Application of the Fractional Reduced Differential Transform Method to aTime-Fractional Heat Equation

Abstract

This work deals with the numerical solution of a time-fractional heat equation where a Caputo fractional derivative of order 0  is used in place of the traditional first-order time derivative. This change improves the model's capacity to represent anomalous diffusion behavior and memory effects, which are frequently seen in intricate engineering and physical systems. Applying and evaluating the Fractional Reduced Differential Transform Method (FRDTM) to solve this fractional-order partial differential equation is the aim of this work. The Fractional Variational Iteration Method (FVIM) was used to validate the findings. For different fractional orders, namely, and the classical case where  with a known exact solution, two numerical examples were performed. The findings demonstrate that FRDTM offers extremely stable and accurate solutions that closely match the exact solution in the classical case. When it comes to capturing the change from rapid decay at lower fractional orders to more sustained solution profiles as the order increases, the FRDTM performs better than the FVIM. The differences between the two methods demonstrate FRDTM's superior convergence and accuracy across all cases considered. Finally, this study demonstrates the effectiveness of FRDTM as a reliable semi-analytical tool for solving fractional heat problems, and it contributes to advancing computational approaches for solving partial differential equations in science and engineering.