An improved path integration method for nonlinear systems under Poisson white noise excitation Abstract   In order to overcome some unsatisfactory trends and limitations of the traditional path integration (PI) method for Poisson white noise, a novel PI method is proposed in this paper, which includes two improved schemes. The first one is a new Transition Probability Density Function (TPDF) approximation which considers the randomness of the impulse happening time during each time intervals. The second one is a transformation of Chapmanxe2x80x93Kolmogorov (CK) equation by a variable substitution instead of directly using it, whose numerical calculation is based on the back stepping Rungexe2x80x93Kutta scheme and the triangulationbased interpolation. Monte Carlo Simulations (MCS) are utilized to measure the accuracy of the improved algorithm with three illustrative nonlinear systems. The results show that compared with the traditional PI method, the improved PI method can give a more accurate description of the TPDF values, and provide more precise stationary Probability Density Function (PDF) results whenever the mean arrival rate is large or small. The improved algorithm has a wider range of choices in time interval values to maintain the accuracy of stationary PDF results. Besides, it is discovered that cubic interpolation deserves to be applied in the improved PI method more than linear and natural interpolations.

Second order linear differential equations with analytic uncertainties: stochastic analysis via the computation of the probability density function Abstract   This paper concerns the analysis of random second order linear differential equations. Usually, solving these equations consists of computing the first statistics of the response process, and that task has been an essential goal in the literature. A more ambitious objective is the computation of the solution probability density function. We present advances on these two aspects in the case of general random nonautonomous second order linear differential equations with analytic data processes. The Frobenius method is employed to obtain the stochastic solution in the form of a mean square convergent power series. We demonstrate that the convergence requires the boundedness of the random input coefficients. Further, the mean square error of the Frobenius method is proved to decrease exponentially with the number of terms in the series, although not uniformly in time. Regarding the probability density function of the solution at a given time, which is the focus of the paper, we rely on the law of total probability to express it in closedform as an expectation. For the computation of this expectation, a sequence of approximating density functions is constructed by reducing the dimensionality of the problem using the truncated power series of the fundamental set. We prove several theoretical results regarding the pointwise convergence of the sequence of density functions and the convergence in total variation. The pointwise convergence turns out to be exponential under a Lipschitz hypothesis. As the density functions are expressed in terms of expectations, we propose a crude Monte Carlo sampling algorithm for their estimation. This algorithm is implemented and applied on several numerical examples designed to illustrate the theoretical findings of the paper. After that, the efficiency of the algorithm is improved by employing the control variates method. Numerical examples corroborate the variance reduction of the Monte Carlo approach.