The Symmetry Method for Fractional Airy Equation

ABSTRACT

In this article, we propose a symmetry‐based analytical framework for solving time‐fractional Airy‐type partial differential equations (PDEs), a class of nonlocal evolution equations for which exact solutions are generally unavailable. By combining Lie symmetry analysis with Erdélyi–Kober fractional operators, the original fractional PDEs are reduced to fractional ordinary differential equations (ODEs) in similarity variables. Formal power‐series solutions are then constructed, with coefficients determined recursively. Two representative models are studied to illustrate the effectiveness of the method. Appropriate initial conditions are incorporated to determine the leading coefficients and ensure consistency of the solutions. Furthermore, truncation error estimates are established to assess the accuracy of the series approximations, and numerical examples are provided to validate the theoretical results. The proposed approach offers an effective and systematic tool for analyzing time‐fractional Airy‐type equations and can be extended to other classes of fractional differential equations.