Numerical Solutions of Time-Fractional N-W-S and Burger’s Equations Using the Tarig Projected Differential Transform Method (TPDTM)

Abstract

Solving time-fractional nonlinear equations, such as the Newell-Whitehead-Segel (N-W-S) and Burger's equations, is inherently complex due to the intricacies of fractional calculus and the limitations of current numerical and analytical techniques. This study introduces the Tarig Projected Differential Transform Method (TPDTM), a hybrid approach that offers a novel solution to time-fractional linear and nonlinear partial differential equations (PDEs) without the need for linearization, perturbation, or variable discretization. TPDTM stands out as a simple yet powerful method, offering remarkable accuracy, computational efficiency, and ease of implementation. In comparative analysis with the Finite Difference Method (FDM) and Laplace Adomian Decomposition Method (LADM), TPDTM demonstrates superior performance, particularly in its handling of Adomian polynomials, all while preserving stability and precision. With its rapid convergence and versatility, TPDTM proves to be a robust tool for tackling complex fractional PDEs, making it highly valuable for applied mathematics and physics. Looking forward, expanding TPDTM's application to coupled fractional systems and multi-scale problems will not only enhance its theoretical depth but also open exciting new possibilities for breakthroughs in engineering, physics, and computational modeling.