Fractional Adomian J‐Transform Method for Time‐Fractional Diffusion‐Wave Equations: Theory and Applications

In this research, we introduce the fractional Adomian ‐transform method , which functions as a robust hybrid analytical‐numerical framework. Diverging from traditional transform techniques, the leverages the specific scaling attributes of the ‐transform to streamline the inversion of nonlocal fractional operators. We establish a comprehensive mathematical structure by deriving original transform identities in Theorems and defining rigorous convergence criteria and error constraints in Theorems. Our comparative assessment indicates that the provides superior computational efficiency when managing power‐law kernels and enhances symbolic processing. The efficacy of this approach is confirmed through various fractional diffusion scenarios, showing enhanced convergence speed compared to standard and models.