Combining conformable operator with Elzaki Adomian decomposition and reduced differential transform for time-fractional diffusion equations

The present paper deals with the numerical solutions of time-fractional diffusion equations that play important roles in modeling anomalous diffusion processes in many physical systems. We explore two semi-analytical methods, the Conformable Elzaki Adomian Decomposition Method (CEADM) and the Conformable Fractional Reduced Differential Transform Method (CFRDTM) which are formulated in the conformable operator framework. The conformable operator also offers a more generalized environment of working with fractional derivatives, simplifying the process of working with terms of the fractional order and enhancing the transparency of the solution process. The applicability of the methods is illustrated using nonlinear time-fractional diffusion problems, the analytical and numerical approximations of which are obtained. The convergence, accuracy, and the efficiency of both CEADM and CFRDTM are compared to the exact solution to test the level of convergence, accuracy, and computational efficiency. In addition, the outcome of the suggested techniques is compared to that of a semi-analytical method, the Formable Transform Adomian Decomposition Method (FTADM) to show the comparative superiority of the suggested techniques. It is found that the conformable operator-based methods offer accurate and effective approximations and the useful insight into the numerical solution of the equations of fractional diffusion, thus forming a basis of its application to more complex real-world systems.