基于Fourier级数的时变周期系数Riccati微分方程精细积分

结合Fourier级数展开方法,本文提出了基于精细积分的时变周期系数Riccati微分方程求解高效算法.首先,利用Fourier级数展开方法将周期系统表示成三角级数形式,在一个积分步内使用精细积分方法得到对应Hamilton系统状态转移矩阵的表达式.然后,通过Riccati变换的方法,得到含有状态转移矩阵的时变周期系数Riccati微分方程解的递推格式.本文方法充分利用了方程本身的周期性特点,文中的数值算例表明算法具有计算效率高、结果可靠等优势.

Solving time-varying periodic coefficient Riccati differential equations via Fourier series and precise integration method

With Fourier series expansions, an efficient numerical algorithm is proposed for solving time-varying periodic coefficient Riccati differential equations. Firstly, periodic coefficients are expanded in terms of Fourier series, and the state transformation matrix of the associated Hamiltonian system is evaluated by the precise integration method. Then, by employing the Riccati transformation method, recursive formulae for time-varying periodic coefficient Riccati differential equations are derived, which consist of blocks of the state transformation matrix. At last, the efficiency and reliability of the periodic character based numerical method is demonstrated by two examples.