A two-step symplectic dimensional perturbation series method for ordinary differential equations with initial conditions

Ordinary differential equations (ODEs) are fundamental in modeling various phenomena across disciplines such as physics, engineering, and economics. This paper presents a novel symplectic method that employs a two-step perturbation process for solving physical problems where governing equations are transformed into first-order ODEs with initial conditions. Due to the superior computational stability of the symplectic algorithm, it is integrated into the solution of ODEs in physical problems and is combined with the perturbation series method. For even-dimensional ODEs, the Hamiltonian form of the equations is derived using the perturbation method, and the symplectic algorithm is then applied in the first perturbation step. For odd-dimensional ODEs, the dimension is adjusted—either increased or decreased—via the perturbation series method before employing the symplectic method, constituting the second perturbation. This approach introduces a new perspective for solving ODEs in physical contexts. The efficacy and precision of the proposed method are demonstrated through two numerical examples. The results confirm the method’s effectiveness and accuracy. (Figure presented.)