用矩阵连分法求解非平稳的FPK方程

FPK方程是求解系统响应概率结构的关键方程,目前其解析解还只限于线性系统响应和少数几种特殊情况非线性系统的平稳响应过程。本文以正交函数展开法为基础,选择了FPK方程的部分算子作为解的展开函数系,同时根据系数方程最小耦合关系式的初始值问题和特征值问题及其两者的关系,提出了求解振动系统转移概率密度函数的近似方法。最后,以Duffin方程为例给出了非线性系统非平稳过程FPK方程的求解过程。计算结果表明,矩阵连分法收敛快、方法规则且精度高,能适用于各种非线性势场。

Solution to FPK Equation by Matrix Continued Fraction Method

The Fokker-Planck-Kolmogorov equation plays an important role infinding the probabilistic structures of systems response. The FPK equationwill be exactly solvable only in the linear systems and in a few special cases ofnonlinear systems in stationary state. In this paper an approximate method offinding the transition probability density is proposed. On the basis of theexpansion of orthogonal function, the part-operator of the FPK equation isselected as an expansion function of solution, and the relationship between theinitial and eigenvalue problems of the least coupling coefficients is established.Finally, as an example, the procedure for solving the FPK equation of a non-linear system-Duffin's equation-in nonstationary state is presented. Itsresults show that the Matrix Continued-Fraction Method proposed in this paperis of fast covergence, high accuracy and regular in calculating patterns. Thusit can be available in various nonlinear potential fields.