Ideal and physical barrier problems for non-linear systems driven by normal and Poissonian white noise via path integral method

•Compute the probability density function for non-linear barrier problems via path integral method.•Unified treatment of ideal and physical barriers.•Applicable to average level crossing problem, first passage problems, vibro-impact problems and elastic-plastic oscillators.•Treat normal and Poissonian white noise with the same computational procedure.•Achieve highly accurate results verified by extensive Monte-Carlo simulation.

In this paper, the probability density evolution of Markov processes is analyzed for a class of barrier problems specified in terms of certain boundary conditions. The standard case of computing the probability density of the response is associated with natural boundary conditions, and the first passage problem is associated with absorbing boundaries. In contrast, herein we consider the more general case of partially reflecting boundaries and the effect of these boundaries on the probability density of the response. In fact, both standard cases can be considered special cases of the general problem. We provide solutions by means of the path integral method for half- and single-degree-of-freedom systems for both normal and Poissonian white noise. Emphasis is put on the considerations of the yielding barrier which is expressed in terms of non-reflecting (but not absorbing) boundary conditions. Comparison with Monte Carlo simulation demonstrates the excellent accuracy of the proposed method.