基于算子分裂法的Riccati方程的近似解

Approximate solution of the Riccati equation based on operator decomposition method

Riccati方程是一类形式上非常简单的微分方程。采用算子分裂法,假设方程的解为级数形式u=i=1∑∞ui,对于Riccati方程中较难处理的非线性项N(u),通过Adomian多项式给出表示N(u)=n=0∑∞Am。通过计算Adomian多项式A_n建立Riccati方程的解析近似解的表达式,并给出了具体的数值计算结果。结果表明:算子分裂法对于计算Riccati方程是非常有效的。

The Riccati equation is a kind of differential equation which is very simple in form. However, in general, it is difficult to present the analytic solution. In this work, the operator decomposition method is used to solve the Riccati equation. The present paper supposes that the solution to the Riccati equation is in the form of u=i=1∑∞ui. The nonlinear part N(u), which is difficult to solve, can be expressed in terms of the Adomian polynomials N(u)=n=0∑∞Am. By calculating the Adomian polynomials, the approximate solution can be obtained. Finally, the numerical example shows that the present method is highly effective for finding the approximate solution to the Riccati equation.