prince of Songkla University, Songkhla 90110, Thailand
Jessada Tanthanuch: School of Mathematical Sciences and Geoinformatics, Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand

Mathematics, 2025, vol. 13, issue 22, 1-16

Abstract: 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.

Keywords: fractional differential equations; series solutions; residual power series method; general Laplace-like transforms; transform invariance (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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