ABSTRACT Throughout this work, we implement a novel method to provide exact and approximate solutions, called the ‐transform Adomian decomposition method (). This method is based on the ‐transform and the Adomian decomposition method (), which can be used to find exact solutions for linear fractional ordinary differential equations (LFODEs) and nonlinear fractional ordinary differential equations (NLFODEs) and linear partial differential equations (LFPDEs) and nonlinear partial differential equations (NLFPDEs). Clearly, from the results obtained, the new scheme proposed in this work is highly accurate and efficient, and the results have shown how powerful and effective this method is and how straightforward it is for solving many types of fractional differential equations. The proposed transform technique is used to explore various aspects of the fractional Caputo and Riemann–Liouville derivatives, including their properties. We also present detailed proofs of new theorems related to the inverse fractional ‐transform method (). We carried out the majority of the numerical calculations using Mathematica 13.
Obeidat et al. (Wed,) studied this question.