基于精细积分技术的非线性动力学方程的同伦摄动法

将精细积分技术（PIM）和同伦摄动方法（HPM）相结合，给出了一种求解非线性动力学方程的新的渐近数值方法。采用精细积分法求解非线性问题时，需要将非线性项对时间参数按Taylor级数展开，在展开项少时，计算精度对时间步长敏感；随着展开项的增加，计算格式会变得越来越复杂。采用同伦摄动法，则具有相对筒单的计算格式，但计算精度较差，应用范围也限于低维非线性微分方程。将这两种方法相结合得到的新的渐近数值方法则同时具备了两者的优点，既使同伦摄动方法的应用范围推广到高维非线性动力学方程的求解，又使精细积分方法在求解非线性问题时具有较简单的计算格式。数值算例表明，该方法具有较高的数值精度和计算效率。

Homotopy perturbation method for nonlinear dynamic equations based on precise integration technology

A new asymptotic numerical method for nonlinear dynamic equations is proposed in this paper by combining the precise integration method(PIM) with the homotopy perturbation method(HPM).For solving nonlinear dynamic equations in PIM,the nonlinear term should be expanded in Taylor series to the time parameter.The computational accuracy is sensitive to the time step if the series is truncated at the first order or second order,and if the series is truncated at the higher order,the computational format will be more complex.Correspondingly,the format derived from the homotopy perturbation method is simpler,but its applicability is limited to one or two dimensional nonlinear differential equations and the computational accuracy is lower.The new asymptotic numerical method obtained by combining above two methods possesses all their merits,that is,not only extend the applicability of the homotopy perturbation method to high dimensional nonlinear dynamic equations,but also simplify the computational format of PIM in solving nonlinear problems.The numerical example shows that the numerical accuracy and the computational efficiency of the new method is higher.