This study presents a comprehensive and integrative examination of modern numerical frameworks for solving ordinary, partial, fractional, and epidemiological differential systems using spectral and differential transform based methodologies. Drawing strictly on the referenced body of work, the article investigates how polynomial based spectral schemes, differential transformation methods, and adaptive time stepping strategies collectively form a powerful computational ecosystem for addressing nonlinear and complex dynamical models. Differential equations are fundamental tools for describing heat conduction, epidemic propagation, biological interactions, and wave phenomena, yet the inherent nonlinearity, fractional order behavior, and multi dimensionality of many real world systems continue to challenge traditional numerical solvers. As demonstrated in computational immunology models, ordinary differential equations provide a flexible representation of interacting biological components, but their accuracy depends heavily on the stability and convergence of the underlying numerical method used for simulation (Hoops et al., 2016). Spectral based schemes, including Gegenbauer, Lucas, Legendre, and Chebyshev polynomial families, have emerged as high precision approximators that can dramatically reduce computational cost while maintaining global accuracy for smooth solutions (Sayed et al., 2024; Youssri et al., 2023). In parallel, differential transformation methods have proven to be efficient semi analytical techniques that convert differential models into algebraic recurrence systems, enabling the study of nonlinear epidemic, chaotic, and damped wave models with strong stability and fast convergence (Ahmad et al., 2017; Odibat et al., 2010).

This article synthesizes these two dominant traditions into a unified narrative that highlights their complementary strengths. By examining how polynomial based spectral algorithms manage spatial and fractional complexities while differential transforms manage temporal and nonlinear expansions, this study demonstrates how advanced numerical solvers can be structured to address the limitations of classical finite difference and Runge Kutta methods. The literature further shows that modified polynomial bases and alleviated spectral shifts enhance numerical conditioning, making these approaches suitable for boundary value problems, time fractional equations, and coupled nonlinear systems (Sayed et al., 2024; Abd Elhameed et al., 2023). At the same time, multistep differential transform schemes overcome the local convergence limitation of classical transforms and provide reliable long time simulations for chaotic and epidemiological models (Odibat et al., 2010).