The Asad Correctional Power Series Method: A Novel Approach to Solving Fractional Differential Equations

Abstract

This paper introduces the Asad Correctional Power Series Method (ACPS), a novel and groundbreaking approach designed to simplify and optimize the solution of fractional differential equations. The ACPS combines algebraic manipulation with iterative refinement to achieve greater accuracy and computational efficiency than mainstream methods. By incorporating principles from both fractional calculus and functional analysis, the method offers a flexible framework capable of addressing a wide range of fractional equations, from linear to highly nonlinear cases. Additionally, a representative counterexample is provided to indicate that the conformable fractional derivative does not fulfill the mathematical criteria for a valid definition of fractional differentiation. The Asad Correctional Power Series (ACPS) method is employed to construct an analytic solution of the fractional SIR model in the form of a rapidly convergent power series. Its performance is validated through comparisons with the classical fourth-order Runge Kutta method, where both numerical and graphical analyses corroborate the method's precision and efficiency. The application of ACPS to the fractional epidemic model highlights its ability to capture memory and hereditary effects, offering more realistic insights into disease transmission dynamics than integer-order models. These findings demonstrate that ACPS can serve as a useful tool for solving fractional differential equations arising in real world applications