Beyond Numerical Discretization: The Differential Transform Method as an Efficient Framework for Solving the Duffing Equation

The paper presents a comparative analysis of the Differential Transform Method (DTM) with respect to classical numerical approaches, such as the Euler and Runge–Kutta methods, using the nonlinear Duffing equation as a representative example. This equation, being a prototypical model of dynamical systems exhibiting chaotic behavior, provides a demanding test environment for techniques used to approximate solutions of ordinary differential equations. The aim of the study is to assess the accuracy, stability, and computational efficiency of the considered methods as functions of the system parameters. The DTM approach, based on differential transform and a series representation of the solution, was compared with classical discretization schemes. In the DTM framework, the symmetry of the system is not imposed explicitly, but emerges from the initial conditions and the recursive structure used to determine the series coefficients. However, the preservation of this symmetry may be disrupted, leading to asymmetry due to truncation of the series expansion and the propagation of numerical errors. The obtained results indicate that DTM can serve as a competitive alternative to conventional methods, particularly in short-term simulations of nonlinear dynamical systems, offering high accuracy at a relatively low computational cost.