Application of Reduced Differential Transform Method to Solve Linear, Non-Linear Convection-Diffusion and Reaction-Diffusion Problems

Convection-diffusion and reaction-diffusion equations are fundamental in describing various physical phenomena, yet their solution, particularly for nonlinear cases, often presents significant mathematical challenges. This study investigates the application of the Reduced Differential Transform Method (RDTM) to obtain analytical solutions for both linear and nonlinear convection-diffusion and reaction-diffusion problems. The RDTM, derived from power series expansion, was systematically applied to four illustrative examples: two linear convection-diffusion equations and two nonlinear reaction-diffusion equations, with initial conditions sourced from existing literature. The core of the method involves an iterative procedure to determine the t-dimensional spectrum functions, which are then used to construct the series solution. The RDTM successfully yielded analytical solutions in convergent series form for all problems considered. For the convection-diffusion problems, the RDTM solutions were found to be identical to the exact solutions obtained via the homotopy perturbation transform method. Similarly, for the reaction-diffusion problems, the solutions precisely matched those derived from the differential transformed method. The accuracy was further validated by graphical analysis of the absolute errors between the RDTM and exact solutions. The findings demonstrate that the RDTM is a powerful, efficient, and direct analytical technique. Notably, it does not require common complex procedures such as linearization, discretization, or perturbation, simplifying the solution process significantly. This research affirms the RDTM as a reliable tool for solving a range of linear and nonlinear partial differential equations, offering a valuable alternative for scientists and engineers.