Advanced Analytical Approaches for Nonlinear Fractional PDEs Reduced Differential Transform Method and Elzaki Transform Homotopy Perturbation Method

This study presents a comparative analysis of two advanced analytical methods—the Elzaki Transform Homotopy Perturbation Method (ETHPM) and the Fractional Reduced Differential Transform Method (FRDTM)-for solving nonlinear fractional partial differential equations (FPDEs) arising in biological population dynamics. After establishing the mathematical foundations of fractional calculus, the Elzaki transform, and homotopy perturbation theory, we demonstrate the applicability of both methods to FPDEs modelling population growth and swarm behaviour. Our results reveal that ETHPM and FRDTM yield highly accurate approximate solutions, underscoring their efficacy as computational tools for complex biological systems. The study highlights the broader implications of these fractional-order models in ecology and population dynamics, bridging theoretical mathematics with practical applications in life sciences. Through systematic comparisons, we provide insights into the strengths and limitations of each method, offering valuable guidance for researchers working with nonlinear fractional systems in biological contexts.