A class of polynomial approximation methods to second-order delay differential equations

This paper studies special high-order methods to numerically solve special second-order ordinary differential equations and second-order differential equations with constant delay, which are based on vector field expansion along the shifted and scaled Legendre polynomial orthonormal basis. In the framework of method design, we obtain the second-order perturbation results by transforming the second-order differential equations into first-order ones, and the error accuracy estimation of numerical solutions is achieved by using perturbation results for the problem under consideration. In addition, some numerical tests on specific second-order delay Hamiltonian systems are shown, the purpose of which is to verify the effectiveness of the theoretical findings.