In this paper we construct high order numerical methods for solving third and fourth orders nonlinear functional differential equations (FDE). They are based on the discretization of iterative methods on continuous level with the use of the trapezoidal quadrature formulas with corrections. Depending on the number of terms in the corrections we obtain methods of $O(h^4)$ and $O(h^6)$ accuracy. Some numerical experiments demonstrate the validity of the obtained theoretical results. The approach used here for the third and fourth orders nonlinear functional differential equations can be applied to functional differential equations of any orders.

翻译：本文针对三阶和四阶非线性函数微分方程构建了高阶数值求解方法。这些方法基于连续层面迭代方法的离散化，并采用了带修正项的梯形求积公式。根据修正项的数量，我们获得了精度为$O(h^4)$和$O(h^6)$的数值方法。数值实验验证了所得理论结果的有效性。本文针对三阶和四阶非线性函数微分方程所采用的方法，可推广应用于任意阶数的函数微分方程。