ANALYTICAL SOLUTIONS OF THE GENERALIZED PROBABILITY DENSITY EVOLUTION EQUATION OF THREE CLASSES STOCHASTIC SYSTEMS

As a gradually improving and developed method, generalized probability density evolution equation(GDEE) provides a new methodology for the analysis and control of stochastic dynamic system.A variety of numerical methods such as finite difference method and meshfree method were introduced to solve the generalized probability density evolution equation.However, the analytical solution of the GDEE corresponding to typical stochastic systems is scarce relatively.In this paper, the explicit solutions of the GDEE corresponding to three classes of nonlinear stochastic systems including Van der Pol oscillator, Riccati system, and Helmholtz oscillator with random parameters are studied by using Lie group method.The results not only can be the benchmark of numerical methods, but also can provide more information for the further research.