Differential Transform Method for Linear and Nonlinear Differential Equations: Convergence, Error Analysis, and Comparative Validation

Abstract

The Differential Transform Method (DTM) is a semi-analytical series technique for constructing approximated series solutions of ordinary differential equations without evaluating high-order derivatives as in the Taylor procedure. This study presents a combined assessment of DTM for linear and nonlinear initial-value problems, with particular emphasis on convergence, truncation, and quantitative error evaluation. In addition to comparisons with exact solutions, the revised analysis introduces an analytic truncation-error bound for solutions that are analytic near the expansion point, an explicit stopping criterion, relative and maximum absolute error measures, and an order-normalized comparison with HAM, HPM, and OHAM for a common test problem. A nonlinear Riccati problem with a finite radius of convergence is also included to examine strength and to illustrate limitation of local power-series approximations. The results show that DTM can produce highly precise approximations within its region of convergence, but its performance depends on truncation order, solution analyticity, and the distance from singularities. The study therefore, emphasizes reproducible accuracy assessment rather than an unconditional claim of superiority over other semi-analytical methods.