Universal Coefficients Formula for the Residual Power Series Method with General Integral Transform

Abstract: This paper introduces a novel approach to address inherent limitations in the Residual Power Series (RPS) method and its variants with Laplace-like transforms when applied to solving time-fractional differential equations. Existing methods, while successful, often require computationally expensive calculations for the coefficients of the series solution. To overcome this limitation, we propose a new framework for the RPS method that utilizes a general integral transform. This framework incorporates an explicit formula for calculating coefficients, thereby eliminating repetitive computations and streamlining the solution process. Moreover, it offers a universally applicable approach, remaining compatible with various RPS methods that employ Laplace-like transform variants.