非线性(时滞)动力学问题的一种新暂态时程积分解法

非线性微分方程很难求得精确解析解,数值方法是求解非线性问题的一种有效手段。针对非线性微分方程,提出一种新的暂态时程积分方法。在暂态时程积分过程中,将非线性项看做非齐次项,在瞬态区间起始时刻处进行Taylor展开,并结合Romberg数值积分进行计算。Taylor展开时,将系统状态方程连续引入到非线性项导数的求解过程中,可简单有效地计算高阶导数。在此基础上,对含有时滞的非线性微分方程数值解法进行了研究,将时滞项同样看做非齐次项,利用线性插值处理后,结合Romberg积分进行计算。实例计算结果表明,该方法对有无时滞的非线性微分方程,均可求得较高精度的数值解。

A Innovative Transient Time-integration Method for Nonlinear Dynamic Problems with or without Time-delay

It is difficult to obtain an exact analytical solution for nonlinear differential equations. Numerical methods are an effective way to solve nonlinear problems. In this paper, a innovative transient time-integration method is proposed for nonlinear differential equations. In the transient time-integration process, the nonlinear term was regarded as a non-homogeneous term. Taylor expansion was performed at the beginning of the transient interval. Romberg numerical integration was used for calculation. The system state equation was continuously introduced into the nonlinear term derivative, then the high-order derivative could be calculated simply and efficiently. On this basis, the numerical solution of nonlinear differential equations with time-delay was studied. The time-delay term was also treated as a non-homogeneous term. Linear interpolation and Romberg integral was used for calculation. The example show that the method can obtain a higher precision numerical solution for nonlinear differential equations with or without time delay.